!!******************************************************************************
!!
!! This file is part of the AMUN source code, a program to perform
!! Newtonian or relativistic magnetohydrodynamical simulations on uniform or
!! adaptive mesh.
!!
!! Copyright (C) 2008-2021 Grzegorz Kowal <grzegorz@amuncode.org>
!!
!! This program is free software: you can redistribute it and/or modify
!! it under the terms of the GNU General Public License as published by
!! the Free Software Foundation, either version 3 of the License, or
!! (at your option) any later version.
!!
!! This program is distributed in the hope that it will be useful,
!! but WITHOUT ANY WARRANTY; without even the implied warranty of
!! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
!! GNU General Public License for more details.
!!
!! You should have received a copy of the GNU General Public License
!! along with this program. If not, see <http://www.gnu.org/licenses/>.
!!
!!******************************************************************************
!!
!! module: EVOLUTION
!!
!! This module provides an interface for temporal integration with
!! the stability handling. New integration methods can be added by
!! implementing more evolve_* subroutines.
!!
!!******************************************************************************
!
module evolution
implicit none
! interfaces for procedure pointers
!
abstract interface
subroutine evolve_iface()
end subroutine
subroutine update_errors_iface(n, m)
integer, intent(in) :: n, m
end subroutine
end interface
! pointer to the temporal integration subroutine
!
procedure(evolve_iface), pointer, save :: evolve => null()
! pointer to the error update subroutine
!
procedure(update_errors_iface), pointer, save :: update_errors => null()
! evolution parameters
!
logical , save :: error_control = .false.
logical , save :: fsal = .false.
logical , save :: fsal_update = .true.
character(len=255), save :: name_int = ""
character(len=255), save :: name_norm = ""
integer , save :: order = 2
integer , save :: stages = 2
integer , save :: registers = 1
real(kind=8) , save :: cfl = 5.0d-01
! coefficients controlling the decay of scalar potential ѱ
!
real(kind=8) , save :: glm_alpha = 5.0d-01
! time variables
!
integer , save :: step = 0
real(kind=8) , save :: time = 0.0d+00
real(kind=8) , save :: dt = 1.0d+00
real(kind=8) , save :: dth = 1.0d+00
real(kind=8) , save :: dte = 0.0d+00
real(kind=8) , save :: dtp = huge(dt) ! dt for precise snapshots
! the absolute and relative tolerances, limiting factors, the maximum error,
! the maximum number of passes for the adaptive step,
! the total number of integration iterations, the number of rejected iterations
!
real(kind=8) , save :: atol = 1.0d-04
real(kind=8) , save :: rtol = 1.0d-04
real(kind=8) , save :: fac = 9.0d-01
real(kind=8) , save :: facmin = 1.0d-01
real(kind=8) , save :: facmax = 5.0d+00
real(kind=8) , save :: errtol = 1.0d+00
real(kind=8) , save :: chi = 1.0d+00
integer , save :: mrej = 5
integer , save :: niterations = 0
integer , save :: nrejections = 0
! coefficients of Runge-Kutta methods
!
real(kind=8), dimension(:) , allocatable, save :: c, bt, dl, bh
real(kind=8), dimension(:,:), allocatable, save :: gm
! errors in three recent steps
!
real(kind=8), dimension(3), save :: betas, errs
! GLM variables
!
real(kind=8), dimension(:), allocatable, save :: adecay, aglm
! by default everything is private
!
private
! declare public subroutines
!
public :: initialize_evolution, finalize_evolution, print_evolution
public :: initialize_time_step, new_time_step, advance
! declare public variables
!
public :: step, time, dt, dth, dte, dtp, cfl, glm_alpha, registers
public :: atol, rtol, mrej, niterations, nrejections, errs, errtol
public :: error_control
!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!
contains
!
!===============================================================================
!!
!!*** PUBLIC SUBROUTINES *****************************************************
!!
!===============================================================================
!
!===============================================================================
!
! subroutine INITIALIZE_EVOLUTION:
! -------------------------------
!
! Subroutine initializes module EVOLUTION by setting its parameters.
!
! Arguments:
!
! verbose - a logical flag turning the information printing;
! status - an integer flag for error return value;
!
!===============================================================================
!
subroutine initialize_evolution(verbose, status)
use coordinates, only : toplev, adx, ady, adz
use forcing , only : initialize_forcing
use gravity , only : initialize_gravity
use helpers , only : print_message
use mpitools , only : check_status
use parameters , only : get_parameter
use schemes , only : initialize_schemes
use shapes , only : initialize_shapes
use sources , only : initialize_sources
implicit none
logical, intent(in) :: verbose
integer, intent(out) :: status
character(len=255) :: integration = "rk2", error_norm = "l2"
integer :: n
character(len=*), parameter :: loc = 'EVOLUTION::initialize_evolution()'
!-------------------------------------------------------------------------------
!
status = 0
! get the integration method and the value of the CFL coefficient
!
call get_parameter("time_advance", integration)
call get_parameter("stages" , stages )
call get_parameter("cfl" , cfl )
call get_parameter("glm_alpha" , glm_alpha )
! get the absolute and relative tolerances, the maximum number of passes for
! the adaptive step, and limiting factors
!
call get_parameter("absolute_tolerance", atol)
call get_parameter("relative_tolerance", rtol)
call get_parameter("maximum_rejections", mrej)
call get_parameter("chi" , chi)
call get_parameter("factor" , fac )
call get_parameter("minimum_factor" , facmin)
call get_parameter("maximum_factor" , facmax)
! select the integration method, check the correctness of the integration
! parameters and adjust the CFL coefficient if necessary
!
select case(trim(integration))
case ("euler", "EULER", "rk1", "RK1")
evolve => evolve_euler
order = 1
registers = 2
name_int = "1st order Euler"
case ("rk2", "RK2")
evolve => evolve_rk2
order = 2
registers = 2
name_int = "2nd order Runge-Kutta"
case ("rk3", "RK3")
evolve => evolve_rk3
order = 3
registers = 2
name_int = "3rd order Runge-Kutta"
case ("rk3.4", "RK3.4", "ssprk(4,3)", "SSPRK(4,3)")
evolve => evolve_ssprk34
order = 3
registers = 2
cfl = 2.0d+00 * cfl
name_int = "3rd order SSPRK(4,3)"
case ("rk3.5", "RK3.5", "ssprk(5,3)", "SSPRK(5,3)")
evolve => evolve_ssprk35
order = 3
registers = 3
cfl = 2.65062919143939d+00 * cfl
name_int = "3rd order SSPRK(5,3)"
case ("rk4.10", "RK4.10", "ssprk(10,4)", "SSPRK(10,4)")
evolve => evolve_ssprk4_10
order = 4
registers = 3
cfl = 6.0d+00 * cfl
name_int = "Optimal 4th order SSPRK(10,4)"
case ("ssprk(m,2)", "SSPRK(m,2)")
error_control = .true.
evolve => evolve_ssprk2_m
order = 2
registers = 3
stages = max(2, min(9, stages))
cfl = (stages - 1) * cfl
write(name_int, "('2ⁿᵈ-order ',i0,'-step embedded " // &
"SSPRK(',i0,',2) method')") stages, stages
allocate(c(stages), dl(stages), bt(stages), bh(stages))
do n = 1, stages
c(n) = n - 1.0d+00
end do
c (:) = c(:) / (stages - 1)
bt(:) = 1.0d+00 / (stages - 1)
bt(stages) = 1.0d+00 / stages
bh(:) = 1.0d+00 / stages
bh(1) = (stages + 1.0d+00) / stages**2
bh(stages) = (stages - 1.0d+00) / stages**2
dl(1) = (stages - 1.0d+00) / stages
dl(2) = 1.0d+00 - dl(1)
betas(:) = [ 5.8d-01, -2.1d-01, 1.0d-01 ]
call get_parameter("beta(1)", betas(1))
call get_parameter("beta(2)", betas(2))
call get_parameter("beta(3)", betas(3))
betas(:) = - betas(:) / 2.0d+00
case ("SSPRK3(2)4", "SSP3(2)4", "ssprk3(2)4", "ssp3(2)4")
error_control = .true.
evolve => evolve_ssp324
order = 3
registers = 3
stages = 4
cfl = 2.0d+00 * cfl
name_int = "3ʳᵈ-order 4-step embedded SSP3(2)4 method"
betas(:) = [ 5.5d-01, -2.7d-01, 5.0d-02 ]
call get_parameter("beta(1)", betas(1))
call get_parameter("beta(2)", betas(2))
call get_parameter("beta(3)", betas(3))
betas(:) = - betas(:) / 3.0d+00
case ("SSPRK3(2)m", "SSP3(2)m", "ssprk3(2)m", "ssp3(2)m", "SSPRK(m,3)", "ssprk(m,3)")
error_control = .true.
evolve => evolve_ssprk32m
order = 3
registers = 4
n = 2
do while(n**2 <= stages)
n = n + 1
end do
n = n - 1
stages = max(4, n**2)
cfl = (n - 1) * n * cfl
write(name_int, "('3ʳᵈ-order order ',i0,'-step embedded SSPRK3(2)n² method')") stages
betas(:) = [ 5.5d-01, -2.7d-01, 5.0d-02 ]
call get_parameter("beta(1)", betas(1))
call get_parameter("beta(2)", betas(2))
call get_parameter("beta(3)", betas(3))
betas(:) = - betas(:) / 3.0d+00
if (stages == 4 .and. verbose) then
write(*,*)
write(*,*) "ADVICE:"
write(*,*) "The method 'SSPRK3(2)m' is primarily designed for n²," // &
" where n > 2. Since you intend to use 4 stages with" // &
" this method, it is highly advised to use method" // &
" 'SSPRK3(2)4', since it slightly better optimized and" // &
" significantly used less memory."
end if
case ("RK3(2)5", "rk3(2)5")
! References:
!
! [1] Ranocha, H., Dalcin, L., Parsani, M., Ketcheson, D. I.
! "Coefficients of Optimized Runge-Kutta Methods with Automatic
! Step Size Control for Compressible Computational Fluid Dynamics",
! 2021, DOI:10.5281/zenodo.4671927,
! https://github.com/ranocha/Optimized-RK-CFD
!
error_control = .true.
evolve => evolve_3sstarplus
registers = 4
order = 3
stages = 5
cfl = 2.5d+00 * cfl
name_int = "Optimized 3ʳᵈ-order 5-step embedded RK3(2)5"
allocate(c(stages+1), dl(stages), bt(stages), bh(stages+1), gm(3,stages))
c (:) = [ 0.000000000000000000000000000000000000d+00, &
2.300285062878154351930669430512780706d-01, &
4.050049049262914975700372321130661410d-01, &
8.947823877926760224705450466361360720d-01, &
7.235108137218888081489570284485201518d-01, &
1.000000000000000000000000000000000000d+00 ]
dl(:) = [ 1.000000000000000000000000000000000000d+00, &
3.407687209321455242558804921815861422d-01, &
3.414399280584625023244387687873774697d-01, &
7.229302732875589702087936723400941329d-01, &
0.000000000000000000000000000000000000d+00 ]
bt(:) = [ 2.300285062878154351930669430512780706d-01, &
3.021457892454169700189445968126242994d-01, &
8.025601039472704213300183888573974531d-01, &
4.362158997637629844305216319994356355d-01, &
1.129268494470295369172265188216779157d-01 ]
bh(:) = [ 1.046363371354093758897668305991705199d-01, &
9.520431574956758809511173383346476348d-02, &
4.482446645568668405072421350300379357d-01, &
2.449030295461310135957132640369862245d-01, &
1.070116530120251819121660365003405564d-01, &
0.000000000000000000000000000000000000d+00 ]
gm(1,:) = [ 0.000000000000000000000000000000000000d+00, &
2.587669070352079020144955303389306026d-01, &
-1.324366873994502973977035353758550057d-01, &
5.055601231460399101814291350373559483d-02, &
5.670552807902877312521811889846000976d-01 ]
gm(2,:) = [ 1.000000000000000000000000000000000000d+00, &
5.528418745102160639901976698795928733d-01, &
6.731844400389673824374042790213570079d-01, &
2.803103804507635075215805236096803381d-01, &
5.521508873507393276457754945308880998d-01 ]
gm(3,:) = [ 0.000000000000000000000000000000000000d+00, &
0.000000000000000000000000000000000000d+00, &
0.000000000000000000000000000000000000d+00, &
2.752585813446636957256614568573008811d-01, &
-8.950548709279785077579454232514633376d-01 ]
betas(:) = [ 6.4d-01, -3.1d-01, 4.0d-02 ]
call get_parameter("beta(1)", betas(1))
call get_parameter("beta(2)", betas(2))
call get_parameter("beta(3)", betas(3))
betas(:) = - betas(:) / 3.0d+00
case ("RK3(2)5F", "rk3(2)5f")
! References:
!
! [1] Ranocha, H., Dalcin, L., Parsani, M., Ketcheson, D. I.
! "Coefficients of Optimized Runge-Kutta Methods with Automatic
! Step Size Control for Compressible Computational Fluid Dynamics",
! 2021, DOI:10.5281/zenodo.4671927,
! https://github.com/ranocha/Optimized-RK-CFD
!
error_control = .true.
fsal = .true.
evolve => evolve_3sstarplus
registers = 4
order = 3
stages = 5
cfl = 2.5d+00 * cfl
name_int = "Optimized 3ʳᵈ-order 5-step embedded RK3(2)5 FSAL"
allocate(c(stages+1), dl(stages), bt(stages), bh(stages+1), gm(3,stages))
c (:) = [ 0.000000000000000000000000000000000000d+00, &
2.300298624518076223899418286314123354d-01, &
4.050046072094990912268498160116125481d-01, &
8.947822893693433545220710894560512805d-01, &
7.235136928826589010272834603680114769d-01, &
1.000000000000000000000000000000000000d+00 ]
dl(:) = [ 1.000000000000000000000000000000000000d+00, &
3.407655879334525365094815965895763636d-01, &
3.414382655003386206551709871126405331d-01, &
7.229275366787987419692007421895451953d-01, &
0.000000000000000000000000000000000000d+00 ]
bt(:) = [ 2.300298624518076223899418286314123354d-01, &
3.021434166948288809034402119555380003d-01, &
8.025606185416310937583009085873554681d-01, &
4.362158943603440930655148245148766471d-01, &
1.129272530455059129782111662594436580d-01 ]
bh(:) = [ 9.484166705035703392326247283838082847d-02, &
1.726371339430353766966762629176676070d-01, &
3.998243189084371024483169698618455770d-01, &
1.718016807580178450618829007973835152d-01, &
5.881914422155740300718268359027168467d-02, &
1.020760551185952388626787099944507877d-01 ]
gm(1,:) = [ 0.000000000000000000000000000000000000d+00, &
2.587771979725733308135192812685323706d-01, &
-1.324380360140723382965420909764953437d-01, &
5.056033948190826045833606441415585735d-02, &
5.670532000739313812633197158607642990d-01 ]
gm(2,:) = [ 1.000000000000000000000000000000000000d+00, &
5.528354909301389892439698870483746541d-01, &
6.731871608203061824849561782794643600d-01, &
2.803103963297672407841316576323901761d-01, &
5.521525447020610386070346724931300367d-01 ]
gm(3,:) = [ 0.000000000000000000000000000000000000d+00, &
0.000000000000000000000000000000000000d+00, &
0.000000000000000000000000000000000000d+00, &
2.752563273304676380891217287572780582d-01, &
-8.950526174674033822276061734289327568d-01 ]
betas(:) = [ 7.0d-01, -2.3d-01, 0.0d+00 ]
call get_parameter("beta(1)", betas(1))
call get_parameter("beta(2)", betas(2))
call get_parameter("beta(3)", betas(3))
betas(:) = - betas(:) / 3.0d+00
case ("RK4(3)9", "rk4(3)9")
! References:
!
! [1] Ranocha, H., Dalcin, L., Parsani, M., Ketcheson, D. I.
! "Coefficients of Optimized Runge-Kutta Methods with Automatic
! Step Size Control for Compressible Computational Fluid Dynamics",
! 2021, DOI:10.5281/zenodo.4671927,
! https://github.com/ranocha/Optimized-RK-CFD
!
error_control = .true.
evolve => evolve_3sstarplus
registers = 4
order = 4
stages = 9
cfl = 3.5d+00 * cfl
name_int = "Optimized 4ᵗʰ-order 9-step embedded RK4(3)9"
allocate(c(stages+1), dl(stages), bt(stages), bh(stages+1), gm(3,stages))
c (:) = [ 0.000000000000000000000000000000000000d+00, &
2.836343531977826022543660465926414772d-01, &
5.484073767552486705240014599676811834d-01, &
3.687229456675706936558667052479014150d-01, &
-6.806119916032093175251948474173648331d-01, &
3.518526451892056368706593492732753284d-01, &
1.665941920204672094647868254892387293d+00, &
9.715276989307335935187466054546761665d-01, &
9.051569554420045339601721625247585643d-01, &
1.000000000000000000000000000000000000d+00 ]
dl(:) = [ 1.000000000000000000000000000000000000d+00, &
1.262923854387806460989545005598562667d+00, &
7.574967177560872438940839460448329992d-01, &
5.163591158111222863455531895152351544d-01, &
-2.746333792042827389548936599648122146d-02, &
-4.382674653941770848797864513655752318d-01, &
1.273587103668392811985704533534301656d+00, &
-6.294740045442794829622796613103492913d-01, &
0.000000000000000000000000000000000000d+00 ]
bt(:) = [ 2.836343531977826022543660465926414772d-01, &
9.736497978646965372894268287659773644d-01, &
3.382358566377620380505126936670933370d-01, &
-3.584937820217850715182820651063453804d-01, &
-4.113955814725134294322006403954822487d-03, &
1.427968962196019024010757034274849198d+00, &
1.808467712038743032991177525728915926d-02, &
1.605771316794521018947553625079465692d-01, &
2.952226811394310028003810072027839487d-01 ]
bh(:) = [ 4.550655927970944948340364817140593012d-02, &
1.175968310492638562142460384341959193d-01, &
3.658257330515213200375475084421083608d-02, &
-5.311555834355629559010061596928357525d-03, &
5.178250012713127329531367677410650996d-03, &
4.954639022118682638697706200022961443d-01, &
-5.999303132737865921441409466809521699d-03, &
9.405093434568315929035250835218733824d-02, &
2.169318087627035072893925375820310602d-01, &
0.000000000000000000000000000000000000d+00 ]
gm(1,:) = [ 0.000000000000000000000000000000000000d+00, &
-4.655641301259180308677051498071354582d+00, &
-7.720264924836063859141482018013692338d-01, &
-4.024423213419724605695005429153112050d+00, &
-2.129685246739018613087466942802498152d-02, &
-2.435022519234470128602335652131234586d+00, &
1.985627480986167686791439120784668251d-02, &
-2.810790112885283952929218377438668784d-01, &
1.689434895835535695524003319503844110d-01 ]
gm(2,:) = [ 1.000000000000000000000000000000000000d+00, &
2.499262752607825957145627300817258023d+00, &
5.866820365436136799319929406678132638d-01, &
1.205141365412670762568835277881144391d+00, &
3.474793796700868848597960521248007941d-01, &
1.321346140128723105871355808477092220d+00, &
3.119636324379370564023292317172847140d-01, &
4.351419055894087609560896967082486864d-01, &
2.359698299440788299161958168555704234d-01 ]
gm(3,:) = [ 0.000000000000000000000000000000000000d+00, &
0.000000000000000000000000000000000000d+00, &
0.000000000000000000000000000000000000d+00, &
7.621037111138170045618771082985664430d-01, &
-1.981182159087218433914909510116664154d-01, &
-6.228960706317566993192689455719570179d-01, &
-3.752246993432626328289874575355102038d-01, &
-3.355436539000946543242869676125143358d-01, &
-4.560963110717484359015342341157302403d-02 ]
betas(:) = [ 2.5d-01, -1.2d-01, 0.0d+00 ]
call get_parameter("beta(1)", betas(1))
call get_parameter("beta(2)", betas(2))
call get_parameter("beta(3)", betas(3))
betas(:) = - betas(:) / 4.0d+00
case ("RK4(3)9F", "rk4(3)9f")
! References:
!
! [1] Ranocha, H., Dalcin, L., Parsani, M., Ketcheson, D. I.
! "Coefficients of Optimized Runge-Kutta Methods with Automatic
! Step Size Control for Compressible Computational Fluid Dynamics",
! 2021, DOI:10.5281/zenodo.4671927,
! https://github.com/ranocha/Optimized-RK-CFD
!
error_control = .true.
fsal = .true.
evolve => evolve_3sstarplus
registers = 4
order = 4
stages = 9
cfl = 3.5d+00 * cfl
name_int = "Optimized 4ᵗʰ-order 9-step embedded RK4(3)9 FSAL"
allocate(c(stages+1), dl(stages), bt(stages), bh(stages+1), gm(3,stages))
c (:) = [ 0.000000000000000000000000000000000000d+00, &
2.836343005184365275160654678626695428d-01, &
5.484076570002894365286665352032296535d-01, &
3.687228761669438493478872632332010073d-01, &
-6.806126440140844191258463830024463902d-01, &
3.518526124230705801739919476290327750d-01, &
1.665941994879593315477304663913129942d+00, &
9.715279295934715835299192116436237065d-01, &
9.051569840159589594903399929316959062d-01, &
1.000000000000000000000000000000000000d+00 ]
dl(:) = [ 1.000000000000000000000000000000000000d+00, &
1.262923876648114432874834923838556100d+00, &
7.574967189685911558308119415539596711d-01, &
5.163589453140728104667573195005629833d-01, &
-2.746327421802609557034437892013640319d-02, &
-4.382673178127944142238606608356542890d-01, &
1.273587294602656522645691372699677063d+00, &
-6.294740283927400326554066998751383342d-01, &
0.000000000000000000000000000000000000d+00 ]
bt(:) = [ 2.836343005184365275160654678626695428d-01, &
9.736500104654741223716056170419660217d-01, &
3.382359225242515288768487569778320563d-01, &
-3.584943611106183357043212309791897386d-01, &
-4.113944068471528211627210454497620358d-03, &
1.427968894048586363415504654313371031d+00, &
1.808470948394314017665968411915568633d-02, &
1.605770645946802213926893453819236685d-01, &
2.952227015964591648775833803635147962d-01 ]
bh(:) = [ 2.483675912451591196775756814283216443d-02, &
1.866327774562103796990092260942180726d-01, &
5.671080795936984495604436622517631183d-02, &
-3.447695439149287702616943808570747099d-03, &
3.602245056516636472203469198006404016d-03, &
4.545570622145088936800484247980581766d-01, &
-2.434665289427612407531544765622888855d-04, &
6.642755361103549971517945063138312147d-02, &
1.613697079523505006226025497715177578d-01, &
4.955424859358438183052504342394102722d-02 ]
gm(1,:) = [ 0.000000000000000000000000000000000000d+00, &
-4.655641447335068552684422206224169103d+00, &
-7.720265099645871829248487209517314217d-01, &
-4.024436690519806086742256154738379161d+00, &
-2.129676284018530966221583708648634733d-02, &
-2.435022509790109546199372365866450709d+00, &
1.985627297131987000579523283542615256d-02, &
-2.810791146791038566946663374735713961d-01, &
1.689434168754859644351230590422137972d-01 ]
gm(2,:) = [ 1.000000000000000000000000000000000000d+00, &
2.499262792574495009336242992898153462d+00, &
5.866820377718875577451517985847920081d-01, &
1.205146086523094569925592464380295241d+00, &
3.474793722186732780030762737753849272d-01, &
1.321346060965113109321230804210670518d+00, &
3.119636464694193615946633676950358444d-01, &
4.351419539684379261368971206040518552d-01, &
2.359698130028753572503744518147537768d-01 ]
gm(3,:) = [ 0.000000000000000000000000000000000000d+00, &
0.000000000000000000000000000000000000d+00, &
0.000000000000000000000000000000000000d+00, &
7.621006678721315291614677352949377871d-01, &
-1.981182504339400567765766904309673119d-01, &
-6.228959218699007450469629366684127462d-01, &
-3.752248380775956442989480369774937099d-01, &
-3.355438309135169811915662336248989661d-01, &
-4.560955005031121479972862973705108039d-02 ]
betas(:) = [ 3.8d-01, -1.8d-01, 1.0d-02 ]
call get_parameter("beta(1)", betas(1))
call get_parameter("beta(2)", betas(2))
call get_parameter("beta(3)", betas(3))
betas(:) = - betas(:) / 4.0d+00
case ("RK5(4)10", "rk5(4)10")
! References:
!
! [1] Ranocha, H., Dalcin, L., Parsani, M., Ketcheson, D. I.
! "Coefficients of Optimized Runge-Kutta Methods with Automatic
! Step Size Control for Compressible Computational Fluid Dynamics",
! 2021, DOI:10.5281/zenodo.4671927,
! https://github.com/ranocha/Optimized-RK-CFD
!
error_control = .true.
evolve => evolve_3sstarplus
registers = 4
order = 5
stages = 10
cfl = 4.2d+00 * cfl
name_int = "Optimized 5ᵗʰ-order 10-step embedded RK5(4)10"
allocate(c(stages+1), dl(stages), bt(stages), bh(stages+1), gm(3,stages))
c (:) = [ 0.000000000000000000000000000000000000d+00, &
2.597883575710995826783320802193635406d-01, &
9.904573115730917688557891428202061598d-02, &
2.155511882303785204133426661931565216d-01, &
5.007950078421880417512789524851012021d-01, &
5.592251914858131230054392022144328176d-01, &
5.449986973408778242805929551952000165d-01, &
7.615224662599497796472095353126697300d-01, &
8.427062083059167761623893618875787414d-01, &
9.152209807185253394871325258038753352d-01, &
1.000000000000000000000000000000000000d+00 ]
dl(:) = [ 1.000000000000000000000000000000000000d+00, &
-1.331778409133849616712007380176762548d-01, &
8.260422785246030254485064732649153253d-01, &
1.513700430513332405798616943654007796d+00, &
-1.305810063177048110528482211982726539d+00, &
3.036678789342507704281817524408221954d+00, &
-1.449458267074592489788800461540171106d+00, &
3.834313873320957483471400258279635203d+00, &
4.122293971923324492772059928094971199d+00, &
0.000000000000000000000000000000000000d+00 ]
bt(:) = [ 2.597883575710995826783320802193635406d-01, &
1.777008800169541694837687556103565007d-02, &
2.481636637328140606807905234325691851d-01, &
7.941736827560429420202759490815682546d-01, &
3.885391296871822541486945325814526190d-01, &
1.455051664264339366757555740296587660d-01, &
1.587517379462528932413419955691782412d-01, &
1.650605631567659573994022720500446501d-01, &
2.118093299943235065178000892467421832d-01, &
1.559392340339606299335442956580114440d-01 ]
bh(:) = [ 5.734588484676193812418453938089759359d-02, &
1.971447518039733870541652912891291496d-02, &
7.215296605683716720707226840456658773d-02, &
1.739659489807939956977075317768151880d-01, &
3.703693600445487815015171515640585668d-01, &
-1.215599039055065009827765147821222534d-01, &
1.180372945491121604465067725859678821d-01, &
4.155688823364870056536983972605056553d-02, &
1.227886627910379901351569893551486490d-01, &
1.456284232223684285998448928597043056d-01, &
0.000000000000000000000000000000000000d+00 ]
gm(1,:) = [ 0.000000000000000000000000000000000000d+00, &
4.043660078504695837542588769963326988d-01, &
-8.503427464263185087039788184485627962d-01, &
-6.950894167072419998080989313353063399d+00, &
9.238765225328278557805080247596562995d-01, &
-2.563178039957404359875124580586147888d+00, &
2.545744869966347362604059848503340890d-01, &
3.125831733863168874151935287174374515d-01, &
-7.007114800567584871263283872289072079d-01, &
4.839620970980726631935174740648996010d-01 ]
gm(2,:) = [ 1.000000000000000000000000000000000000d+00, &
6.871467069752345566001768382316915820d-01, &
1.093024760468898686510433898645775908d+00, &
3.225975382330161123625348062949430509d+00, &
1.041153700841396427100436517666787823d+00, &
1.292821488864702752767390075072674807d+00, &
7.391462769297006312785029455392854586d-01, &
1.239129257039300081860496157739352186d-01, &
1.842753479366766790220633908793933781d-01, &
5.712788942697077644959290025755003720d-02 ]
gm(3,:) = [ 0.000000000000000000000000000000000000d+00, &
0.000000000000000000000000000000000000d+00, &
0.000000000000000000000000000000000000d+00, &
-2.393405159342139386425044844626597490d+00, &
-1.902854422095986544338294743445530533d+00, &
-2.820042210583207174321941694153843259d+00, &
-1.832698464130564949123807896975136336d+00, &
-2.199094510750697865007677774395365522d-01, &
-4.082430660384876496971887725512427800d-01, &
-1.377669791121207993339861855818881150d-01 ]
betas(:) = [ 4.7d-01, -2.0d-01, 6.0d-02 ]
call get_parameter("beta(1)", betas(1))
call get_parameter("beta(2)", betas(2))
call get_parameter("beta(3)", betas(3))
betas(:) = - betas(:) / 5.0d+00
case ("RK5(4)10F", "rk5(4)10f")
! References:
!
! [1] Ranocha, H., Dalcin, L., Parsani, M., Ketcheson, D. I.
! "Coefficients of Optimized Runge-Kutta Methods with Automatic
! Step Size Control for Compressible Computational Fluid Dynamics",
! 2021, DOI:10.5281/zenodo.4671927,
! https://github.com/ranocha/Optimized-RK-CFD
!
error_control = .true.
fsal = .true.
evolve => evolve_3sstarplus
registers = 4
order = 5
stages = 10
cfl = 4.2d+00 * cfl
name_int = "Optimized 5ᵗʰ-order 10-step embedded RK5(4)10 FSAL"
allocate(c(stages+1), dl(stages), bt(stages), bh(stages+1), gm(3,stages))
c (:) = [ 0.000000000000000000000000000000000000d+00, &
2.597883554788674084039539165398464630d-01, &
9.904573247592460887087003212056568980d-02, &
2.155511890524058691860390281856497503d-01, &
5.007950088969676776844289399972611534d-01, &
5.592251911688643533787800688765883636d-01, &
5.449986978853637084972622392134732553d-01, &
7.615224694532590139829150720490417596d-01, &
8.427062083267360939805493320684741215d-01, &
9.152209805057669959657927210873423883d-01, &
1.000000000000000000000000000000000000d+00 ]
dl(:) = [ 1.000000000000000000000000000000000000d+00, &
-1.331778419508803397033287009506932673d-01, &
8.260422814750207498262063505871077303d-01, &
1.513700425755728332485300719652378197d+00, &
-1.305810059935023735972298885749903694d+00, &
3.036678802924163246003321318996156380d+00, &
-1.449458274398895177922690618003584514d+00, &
3.834313899176362315089976408899373409d+00, &
4.122293760012985409330881631526514714d+00, &
0.000000000000000000000000000000000000d+00 ]
bt(:) = [ 2.597883554788674084039539165398464630d-01, &
1.777008889438867858759149597539211023d-02, &
2.481636629715501931294746189266601496d-01, &
7.941736871152005775821844297293296135d-01, &
3.885391285642019129575902994397298066d-01, &
1.455051657916305055730603387469193768d-01, &
1.587517385964749337690916959584348979d-01, &
1.650605617880053419242434594242509601d-01, &
2.118093284937153836908655490906875007d-01, &
1.559392342362059886106995325687547506d-01 ]
bh(:) = [ -2.019255440012066080909442770590267512d-02, &
2.737903480959184339932730854141598275d-02, &
3.028818636145965534365173822296811090d-01, &
-3.656843880622222190071445247906780540d-02, &
3.982664774676767729863101188528827405d-01, &
-5.715959421140685436681459970502471634d-02, &
9.849855103848558320961101178888983150d-02, &
6.654601552456084978615342374581437947d-02, &
9.073479542748112726465375642050504556d-02, &
8.432289325330803924891866923939606351d-02, &
4.529095628204896774513180907141004447d-02 ]
gm(1,:) = [ 0.000000000000000000000000000000000000d+00, &
4.043660121685749695640462197806189975d-01, &
-8.503427289575839690883191973980814832d-01, &
-6.950894175262117526410215315179482885d+00, &
9.238765192731084931855438934978371889d-01, &
-2.563178056509891340215942413817786020d+00, &
2.545744879365226143946122067064118430d-01, &
3.125831707411998258746812355492206137d-01, &
-7.007114414440507927791249989236719346d-01, &
4.839621016023833375810172323297465039d-01 ]
gm(2,:) = [ 1.000000000000000000000000000000000000d+00, &
6.871467028161416909922221357014564412d-01, &
1.093024748914750833700799552463885117d+00, &
3.225975379607193001678365742708874597d+00, &
1.041153702510101386914019859778740444d+00, &
1.292821487912164945157744726076279306d+00, &
7.391462755788122847651304143259254381d-01, &
1.239129251371800313941948224441873274d-01, &
1.842753472370123193132193302369345580d-01, &
5.712788998796583446479387686662738843d-02 ]
gm(3,:) = [ 0.000000000000000000000000000000000000d+00, &
0.000000000000000000000000000000000000d+00, &
0.000000000000000000000000000000000000d+00, &
-2.393405133244194727221124311276648940d+00, &
-1.902854422421760920850597670305403139d+00, &
-2.820042207399977261483046412236557428d+00, &
-1.832698465277380999601896111079977378d+00, &
-2.199094483084671192328083958346519535d-01, &
-4.082430635847870963724591602173546218d-01, &
-1.377669797880289713535665985132703979d-01 ]
betas(:) = [ 4.5d-01, -1.3d-01, 0.0d+00 ]
call get_parameter("beta(1)", betas(1))
call get_parameter("beta(2)", betas(2))
call get_parameter("beta(3)", betas(3))
betas(:) = - betas(:) / 5.0d+00
case default
if (verbose) then
write(*,*)
write(*,"(1x,a)") "ERROR!"
write(*,"(1x,a)") "The selected time advance method is not " // &
"implemented: " // trim(integration)
write(*,"(1x,a)") "Available methods: 'euler' 'rk2', 'rk3'," // &
" 'ssprk(m,2)', 'ssprk(4,3)', 'ssprk(5,3)'," // &
" 'ssprk(m,3)', 'ssprk(10,4)'."
write(*,"(1x,a)") "Available embedded methods: 'ssprk3(2)4'," // &
" 'ssprk3(2)m', 'rk3(2)5', 'rk4(3)9', 'rk5(4)10',"// &
"'rk3(2)5f', 'rk4(3)9f', 'rk5(4)10f'."
end if
status = 1
end select
! select error norm
!
call get_parameter("error_norm", error_norm)
select case(trim(error_norm))
case("max", "maximum", "infinity")
update_errors => update_errors_max
name_norm = "maximum norm"
case default
update_errors => update_errors_l2
name_norm = "L2-norm"
end select
! reset recent error history
!
errs = 1.0d+00
! GLM coefficients for all refinement levels
!
allocate(adecay(toplev), aglm(toplev))
#if NDIMS == 3
aglm(:) = - glm_alpha / min(adx(:), ady(:), adz(:))
#else /* NDIMS == 3 */
aglm(:) = - glm_alpha / min(adx(:), ady(:))
#endif /* NDIMS == 3 */
! initialize other modules
!
call initialize_schemes(verbose, status)
if (status /=0) &
call print_message(loc, "Could not initialize module SCHEMES!")
if (check_status(status /= 0)) return
call initialize_sources(verbose, status)
if (status /=0) &
call print_message(loc, "Could not initialize module SOURCES!")
if (check_status(status /= 0)) return
call initialize_forcing(verbose, status)
if (status /=0) &
call print_message(loc, "Could not initialize module FORCING!")
if (check_status(status /= 0)) return
call initialize_gravity(verbose, status)
if (status /=0) &
call print_message(loc, "Could not initialize module GRAVITY!")
if (check_status(status /= 0)) return
call initialize_shapes(status)
if (status /=0) &
call print_message(loc, "Could not initialize module SHAPES!")
if (check_status(status /= 0)) return
!-------------------------------------------------------------------------------
!
end subroutine initialize_evolution
!
!===============================================================================
!
! subroutine FINALIZE_EVOLUTION:
! -----------------------------
!
! Subroutine releases memory used by the module.
!
! Arguments:
!
! status - an integer flag for error return value;
!
!===============================================================================
!
subroutine finalize_evolution(verbose, status)
use forcing, only : finalize_forcing
use gravity, only : finalize_gravity
use helpers, only : print_section, print_parameter, print_message
use schemes, only : finalize_schemes
use shapes , only : finalize_shapes
use sources, only : finalize_sources
implicit none
logical, intent(in) :: verbose
integer, intent(out) :: status
character(len=*), parameter :: loc = 'EVOLUTION::finalize_evolution()'
!-------------------------------------------------------------------------------
!
status = 0
call finalize_shapes(status)
if (status /=0) &
call print_message(loc, "Could not finalize module SHAPES!")
call finalize_gravity(status)
if (status /=0) &
call print_message(loc, "Could not finalize module GRAVITY!")
call finalize_forcing(status)
if (status /=0) &
call print_message(loc, "Could not finalize module FORCING!")
call finalize_sources(status)
if (status /=0) &
call print_message(loc, "Could not finalize module SOURCES!")
call finalize_schemes(status)
if (status /=0) &
call print_message(loc, "Could not finalize module SCHEMES!")
if (allocated(adecay)) deallocate(adecay)
if (allocated(aglm)) deallocate(aglm)
if (allocated(c)) deallocate(c)
if (allocated(dl)) deallocate(dl)
if (allocated(bt)) deallocate(bt)
if (allocated(bh)) deallocate(bh)
if (allocated(gm)) deallocate(gm)
! nullify pointers
!
nullify(evolve)
nullify(update_errors)
! print integration summary
!
if (niterations > 0) then
call print_section(verbose, "Integration summary")
call print_parameter(verbose, "Number of iterations", niterations)
call print_parameter(verbose, "Number of rejections", nrejections)
end if
!-------------------------------------------------------------------------------
!
end subroutine finalize_evolution
!
!===============================================================================
!
! subroutine PRINT_EVOLUTION:
! --------------------------
!
! Subroutine prints module parameters.
!
! Arguments:
!
! verbose - a logical flag turning the information printing;
!
!===============================================================================
!
subroutine print_evolution(verbose)
use equations, only : magnetized
use forcing , only : print_forcing
use helpers , only : print_section, print_parameter
use schemes , only : print_schemes
use shapes , only : print_shapes
use sources , only : print_sources
implicit none
logical, intent(in) :: verbose
!-------------------------------------------------------------------------------
!
call print_sources(verbose)
call print_forcing(verbose)
call print_shapes(verbose)
if (verbose) then
call print_section(verbose, "Evolution")
call print_parameter(verbose, "time advance", name_int)
call print_parameter(verbose, "no. of registers", registers)
call print_parameter(verbose, "CFL coefficient", cfl)
if (magnetized) then
call print_parameter(verbose, "GLM alpha coefficient", glm_alpha)
end if
if (error_control) then
call print_parameter(verbose, "absolute tolerance", atol)
call print_parameter(verbose, "relative tolerance", rtol)
call print_parameter(verbose, "error norm" , name_norm)
call print_parameter(verbose, "maximum rejections", mrej)
call print_parameter(verbose, "chi" , chi)
call print_parameter(verbose, "factor" , fac)
call print_parameter(verbose, "minimum factor" , facmin)
call print_parameter(verbose, "maximum factor" , facmax)
end if
end if
call print_schemes(verbose)
!-------------------------------------------------------------------------------
!
end subroutine print_evolution
!
!===============================================================================
!
! subroutine ADVANCE:
! ------------------
!
! Subroutine advances the solution by one time step using the selected time
! integration method.
!
! Arguments:
!
! status - the subroutine call status: 0 for success, otherwise failure;
!
!===============================================================================
!
subroutine advance(status)
! references
!
use blocks , only : set_blocks_update
use coordinates, only : toplev
use forcing , only : update_forcing, forcing_enabled
use mesh , only : update_mesh
! local variables are not implicit by default
!
implicit none
! input variables
!
integer, intent(out) :: status
!
!-------------------------------------------------------------------------------
!
! advance the solution using the selected method
!
call evolve()
! add forcing contribution
!
if (forcing_enabled) call update_forcing(dt)
! check if we need to perform the refinement step
!
if (toplev > 1) then
! set all meta blocks to not be updated
!
call set_blocks_update(.false.)
! check refinement and refine
!
call update_mesh(status)
if (status /= 0) go to 100
! update primitive variables
!
call update_variables(time + dt, 0.0d+00, status)
if (status /= 0) go to 100
! set all meta blocks to be updated
!
call set_blocks_update(.true.)
end if ! toplev > 1
#ifdef DEBUG
! check variables for NaNs
!
call check_variables()
#endif /* DEBUG */
! error entry point
!
100 continue
!-------------------------------------------------------------------------------
!
end subroutine advance
!
!===============================================================================
!
! subroutine INITIALIZE_TIME_STEP:
! -------------------------------
!
! Subroutine estimates the initial time step.
!
! References:
!
! [1] Courant, R., Friedrichs, K., Lewy, H.,
! "Über die partiellen Differenzengleichungen der mathematischen Physik",
! Mathematische Annalen, 1928, volume 100, pages 32–74,
! DOI: 10.1007/BF01448839
! [2] Hairer, E., Nørsett, S. P., Wanner, G.,
! "Solving Ordinary Differential Equations I: Nonstiff Problems",
! Springer-Verlag Berlin Heidelberg, 1993,
! ISBN: 978-3-540-56670-0, DOI: 10.1007/978-3-540-78862-1
!
!===============================================================================
!
subroutine initialize_time_step()
use blocks , only : block_data, list_data, get_nleafs
use coordinates, only : nc => ncells, nb, ne, adx, ady
#if NDIMS == 3
use coordinates, only : adz
#endif /* NDIMS == 3 */
use equations , only : nf, maxspeed, cmax, cmax2
use helpers , only : print_message
#ifdef MPI
use mpitools , only : reduce_maximum, reduce_sum
#endif /* MPI */
use sources , only : viscosity, resistivity, update_sources
implicit none
type(block_data), pointer :: pdata
logical :: flag
integer :: mlev, n, status
real(kind=8) :: dx_min, fnorm, h0, h1
real(kind=8), dimension(3) :: d
real(kind=8), dimension(:,:,:,:), allocatable :: sc, df
real(kind=8), parameter :: eps = tiny(cmax)
character(len=*), parameter :: loc = 'EVOLUTION:initialize_time_step()'
!
!-------------------------------------------------------------------------------
!
cmax = eps
mlev = 1
pdata => list_data
do while (associated(pdata))
mlev = max(mlev, pdata%meta%level)
cmax = max(cmax, maxspeed(pdata%q(:,:,:,:)))
pdata => pdata%next
end do
#ifdef MPI
call reduce_maximum(cmax)
call reduce_maximum(mlev)
#endif /* MPI */
cmax2 = cmax * cmax
#if NDIMS == 2
dx_min = min(adx(mlev), ady(mlev))
#endif /* NDIMS == 2 */
#if NDIMS == 3
dx_min = min(adx(mlev), ady(mlev), adz(mlev))
#endif /* NDIMS == 3 */
! determine the time step due to the stability
!
dth = dx_min / max(cmax &
+ 2.0d+00 * max(viscosity, resistivity) / dx_min, eps)
dt = cfl * dth
! determine the time step due to the error
!
if (error_control) then
#if NDIMS == 3
allocate(sc(nf,nc,nc,nc),df(nf,nc,nc,nc), stat=status)
#else /* NDIMS == 3 */
allocate(sc(nf,nc,nc, 1),df(nf,nc,nc, 1), stat=status)
#endif /* NDIMS == 3 */
#ifdef MPI
call reduce_maximum(status)
#endif /* MPI */
if (status > 0) then
call print_message(loc, &
"Cannot allocate memory for the time step estimation!")
go to 100
end if
fnorm = 1.0d+00 / (get_nleafs() * nc**NDIMS)
d(:) = 0.0d+00
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, 0.0d+00, 0.0d+00, pdata%du(:,:,:,:))
#if NDIMS == 3
sc(:,:,:,:) = atol + rtol * abs(pdata%uu(:,nb:ne,nb:ne,nb:ne,1))
d(1) = d(1) + sum((pdata%uu(:,nb:ne,nb:ne,nb:ne,1) / sc(:,:,:,:))**2)
d(2) = d(2) + sum((pdata%du(:,nb:ne,nb:ne,nb:ne) / sc(:,:,:,:))**2)
#else /* NDIMS == 3 */
sc(:,:,:,:) = atol + rtol * abs(pdata%uu(:,nb:ne,nb:ne, : ,1))
d(1) = d(1) + sum((pdata%uu(:,nb:ne,nb:ne, : ,1) / sc(:,:,:,:))**2)
d(2) = d(2) + sum((pdata%du(:,nb:ne,nb:ne, : ) / sc(:,:,:,:))**2)
#endif /* NDIMS == 3 */
pdata => pdata%next
end do
#ifdef MPI
call reduce_sum(d(1:2))
#endif /* MPI */
d(1:2) = sqrt(fnorm * d(1:2))
if (minval(d(1:2)) >= 1.0d-05) then
h0 = 1.0d-02 * d(1) / d(2)
else
h0 = 1.0d-06
end if
n = 10
flag = .true.
do while(flag)
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,1) + h0 * pdata%du(:,:,:,:)
pdata%u => pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
call update_variables(time + h0, h0, status)
if (status /= 0) then
h0 = 2.5d-01 * h0
flag = n > 0
n = n - 1
else
flag = .false.
end if
end do
if (status /= 0) then
call print_message(loc, "Could not estimate the initial time step " // &
"due to the error. Setting it to the CFL time step.")
dte = dt
else
pdata => list_data
do while (associated(pdata))
#if NDIMS == 3
df(:,:,:,:) = pdata%du(:,nb:ne,nb:ne,nb:ne)
sc(:,:,:,:) = atol + rtol * abs(pdata%uu(:,nb:ne,nb:ne,nb:ne,1))
#else /* NDIMS == 3 */
df(:,:,:,:) = pdata%du(:,nb:ne,nb:ne, : )
sc(:,:,:,:) = atol + rtol * abs(pdata%uu(:,nb:ne,nb:ne, : ,1))
#endif /* NDIMS == 3 */
call update_increment(pdata)
call update_sources(pdata, time + h0, 0.0d+00, pdata%du(:,:,:,:))
#if NDIMS == 3
df(:,:,:,:) = df(:,:,:,:) - pdata%du(:,nb:ne,nb:ne,nb:ne)
#else /* NDIMS == 3 */
df(:,:,:,:) = df(:,:,:,:) - pdata%du(:,nb:ne,nb:ne, : )
#endif /* NDIMS == 3 */
d(3) = d(3) + sum((df(:,:,:,:) / sc(:,:,:,:))**2)
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
#ifdef MPI
call reduce_sum(d(3:3))
#endif /* MPI */
d(3) = sqrt(fnorm * d(3)) / h0
if (max(d(2), d(3)) > 1.0d-15) then
h1 = (1.0d-02 / max(d(2), d(3)))**(1.0d+00 / (order + 1))
else
h1 = max(1.0d-06, 1.0d-03 * h0)
end if
dte = min(1.0d+02 * h0, h1)
dt = min(dt, dte)
end if
if (allocated(sc)) deallocate(sc)
if (allocated(df)) deallocate(df)
100 continue
end if
!-------------------------------------------------------------------------------
!
end subroutine initialize_time_step
!
!===============================================================================
!
! subroutine NEW_TIME_STEP:
! ------------------------
!
! Subroutine estimates the new time step from the maximum speed in the system
! and source term constraints.
!
! References:
!
! [1] Courant, R., Friedrichs, K., Lewy, H.,
! "Über die partiellen Differenzengleichungen der mathematischen Physik",
! Mathematische Annalen, 1928, volume 100, pages 32–74,
! DOI: 10.1007/BF01448839
!
!===============================================================================
!
subroutine new_time_step()
use blocks , only : block_data, list_data
use coordinates , only : adx, ady
#if NDIMS == 3
use coordinates , only : adz
#endif /* NDIMS == 3 */
use equations , only : maxspeed, cmax, cmax2
#ifdef MPI
use mpitools , only : reduce_maximum
#endif /* MPI */
use sources , only : viscosity, resistivity
implicit none
type(block_data), pointer :: pdata
integer :: mlev
real(kind=8) :: cm, dx_min
real(kind=8), parameter :: eps = tiny(cmax)
!
!-------------------------------------------------------------------------------
!
cmax = eps
mlev = 1
pdata => list_data
do while (associated(pdata))
mlev = max(mlev, pdata%meta%level)
cm = maxspeed(pdata%q(:,:,:,:))
cmax = max(cmax, cm)
pdata => pdata%next
end do
#ifdef MPI
call reduce_maximum(cmax)
call reduce_maximum(mlev)
#endif /* MPI */
cmax2 = cmax * cmax
#if NDIMS == 2
dx_min = min(adx(mlev), ady(mlev))
#endif /* NDIMS == 2 */
#if NDIMS == 3
dx_min = min(adx(mlev), ady(mlev), adz(mlev))
#endif /* NDIMS == 3 */
dth = dx_min / max(cmax &
+ 2.0d+00 * max(viscosity, resistivity) / dx_min, eps)
dt = cfl * dth
dt = min(dt, dtp)
if (error_control) dt = min(dt, dte)
!-------------------------------------------------------------------------------
!
end subroutine new_time_step
!
!===============================================================================
!!
!!*** PRIVATE SUBROUTINES ****************************************************
!!
!===============================================================================
!
!===============================================================================
!
! subroutine EVOLVE_EULER:
! -----------------------
!
! Subroutine advances the solution by one time step using the 1st order
! Euler integration method.
!
! References:
!
! [1] Press, W. H, Teukolsky, S. A., Vetterling, W. T., Flannery, B. P.,
! "Numerical Recipes in Fortran",
! Cambridge University Press, Cambridge, 1992
!
!===============================================================================
!
subroutine evolve_euler()
use blocks , only : block_data, list_data
use boundaries, only : boundary_fluxes
use equations , only : ibp, cmax
use sources , only : update_sources
implicit none
type(block_data), pointer :: pdata
integer :: status
real(kind=8) :: t
!
!-------------------------------------------------------------------------------
!
status = 1
do while(status /= 0)
t = time + dt
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,1)
pdata%u => pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, dt, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,2) + dt * pdata%du(:,:,:,:)
pdata => pdata%next
end do
if (ibp > 0) then
adecay(:) = exp(aglm(:) * cmax * dt)
pdata => list_data
do while (associated(pdata))
pdata%u(ibp,:,:,:) = adecay(pdata%meta%level) * pdata%u(ibp,:,:,:)
pdata => pdata%next
end do
end if
call update_variables(t, dt, status)
if (status /= 0) dt = 2.5d-01 * dt
end do
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,1) = pdata%uu(:,:,:,:,2)
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
!-------------------------------------------------------------------------------
!
end subroutine evolve_euler
!
!===============================================================================
!
! subroutine EVOLVE_RK2:
! ---------------------
!
! Subroutine advances the solution by one time step using the 2nd order
! Runge-Kutta time integration method.
!
! References:
!
! [1] Press, W. H, Teukolsky, S. A., Vetterling, W. T., Flannery, B. P.,
! "Numerical Recipes in Fortran",
! Cambridge University Press, Cambridge, 1992
!
!===============================================================================
!
subroutine evolve_rk2()
use blocks , only : block_data, list_data
use boundaries, only : boundary_fluxes
use equations , only : ibp, cmax
use sources , only : update_sources
implicit none
type(block_data), pointer :: pdata
integer :: status
real(kind=8) :: t, ds
!
!-------------------------------------------------------------------------------
!
status = 1
do while(status /= 0)
ds = 5.0d-01 * dt
!= 1st step: U(1) = U(n) + dt * F[U(n)]
!
t = time + dt
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, 0.0d+00, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,1) + dt * pdata%du(:,:,:,:)
pdata%u => pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
call update_variables(t, dt, status)
if (status /= 0) go to 100
!= 2nd step: U(n+1) = 1/2 U(n) + 1/2 U(1) + 1/2 dt * F[U(1)]
!
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, ds, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = 5.0d-01 * (pdata%uu(:,:,:,:,1) &
+ pdata%uu(:,:,:,:,2)) &
+ ds * pdata%du(:,:,:,:)
pdata => pdata%next
end do
if (ibp > 0) then
adecay(:) = exp(aglm(:) * cmax * dt)
pdata => list_data
do while (associated(pdata))
pdata%u(ibp,:,:,:) = adecay(pdata%meta%level) * pdata%u(ibp,:,:,:)
pdata => pdata%next
end do
end if
call update_variables(t, ds, status)
100 continue
if (status /= 0) dt = 2.5d-01 * dt
end do
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,1) = pdata%uu(:,:,:,:,2)
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
!-------------------------------------------------------------------------------
!
end subroutine evolve_rk2
!
!===============================================================================
!
! subroutine EVOLVE_RK3:
! ---------------------
!
! Subroutine advances the solution by one time step using the 3rd order
! Runge-Kutta time integration method.
!
! References:
!
! [1] Press, W. H, Teukolsky, S. A., Vetterling, W. T., Flannery, B. P.,
! "Numerical Recipes in Fortran",
! Cambridge University Press, Cambridge, 1992
!
!===============================================================================
!
subroutine evolve_rk3()
use blocks , only : block_data, list_data
use boundaries, only : boundary_fluxes
use equations , only : ibp, cmax
use sources , only : update_sources
implicit none
type(block_data), pointer :: pdata
integer :: status
real(kind=8) :: t, ds
real(kind=8), parameter :: f21 = 3.0d+00 / 4.0d+00, f22 = 1.0d+00 / 4.0d+00
real(kind=8), parameter :: f31 = 1.0d+00 / 3.0d+00, f32 = 2.0d+00 / 3.0d+00
!
!-------------------------------------------------------------------------------
!
status = 1
do while(status /= 0)
!= 1st substep: U(1) = U(n) + dt F[U(n)]
!
t = time + dt
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, 0.0d+00, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,1) + dt * pdata%du(:,:,:,:)
pdata%u => pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
call update_variables(t, dt, status)
if (status /= 0) go to 100
!= 2nd step: U(2) = 3/4 U(n) + 1/4 U(1) + 1/4 dt F[U(1)]
!
ds = f22 * dt
t = time + 0.5d+00 * dt
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, ds, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = f21 * pdata%uu(:,:,:,:,1) &
+ f22 * pdata%uu(:,:,:,:,2) + ds * pdata%du(:,:,:,:)
pdata => pdata%next
end do
call update_variables(t, ds, status)
if (status /= 0) go to 100
!= 3rd step: U(n+1) = 1/3 U(n) + 2/3 U(2) + 2/3 dt F[U(2)]
!
ds = f32 * dt
t = time + dt
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, ds, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = f31 * pdata%uu(:,:,:,:,1) &
+ f32 * pdata%uu(:,:,:,:,2) + ds * pdata%du(:,:,:,:)
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
if (ibp > 0) then
adecay(:) = exp(aglm(:) * cmax * dt)
pdata => list_data
do while (associated(pdata))
pdata%u(ibp,:,:,:) = adecay(pdata%meta%level) * pdata%u(ibp,:,:,:)
pdata => pdata%next
end do
end if
call update_variables(t, dt, status)
100 continue
if (status /= 0) dt = 2.5d-01 * dt
end do
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,1) = pdata%uu(:,:,:,:,2)
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
!-------------------------------------------------------------------------------
!
end subroutine evolve_rk3
!
!===============================================================================
!
! subroutine EVOLVE_SSPRK34:
! -------------------------
!
! Subroutine advances the solution by one time step using the 3rd order
! 4-stage Strong Stability Preserving Runge-Kutta time integration method.
!
! References:
!
! [1] Ruuth, S. J.,
! "Global Optimization of Explicit Strong-Stability-Preserving
! Runge-Kutta methods",
! Mathematics of Computation,
! 2006, vol. 75, no. 253, pp. 183-207
!
!===============================================================================
!
subroutine evolve_ssprk34()
use blocks , only : block_data, list_data
use boundaries, only : boundary_fluxes
use equations , only : ibp, cmax
use sources , only : update_sources
implicit none
type(block_data), pointer :: pdata
integer :: status
real(kind=8) :: t, ds
!
!-------------------------------------------------------------------------------
!
status = 1
do while(status /= 0)
!= 1st step: U(1) = U(n) + 1/2 dt F[U(n)]
!
ds = 5.0d-01 * dt
t = time + ds
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, 0.0d+00, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,1) + ds * pdata%du(:,:,:,:)
pdata%u => pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
call update_variables(t, ds, status)
if (status /= 0) go to 100
!= 2nd step: U(2) = U(2) + 1/2 dt F[U(1)]
!
t = time + dt
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, ds, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,2) + ds * pdata%du(:,:,:,:)
pdata => pdata%next
end do
call update_variables(t, ds, status)
if (status /= 0) go to 100
!= 3rd step: U(3) = 2/3 U(n) + 1/3 (U(2) + 1/2 dt F[U(2)])
!
t = time + ds
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, ds, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = (2.0d+00 * pdata%uu(:,:,:,:,1) &
+ pdata%uu(:,:,:,:,2) &
+ ds * pdata%du(:,:,:,:)) / 3.0d+00
pdata => pdata%next
end do
call update_variables(t, ds, status)
if (status /= 0) go to 100
!= the final step: U(n+1) = U(3) + 1/2 dt F[U(3)]
!
t = time + dt
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, ds, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,2) + ds * pdata%du(:,:,:,:)
pdata => pdata%next
end do
if (ibp > 0) then
adecay(:) = exp(aglm(:) * cmax * dt)
pdata => list_data
do while (associated(pdata))
pdata%u(ibp,:,:,:) = adecay(pdata%meta%level) * pdata%u(ibp,:,:,:)
pdata => pdata%next
end do
end if
call update_variables(t, dt, status)
100 continue
if (status /= 0) dt = 2.5d-01 * dt
end do
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,1) = pdata%uu(:,:,:,:,2)
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
!-------------------------------------------------------------------------------
!
end subroutine evolve_ssprk34
!
!===============================================================================
!
! subroutine EVOLVE_SSPRK35:
! -------------------------
!
! Subroutine advances the solution by one time step using the 3rd order
! 5-stage Strong Stability Preserving Runge-Kutta time integration method.
!
! References:
!
! [1] Ruuth, S. J.,
! "Global Optimization of Explicit Strong-Stability-Preserving
! Runge-Kutta methods",
! Mathematics of Computation,
! 2006, vol. 75, no. 253, pp. 183-207
!
!===============================================================================
!
subroutine evolve_ssprk35()
use blocks , only : block_data, list_data
use boundaries, only : boundary_fluxes
use equations , only : ibp, cmax
use sources , only : update_sources
implicit none
type(block_data), pointer :: pdata
integer :: status
real(kind=8) :: t, ds
real(kind=8), parameter :: b1 = 3.77268915331368d-01
real(kind=8), parameter :: b3 = 2.42995220537396d-01
real(kind=8), parameter :: b4 = 2.38458932846290d-01
real(kind=8), parameter :: b5 = 2.87632146308408d-01
real(kind=8), parameter :: a31 = 3.55909775063327d-01
real(kind=8), parameter :: a33 = 6.44090224936673d-01
real(kind=8), parameter :: a41 = 3.67933791638137d-01
real(kind=8), parameter :: a44 = 6.32066208361863d-01
real(kind=8), parameter :: a53 = 2.37593836598569d-01
real(kind=8), parameter :: a55 = 7.62406163401431d-01
!
!-------------------------------------------------------------------------------
!
status = 1
do while(status /= 0)
!= 1st step: U(1) = U(n) + b1 dt F[U(n)]
!
ds = b1 * dt
t = time + ds
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, 0.0d+00, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,1) + ds * pdata%du(:,:,:,:)
pdata%u => pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
call update_variables(t, ds, status)
if (status /= 0) go to 100
!= 2nd step: U(2) = U(1) + b1 dt F[U(1)]
!
t = time + 2.0d+00 * ds
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, ds, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,2) + ds * pdata%du(:,:,:,:)
pdata => pdata%next
end do
call update_variables(t, ds, status)
if (status /= 0) go to 100
!= 3rd step: U(3) = a31 U(n) + a33 U(2) + b3 dt F[U(2)]
!
ds = b3 * dt
t = time + (2.0d+00 * a33 * b1 + b3) * dt
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, ds, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = a31 * pdata%uu(:,:,:,:,1) &
+ a33 * pdata%uu(:,:,:,:,2) + ds * pdata%du(:,:,:,:)
pdata => pdata%next
end do
call update_variables(t, ds, status)
if (status /= 0) go to 100
!= 4th step: U(4) = a41 U(n) + a44 U(3) + b4 dt F[U(3)]
!
ds = b4 * dt
t = time + ((2.0d+00 * b1 * a33 + b3) * a44 + b4) * dt
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, ds, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,3) = a41 * pdata%uu(:,:,:,:,1) &
+ a44 * pdata%uu(:,:,:,:,2) + ds * pdata%du(:,:,:,:)
pdata%u => pdata%uu(:,:,:,:,3)
pdata => pdata%next
end do
call update_variables(t, ds, status)
if (status /= 0) go to 100
!= the final step: U(n+1) = a53 U(2) + a55 U(4) + b5 dt F[U(4)]
!
ds = b5 * dt
t = time + dt
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, ds, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = a53 * pdata%uu(:,:,:,:,2) &
+ a55 * pdata%uu(:,:,:,:,3) + ds * pdata%du(:,:,:,:)
pdata%u => pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
if (ibp > 0) then
adecay(:) = exp(aglm(:) * cmax * dt)
pdata => list_data
do while (associated(pdata))
pdata%u(ibp,:,:,:) = adecay(pdata%meta%level) * pdata%u(ibp,:,:,:)
pdata => pdata%next
end do
end if
call update_variables(t, dt, status)
100 continue
if (status /= 0) dt = 2.5d-01 * dt
end do
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,1) = pdata%uu(:,:,:,:,2)
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
!-------------------------------------------------------------------------------
!
end subroutine evolve_ssprk35
!
!===============================================================================
!
! subroutine EVOLVE_SSPRK4_10:
! ---------------------------
!
! Subroutine advances the solution by one time step using the 4rd order
! 10-stage Strong Stability Preserving Runge-Kutta time integration method.
!
! References:
!
! [1] Gottlieb, S., Ketcheson, D., Shu C. - W.,
! "Strong stability preserving Runge-Kutta and multistep
! time discretizations",
! World Scientific Publishing, 2011
!
!===============================================================================
!
subroutine evolve_ssprk4_10()
use blocks , only : block_data, list_data
use boundaries, only : boundary_fluxes
use equations , only : ibp, cmax
use sources , only : update_sources
implicit none
type(block_data), pointer :: pdata
integer :: i, status
real(kind=8) :: t
real(kind=8), dimension(9) :: ds
real(kind=8), dimension(9), parameter :: c = &
(/ 1.0d+00, 2.0d+00, 3.0d+00, 4.0d+00, 2.0d+00, &
3.0d+00, 4.0d+00, 5.0d+00, 6.0d+00 /) / 6.0d+00
!
!-------------------------------------------------------------------------------
!
ds(:) = c(:) * dt
status = 1
do while(status /= 0)
!= 1st step: U(2) = U(1)
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,1)
pdata%uu(:,:,:,:,3) = pdata%uu(:,:,:,:,1)
pdata%u => pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
!= 2nd step: U(1) = [1 + dt/6 L] U(1), for i = 1, ..., 5
!
do i = 1, 5
t = time + ds(i)
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, 0.0d+00, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,2) + ds(1) * pdata%du(:,:,:,:)
pdata => pdata%next
end do
call update_variables(t, ds(1), status)
if (status /= 0) go to 100
end do ! i = 1, 5
!= 3rd step: U(2) = U(2)/25 + 9/25 U(1), U(1) = 15 U(2) - 5 U(1)
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,3) = (pdata%uu(:,:,:,:,3) &
+ 9.0d+00 * pdata%uu(:,:,:,:,2)) / 2.5d+01
pdata%uu(:,:,:,:,2) = 1.5d+01 * pdata%uu(:,:,:,:,3) &
- 5.0d+00 * pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
!= 4th step: U(1) = [1 + dt/6 L] U(1), for i = 6, ..., 9
!
! integrate the intermediate steps
!
do i = 6, 9
t = time + ds(i)
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, ds(1), pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,2) + ds(1) * pdata%du(:,:,:,:)
pdata => pdata%next
end do
call update_variables(t, ds(1), status)
if (status /= 0) go to 100
end do ! i = 6, 9
!= the final step: U(n+1) = U(2) + 3/5 U(1) + 1/10 dt F[U(1)]
!
t = time + dt
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, t, 1.0d-01 * dt, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,3) &
+ 6.0d-01 * pdata%uu(:,:,:,:,2) &
+ 1.0d-01 * dt * pdata%du(:,:,:,:)
pdata => pdata%next
end do
if (ibp > 0) then
adecay(:) = exp(aglm(:) * cmax * dt)
pdata => list_data
do while (associated(pdata))
pdata%u(ibp,:,:,:) = adecay(pdata%meta%level) * pdata%u(ibp,:,:,:)
pdata => pdata%next
end do
end if
call update_variables(t, dt, status)
100 continue
if (status /= 0) dt = 2.5d-01 * dt
end do
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,1) = pdata%uu(:,:,:,:,2)
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
!-------------------------------------------------------------------------------
!
end subroutine evolve_ssprk4_10
!
!===============================================================================
!
! EMBEDDED METHODS
!
!===============================================================================
!
! subroutine EVOLVE_SSPRK2_M:
! --------------------------
!
! Subroutine advances the solution by one time step using the 2nd order
! m-stage Strong Stability Preserving Runge-Kutta time integration method.
! Up to 9 stages are allowed, due to stability problems with more stages.
!
! References:
!
! [1] Conde, S., Fekete, I., Shadid, J. N.,
! "Embedded error estimation and adaptive step-size control for
! optimal explicit strong stability preserving Runge–Kutta methods"
! 2018, arXiv:1806.08693v1
! [2] Gottlieb, S. and Gottlieb, L.-A., J.
! "Strong Stability Preserving Properties of Runge-Kutta Time
! Discretization Methods for Linear Constant Coefficient Operators",
! Journal of Scientific Computing,
! 2003, vol. 18, no. 1, pp. 83-109
!
!===============================================================================
!
subroutine evolve_ssprk2_m()
use blocks , only : block_data, list_data
use boundaries , only : boundary_fluxes
use coordinates, only : nb, ne
use equations , only : errors, ibp, cmax
use helpers , only : print_message
use sources , only : update_sources
implicit none
type(block_data), pointer :: pdata
logical :: test
integer :: nrej, i, status
real(kind=8) :: tm, dtm, dh, fc
character(len=*), parameter :: loc = 'EVOLUTION:evolve_ssprk2_m()'
!-------------------------------------------------------------------------------
!
test = .true.
nrej = 0
! at the entry point we assume the previous solution of conserved variables U(n)
! is stored in pdata%u(:,:,:,:,1) and the primitive variables are already
! updated from this array;
!
! pdata%uu(:,:,:,:,1) - the previous solution, U(n)
! pdata%uu(:,:,:,:,2) - the new 2nd order solution, U(1)
! pdata%uu(:,:,:,:,3) - the new 1st order solution, U(2)
!
do while(test)
!= preparatory step: U(1) = U(n), U(2) = 0, U(3) = U(n)
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,1)
pdata%uu(:,:,:,:,3) = pdata%uu(:,:,:,:,1)
pdata%u => pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
!= 1st step: U(1) = U(1) + dt/(m-1) F[U(1)], for i = 1, ..., m-1
!
do i = 1, stages - 1
dtm = c(i) * dt
tm = time + dtm
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, tm, dtm, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,2) &
+ bt(i) * dt * pdata%du(:,:,:,:)
pdata%uu(:,:,:,:,3) = pdata%uu(:,:,:,:,3) &
+ bh(i) * dt * pdata%du(:,:,:,:)
pdata => pdata%next
end do
call update_variables(tm, dtm, status)
if (status /= 0) go to 100
end do ! i = 1, stages - 1
!= 2nd step: U(1) = (m-1)/m U(1) + 1/m U(2) + dt/m F[U(1)]
!
i = stages
dtm = c(i) * dt
tm = time + dtm
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, tm, dtm, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = dl(1) * pdata%uu(:,:,:,:,2) &
+ dl(2) * pdata%uu(:,:,:,:,1) &
+ bt(i) * dt * pdata%du(:,:,:,:)
pdata%uu(:,:,:,:,3) = pdata%uu(:,:,:,:,3) &
+ bh(i) * dt * pdata%du(:,:,:,:)
pdata => pdata%next
end do
call update_variables(tm, dtm, status)
if (status /= 0) go to 100
!= 3rd step: decay the magnetic divergence potential of both solutions
!
if (ibp > 0) then
adecay(:) = exp(aglm(:) * cmax * dt)
pdata => list_data
do while (associated(pdata))
pdata%uu(ibp,:,:,:,2) = adecay(pdata%meta%level) &
* pdata%uu(ibp,:,:,:,2)
pdata%uu(ibp,:,:,:,3) = adecay(pdata%meta%level) &
* pdata%uu(ibp,:,:,:,3)
pdata => pdata%next
end do
end if
! update the vector of errors
!
call update_errors(2, 3)
! calculate the tolerance and estimate the next time step due to the error
!
errtol = maxval(errors)
errs(1) = errtol
fc = product(errs(:)**betas(:))
fc = 1.0d+00 + chi * atan((fc - 1.0d+00) / chi)
dte = dt * fc
100 continue
if ((fc > fac .or. nrej >= mrej) .and. status == 0) then
test = .false.
errs(3) = errs(2)
errs(2) = errs(1)
else
if (status == 0) then
dt = dte
nrej = nrej + 1 ! rejection count in the current step
else
dt = 2.5d-01 * dt
end if
nrejections = nrejections + 1
! since the solution was rejected, we have to revert the primitive variables
! to the previous state
!
pdata => list_data
do while (associated(pdata))
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
call update_variables(tm, dtm, status)
if (status /= 0) &
call print_message(loc, "Possible bug: the revert to " // &
"the previous state found it unphysical.")
end if
niterations = niterations + 1
end do
!= final step: U(n+1) = U(1) - update the accepted solution
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,1) = pdata%uu(:,:,:,:,2)
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
!-------------------------------------------------------------------------------
!
end subroutine evolve_ssprk2_m
!
!===============================================================================
!
! subroutine EVOLVE_SSP324:
! --------------------------
!
! Subroutine advances the solution by one time step using the 3ʳᵈ-order
! 4-stage embedded Strong Stability Preserving Runge-Kutta time integration
! method SSP3(2)4 with the error control.
!
! References:
!
! [1] Ranocha, H., Dalcin, L., Parsani, M., Ketcheson, D. I.,
! "Optimized Runge-Kutta Methods with Automatic Step Size Control
! for Compressible Computational Fluid Dynamics",
! 2021, arXiv:2104.06836v2
! [2] Conde, S., Fekete, I., Shadid, J. N.,
! "Embedded error estimation and adaptive step-size control for
! optimal explicit strong stability preserving Runge–Kutta methods"
! 2018, arXiv:1806.08693v1
! [3] Gottlieb, S., Ketcheson, D., Shu, C.-W.,
! "Strong stability preserving Runge-Kutta and multistep time
! discretizations",
! 2011, World Scientific Publishing
!
!===============================================================================
!
subroutine evolve_ssp324()
use blocks , only : block_data, list_data
use boundaries , only : boundary_fluxes
use coordinates, only : nb, ne
use equations , only : errors, ibp, cmax
use helpers , only : print_message
use sources , only : update_sources
implicit none
type(block_data), pointer :: pdata
logical :: test
integer :: nrej, i, status
real(kind=8) :: tm, dtm, dh, fc
real(kind=8), parameter :: onethird = 1.0d+00 / 3.0d+00, &
twothird = 2.0d+00 / 3.0d+00
character(len=*), parameter :: loc = 'EVOLUTION:evolve_ssp324()'
!-------------------------------------------------------------------------------
!
test = .true.
nrej = 0
! at the entry point we assume the previous solution of conserved variables U(n)
! is stored in pdata%u(:,:,:,:,1) and the primitive variables are already
! updated from this array;
!
! pdata%uu(:,:,:,:,1) - the previous solution, U(n)
! pdata%uu(:,:,:,:,2) - the new 3rd order solution, U(1)
! pdata%uu(:,:,:,:,3) - the new 2nd order solution, U(2)
!
do while(test)
! initiate the fractional time step
!
dh = 5.0d-01 * dt
!= preparation step: U(1) = U(n)
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,1)
pdata%u => pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
!= 1st to 3rd steps: U(1) = U(1) + ½ dt F[U(1)], for i = 1...3
!
do i = 1, 3
tm = time + (i - 1) * dh
dtm = dh
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, tm, dtm, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,2) + dh * pdata%du(:,:,:,:)
pdata => pdata%next
end do
call update_variables(tm, dtm, status)
if (status /= 0) go to 100
end do ! i = 1, 3
!= 4th step: U(1) = ⅔ U(n) + ⅓ U(1), U(2) = ⅓ U(n) + ⅔ U(1)
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,3) = onethird * pdata%uu(:,:,:,:,1) &
+ twothird * pdata%uu(:,:,:,:,2)
pdata%uu(:,:,:,:,2) = twothird * pdata%uu(:,:,:,:,1) &
+ onethird * pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
call update_variables(tm, dtm, status)
if (status /= 0) go to 100
!= 5th step: U(1) = U(1) + ½ dt F[U(1)] <- 3ʳᵈ-order candidate
!
tm = time + dh
dtm = dh
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, tm, dtm, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,2) + dh * pdata%du(:,:,:,:)
pdata => pdata%next
end do
call update_variables(tm, dtm, status)
if (status /= 0) go to 100
!= 6th step: U(2) = ½ (U(1) + U(2)) <- 2ⁿᵈ-order approximation
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,3) = 5.0d-01 * (pdata%uu(:,:,:,:,3) &
+ pdata%uu(:,:,:,:,2))
pdata => pdata%next
end do
!= 7th step: decay the magnetic divergence potential of both solutions
!
if (ibp > 0) then
adecay(:) = exp(aglm(:) * cmax * dt)
pdata => list_data
do while (associated(pdata))
pdata%uu(ibp,:,:,:,2) = adecay(pdata%meta%level) &
* pdata%uu(ibp,:,:,:,2)
pdata%uu(ibp,:,:,:,3) = adecay(pdata%meta%level) &
* pdata%uu(ibp,:,:,:,3)
pdata => pdata%next
end do
end if
! update the vector of errors
!
call update_errors(2, 3)
! calculate the tolerance and estimate the next time step due to the error
!
errtol = maxval(errors)
errs(1) = errtol
fc = product(errs(:)**betas(:))
fc = 1.0d+00 + chi * atan((fc - 1.0d+00) / chi)
dte = dt * fc
100 continue
if ((fc > fac .or. nrej >= mrej) .and. status == 0) then
test = .false.
errs(3) = errs(2)
errs(2) = errs(1)
else
if (status == 0) then
dt = dte
nrej = nrej + 1 ! rejection count in the current step
else
dt = 2.5d-01 * dt
end if
nrejections = nrejections + 1
! since the solution was rejected, we have to revert the primitive variables
! to the previous state
!
pdata => list_data
do while (associated(pdata))
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
call update_variables(tm, dtm, status)
if (status /= 0) &
call print_message(loc, "Possible bug: the revert to " // &
"the previous state found it unphysical.")
end if
niterations = niterations + 1
end do
!= final step: U(n+1) = U(1) - update the accepted solution
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,1) = pdata%uu(:,:,:,:,2)
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
!-------------------------------------------------------------------------------
!
end subroutine evolve_ssp324
!
!===============================================================================
!
! subroutine EVOLVE_SSPRK32M:
! --------------------------
!
! Subroutine advances the solution by one time step using the 3ʳᵈ-order
! m-stage (m=n²) embedded Strong Stability Preserving Runge-Kutta time
! integration method SSP3(2)m with the error control.
!
! References:
!
! [1] Conde, S., Fekete, I., Shadid, J. N.,
! "Embedded error estimation and adaptive step-size control for
! optimal explicit strong stability preserving Runge–Kutta methods"
! 2018, arXiv:1806.08693v1
! [2] Gottlieb, S., Ketcheson, D., Shu, C.-W.,
! "Strong stability preserving Runge-Kutta and multistep time
! discretizations",
! 2011, World Scientific Publishing
!
!===============================================================================
!
subroutine evolve_ssprk32m()
use blocks , only : block_data, list_data
use boundaries, only : boundary_fluxes
use equations , only : errors, ibp, cmax
use helpers , only : print_message
use sources , only : update_sources
implicit none
type(block_data), pointer :: pdata
logical , save :: first = .true.
integer , save :: i1, i2, i3, i4, n, m
real(kind=8), save :: f1, f2, f3, f4
logical :: test
integer :: nrej, i, status
real(kind=8) :: tm, dtm, dh, fc
character(len=*), parameter :: loc = 'EVOLUTION:evolve_ssprk32m()'
!-------------------------------------------------------------------------------
!
if (first) then
n = 2
do while(n**2 < stages)
n = n + 1
end do
m = stages - n
i1 = (n - 1) * (n - 2) / 2
i2 = i1 + 1
i3 = n * (n + 1) / 2
i4 = i3 + 1
fc = real(2 * n - 1, kind=8)
f1 = real(n - 1, kind=8) / fc
f2 = real(n , kind=8) / fc
f4 = real(m , kind=8) / stages
f3 = 1.0d+00 - f4
first = .false.
end if
test = .true.
nrej = 0
! at the entry point we assume the previous solution of conserved variables U(n)
! is stored in pdata%u(:,:,:,:,1) and the primitive variables are already
! updated from this array;
!
! pdata%uu(:,:,:,:,1) - the previous solution, U(n)
! pdata%uu(:,:,:,:,2) - the new 3rd order solution, U(1)
! pdata%uu(:,:,:,:,3) - the intermediate solution, U(2)
! pdata%uu(:,:,:,:,4) - the new 2nd order solution, U(3)
!
do while(test)
! initiate the fractional time step
!
dh = dt / m
!= preparation step: U(1) = U(n)
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,1)
pdata%u => pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
!= 1st step: U(1) = U(1) + dt/r F[U(1)], for i = 1...(n-1)*(n-2)/2
!
do i = 1, i1
tm = time + (i - 1) * dh
dtm = dh
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, tm, dtm, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,2) + dh * pdata%du(:,:,:,:)
pdata => pdata%next
end do
call update_variables(tm, dtm, status)
if (status /= 0) go to 100
end do ! i = 1, i1
!= 2nd step: U(2) = U(1)
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,3) = pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
!= 3rd step: U(1) = U(1) + dt/r F[U(1)], for i = (n-1)*(n-2)/2+1...n*(n+1)/2
!
do i = i2, i3
tm = time + (i - 1) * dh
dtm = dh
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, tm, dtm, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,2) + dh * pdata%du(:,:,:,:)
pdata => pdata%next
end do
call update_variables(tm, dtm, status)
if (status /= 0) go to 100
end do ! i = i2, i3
!= 4th step: U(3) = (1 - r/s) * U(n) + r/s * U(1),
! U(1) = [(n - 1) * U(1) + n * U(2)] / (2*n - 1),
! U(3) = U(3) - r/s * U(1)
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,4) = f3 * pdata%uu(:,:,:,:,1) &
+ f4 * pdata%uu(:,:,:,:,2)
pdata%uu(:,:,:,:,2) = f1 * pdata%uu(:,:,:,:,2) &
+ f2 * pdata%uu(:,:,:,:,3)
pdata%uu(:,:,:,:,4) = pdata%uu(:,:,:,:,4) &
- f4 * pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
call update_variables(tm, dtm, status)
if (status /= 0) go to 100
!= 6th step: U(1) = U(1) + dt/r F[U(1)], for i = n*(n+1)/2+1...stages
!
do i = i4, stages
tm = time + (i - 1 - n) * dh
dtm = dh
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, tm, dtm, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,2) + dh * pdata%du(:,:,:,:)
pdata => pdata%next
end do
call update_variables(tm, dtm, status)
if (status /= 0) go to 100
end do ! i = i4, stages
!= 7th step: U(3) = U(3) + r/s * U(1)
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,4) = pdata%uu(:,:,:,:,4) + f4 * pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
!= 8th step: decay the magnetic divergence potential of both solutions
!
if (ibp > 0) then
adecay(:) = exp(aglm(:) * cmax * dt)
pdata => list_data
do while (associated(pdata))
pdata%uu(ibp,:,:,:,2) = adecay(pdata%meta%level) &
* pdata%uu(ibp,:,:,:,2)
pdata%uu(ibp,:,:,:,4) = adecay(pdata%meta%level) &
* pdata%uu(ibp,:,:,:,4)
pdata => pdata%next
end do
end if
! update the vector of errors
!
call update_errors(2, 4)
! calculate the tolerance and estimate the next time step due to the error
!
errtol = maxval(errors)
errs(1) = errtol
fc = product(errs(:)**betas(:))
fc = 1.0d+00 + chi * atan((fc - 1.0d+00) / chi)
dte = dt * fc
100 continue
if ((fc > fac .or. nrej >= mrej) .and. status == 0) then
test = .false.
errs(3) = errs(2)
errs(2) = errs(1)
else
if (status == 0) then
dt = dte
nrej = nrej + 1 ! rejection count in the current step
else
dt = 2.5d-01 * dt
end if
nrejections = nrejections + 1
! since the solution was rejected, we have to revert the primitive variables
! to the previous state
!
pdata => list_data
do while (associated(pdata))
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
call update_variables(tm, dtm, status)
if (status /= 0) &
call print_message(loc, "Possible bug: the revert to " // &
"the previous state found it unphysical.")
end if
niterations = niterations + 1
end do
!= final step: U(n+1) = U(1) - update the accepted solution
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,1) = pdata%uu(:,:,:,:,2)
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
!-------------------------------------------------------------------------------
!
end subroutine evolve_ssprk32m
!
!===============================================================================
!
! subroutine EVOLVE_3SSTARPLUS:
! ----------------------------
!
! Subroutine advances the solution by one time step using the minimum
! storage implementation of 3S*+ methods.
!
! References:
!
! [1] Ranocha, H., Dalcin, L., Parsani, M., Ketcheson, D. I.
! "Optimized Runge-Kutta Methods with Automatic Step Size Control
! for Compressible Computational Fluid Dynamics",
! 2021, arXiv:2104.06836
! [2] Gottlieb, S., Ketcheson, D., Shu C. - W.,
! "Strong stability preserving Runge-Kutta and multistep
! time discretizations",
! World Scientific Publishing, 2011
!
!===============================================================================
!
subroutine evolve_3sstarplus()
use blocks , only : block_data, list_data
use boundaries, only : boundary_fluxes
use equations , only : errors, ibp, cmax
use helpers , only : print_message
use sources , only : update_sources
implicit none
type(block_data), pointer :: pdata
logical :: test
integer :: i, l, nrej, status
real(kind=8) :: tm, dtm, fc
character(len=*), parameter :: loc = 'EVOLUTION:evolve_3sstarplus()'
!-------------------------------------------------------------------------------
!
test = .true.
nrej = 0
! at the entry point we assume the previous solution of conserved variables U(n)
! is stored in pdata%u(:,:,:,:,1) and the primitive variables are already
! updated from this array;
!
! pdata%uu(:,:,:,:,1) - the previous solution, U(n)
! pdata%uu(:,:,:,:,2) - the new 3rd order solution, U(1)
! pdata%uu(:,:,:,:,3) - the intermediate solution, U(2)
! pdata%uu(:,:,:,:,4) - the new 2nd order solution, U(3)
!
do while(test)
!= preparatory step: U(1) = U(n), U(2) = 0, U(3) = U(n)
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,1)
pdata%uu(:,:,:,:,3) = pdata%uu(:,:,:,:,1)
pdata%uu(:,:,:,:,4) = pdata%uu(:,:,:,:,1)
pdata%u => pdata%uu(:,:,:,:,2)
pdata => pdata%next
end do
!= 1st stage
!
if (fsal_update) then
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, time, 0.0d+00, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
end if
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,2) = pdata%uu(:,:,:,:,1) &
+ bt(1) * dt * pdata%du(:,:,:,:)
pdata%uu(:,:,:,:,4) = pdata%uu(:,:,:,:,4) &
+ bh(1) * dt * pdata%du(:,:,:,:)
pdata => pdata%next
end do
dtm = c(2) * dt
call update_variables(time, dtm, status)
if (status /= 0) go to 100
!= remaining stages
!
do i = 2, stages
dtm = c(i) * dt
tm = time + dtm
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, tm, dtm, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,3) = pdata%uu(:,:,:,:,3) &
+ dl(i) * pdata%uu(:,:,:,:,2)
pdata%uu(:,:,:,:,2) = gm(1,i) * pdata%uu(:,:,:,:,2) &
+ gm(2,i) * pdata%uu(:,:,:,:,3) &
+ gm(3,i) * pdata%uu(:,:,:,:,1) &
+ bt(i) * dt * pdata%du(:,:,:,:)
pdata%uu(:,:,:,:,4) = pdata%uu(:,:,:,:,4) &
+ bh(i) * dt * pdata%du(:,:,:,:)
pdata => pdata%next
end do
dtm = c(i+1) * dt
call update_variables(tm, dtm, status)
if (status /= 0) go to 100
end do ! i = 2, stages
!= extra step: decay the magnetic divergence potential of both solutions
!
if (ibp > 0) then
adecay(:) = exp(aglm(:) * cmax * dt)
pdata => list_data
do while (associated(pdata))
pdata%uu(ibp,:,:,:,2) = adecay(pdata%meta%level) &
* pdata%uu(ibp,:,:,:,2)
pdata%uu(ibp,:,:,:,4) = adecay(pdata%meta%level) &
* pdata%uu(ibp,:,:,:,4)
pdata => pdata%next
end do
end if
!= extra step: FSAL
!
if (fsal) then
tm = time + dt
pdata => list_data
do while (associated(pdata))
call update_increment(pdata)
call update_sources(pdata, tm, 0.0d+00, pdata%du(:,:,:,:))
pdata => pdata%next
end do
call boundary_fluxes()
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,4) = pdata%uu(:,:,:,:,4) &
+ bh(stages+1) * dt * pdata%du(:,:,:,:)
pdata => pdata%next
end do
end if
!= error estimation
!
call update_errors(2, 4)
! calculate the tolerance and estimate the next time step due to the error
!
errtol = maxval(errors)
errs(1) = errtol
fc = product(errs(:)**betas(:))
fc = 1.0d+00 + chi * atan((fc - 1.0d+00) / chi)
dte = dt * fc
100 continue
if ((fc > fac .or. nrej >= mrej) .and. status == 0) then
test = .false.
errs(3) = errs(2)
errs(2) = errs(1)
if (fsal) fsal_update = .true.
else
if (status == 0) then
dt = dte
nrej = nrej + 1 ! rejection count in the current step
else
dt = 2.5d-01 * dt
end if
nrejections = nrejections + 1
! since the solution was rejected, we have to revert the primitive variables
! to the previous state
!
pdata => list_data
do while (associated(pdata))
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
call update_variables(tm, dtm, status)
if (status /= 0) &
call print_message(loc, "Possible bug: the revert to " // &
"the previous state found it unphysical.")
if (fsal) fsal_update = .false.
end if
niterations = niterations + 1
end do
!= final step: U(n+1) = U(1)
!
pdata => list_data
do while (associated(pdata))
pdata%uu(:,:,:,:,1) = pdata%uu(:,:,:,:,2)
pdata%u => pdata%uu(:,:,:,:,1)
pdata => pdata%next
end do
tm = time + dt
call update_variables(tm, dt, status)
if (status /= 0) &
call print_message(loc, "Possible bug: the new state has been " // &
"accepted, nevertheless the variable " // &
"update found it unphysical.")
!-------------------------------------------------------------------------------
!
end subroutine evolve_3sstarplus
!
!===============================================================================
!
! subroutine UPDATE_INCREMENT:
! ---------------------------
!
! Subroutine calculates the conservative variable increment from
! directional fluxes.
!
! Arguments:
!
! pdata - the point to data block storing the directional fluxes;
!
!===============================================================================
!
subroutine update_increment(pdata)
use blocks , only : block_data
use coordinates, only : nbl, ne, neu, nn => bcells
use coordinates, only : adx, ady, adxi, adyi
#if NDIMS == 3
use coordinates, only : adz, adzi
#endif /* NDIMS == 3 */
use equations , only : nf, ns
use equations , only : idn, isl, isu
use helpers , only : print_message
use schemes , only : update_flux
use workspace , only : resize_workspace, work, work_in_use
implicit none
type(block_data), pointer, intent(inout) :: pdata
logical, save :: first = .true.
integer :: i , j , k = 1, p
integer :: im1, ip1
integer :: jm1, jp1
#if NDIMS == 3
integer :: km1, kp1
#endif /* NDIMS == 3 */
integer :: status
real(kind=8) :: df
real(kind=8), dimension(NDIMS) :: dh, dhi
real(kind=8), dimension(:,:,:,:,:) , pointer, save :: f
real(kind=8), dimension(:,:,:,:,:,:), pointer, save :: s
character(len=*), parameter :: loc = 'EVOLUTION:update_increment()'
!-------------------------------------------------------------------------------
!
if (first) then
i = NDIMS * nf * nn**NDIMS
j = 3 * i
call resize_workspace(j, status)
if (status /= 0) then
call print_message(loc, "Could not resize the workspace!")
go to 100
end if
#if NDIMS == 3
f(1:nf,1:nn,1:nn,1:nn,1:3) => work( 1:i)
s(1:nf,1:nn,1:nn,1:nn,1:2,1:3) => work(i+1:j)
#else /* NDIMS == 3 */
f(1:nf,1:nn,1:nn,1: 1,1:2) => work( 1:i)
s(1:nf,1:nn,1:nn,1: 1,1:2,1:2) => work(i+1:j)
#endif /* NDIMS == 3 */
first = .false.
end if
if (work_in_use) &
call print_message(loc, "Workspace is being used right now! " // &
"Corruptions can occur!")
work_in_use = .true.
dh(1) = adx(pdata%meta%level)
dh(2) = ady(pdata%meta%level)
#if NDIMS == 3
dh(3) = adz(pdata%meta%level)
#endif /* NDIMS == 3 */
dhi(1) = adxi(pdata%meta%level)
dhi(2) = adyi(pdata%meta%level)
#if NDIMS == 3
dhi(3) = adzi(pdata%meta%level)
#endif /* NDIMS == 3 */
! calculate the interface fluxes
!
call update_flux(dh(1:NDIMS), pdata%q(:,:,:,:), &
s(:,:,:,:,:,:), f(:,:,:,:,:))
! store the block interface fluxes
!
pdata%fx(:,1,:,:) = f(:,nbl,:,:,1)
pdata%fx(:,2,:,:) = f(:,ne ,:,:,1)
pdata%fy(:,:,1,:) = f(:,:,nbl,:,2)
pdata%fy(:,:,2,:) = f(:,:,ne ,:,2)
#if NDIMS == 3
pdata%fz(:,:,:,1) = f(:,:,:,nbl,3)
pdata%fz(:,:,:,2) = f(:,:,:,ne ,3)
#endif /* NDIMS == 3 */
! calculate the variable update from the directional fluxes
!
#if NDIMS == 3
do k = nbl, neu
km1 = k - 1
#endif /* NDIMS == 3 */
do j = nbl, neu
jm1 = j - 1
do i = nbl, neu
im1 = i - 1
#if NDIMS == 3
pdata%du(1:nf,i,j,k) = - dhi(1) * (f(:,i,j,k,1) - f(:,im1,j,k,1)) &
- dhi(2) * (f(:,i,j,k,2) - f(:,i,jm1,k,2)) &
- dhi(3) * (f(:,i,j,k,3) - f(:,i,j,km1,3))
#else /* NDIMS == 3 */
pdata%du(1:nf,i,j,k) = - dhi(1) * (f(:,i,j,k,1) - f(:,im1,j,k,1)) &
- dhi(2) * (f(:,i,j,k,2) - f(:,i,jm1,k,2))
#endif /* NDIMS == 3 */
end do ! i = nbl, neu
end do ! j = nbl, neu
#if NDIMS == 3
end do ! k = nbl, neu
#endif /* NDIMS == 3 */
! update passive scalars
!
if (ns > 0) then
pdata%du(isl:isu,:,:,:) = 0.0d+00
#if NDIMS == 3
do k = nbl - 1, neu
kp1 = k + 1
#endif /* NDIMS == 3 */
do j = nbl - 1, neu
jp1 = j + 1
do i = nbl - 1, neu
ip1 = i + 1
! X-face
!
if (f(idn,i,j,k,1) >= 0.0d+00) then
do p = isl, isu
df = dhi(1) * f(idn,i,j,k,1) * pdata%q(p,i ,j,k)
pdata%du(p,i ,j,k) = pdata%du(p,i ,j,k) - df
pdata%du(p,ip1,j,k) = pdata%du(p,ip1,j,k) + df
end do
else
do p = isl, isu
df = dhi(1) * f(idn,i,j,k,1) * pdata%q(p,ip1,j,k)
pdata%du(p,i ,j,k) = pdata%du(p,i ,j,k) - df
pdata%du(p,ip1,j,k) = pdata%du(p,ip1,j,k) + df
end do
end if
! Y-face
!
if (f(idn,i,j,k,2) >= 0.0d+00) then
do p = isl, isu
df = dhi(2) * f(idn,i,j,k,2) * pdata%q(p,i,j ,k)
pdata%du(p,i,j ,k) = pdata%du(p,i,j ,k) - df
pdata%du(p,i,jp1,k) = pdata%du(p,i,jp1,k) + df
end do
else
do p = isl, isu
df = dhi(2) * f(idn,i,j,k,2) * pdata%q(p,i,jp1,k)
pdata%du(p,i,j ,k) = pdata%du(p,i,j ,k) - df
pdata%du(p,i,jp1,k) = pdata%du(p,i,jp1,k) + df
end do
end if
#if NDIMS == 3
! Z-face
!
if (f(idn,i,j,k,3) >= 0.0d+00) then
do p = isl, isu
df = dhi(3) * f(idn,i,j,k,3) * pdata%q(p,i,j,k )
pdata%du(p,i,j,k ) = pdata%du(p,i,j,k ) - df
pdata%du(p,i,j,kp1) = pdata%du(p,i,j,kp1) + df
end do
else
do p = isl, isu
df = dhi(3) * f(idn,i,j,k,3) * pdata%q(p,i,j,kp1)
pdata%du(p,i,j,k ) = pdata%du(p,i,j,k ) - df
pdata%du(p,i,j,kp1) = pdata%du(p,i,j,kp1) + df
end do
end if
#endif /* NDIMS == 3 */
end do ! i = nbl, neu
end do ! j = nbl, neu
#if NDIMS == 3
end do ! k = nbl, neu
#endif /* NDIMS == 3 */
end if
work_in_use = .false.
100 continue
!-------------------------------------------------------------------------------
!
end subroutine update_increment
!
!===============================================================================
!
! subroutine UPDATE_VARIABLES:
! ---------------------------
!
! Subroutine iterates over all data blocks and converts the conservative
! variables to their primitive representation.
!
! Arguments:
!
! tm - time at the moment of update;
! dtm - time step since the last update;
! status - status flag indicating if the update was successful;
!
!===============================================================================
!
subroutine update_variables(tm, dtm, status)
use blocks , only : block_meta, list_meta
use blocks , only : block_data, list_data
use blocks , only : set_neighbors_update
use boundaries, only : boundary_variables
use equations , only : update_primitive_variables
use equations , only : fix_unphysical_cells, correct_unphysical_states
#ifdef MPI
use mpitools , only : reduce_maximum
#endif /* MPI */
use shapes , only : update_shapes
implicit none
real(kind=8), intent(in) :: tm, dtm
integer , intent(out) :: status
type(block_meta), pointer :: pmeta
type(block_data), pointer :: pdata
!-------------------------------------------------------------------------------
!
status = 0
pdata => list_data
do while (associated(pdata))
pmeta => pdata%meta
if (pmeta%update) then
call update_primitive_variables(pdata%u, pdata%q, status)
if (status /= 0) go to 100
end if
pdata => pdata%next
end do
100 continue
#ifdef MPI
call reduce_maximum(status)
#endif /* MPI */
if (status /= 0) go to 200
call boundary_variables(tm, dtm)
pdata => list_data
do while (associated(pdata))
pmeta => pdata%meta
if (pmeta%update) call update_shapes(pdata, tm, dtm)
pdata => pdata%next
end do
if (fix_unphysical_cells) then
! if an unphysical cell appeared in a block while updating its primitive
! variables it could be propagated to its neighbors through boundary update;
! mark all neighbors of such a block to be verified and corrected for
! unphysical cells too
!
pmeta => list_meta
do while (associated(pmeta))
if (pmeta%update) call set_neighbors_update(pmeta)
pmeta => pmeta%next
end do
! verify and correct, if necessary, unphysical cells in recently updated blocks
!
pdata => list_data
do while (associated(pdata))
pmeta => pdata%meta
if (pmeta%update) &
call correct_unphysical_states(step, pmeta%id, pdata%q, pdata%u)
pdata => pdata%next
end do
end if
200 continue
!-------------------------------------------------------------------------------
!
end subroutine update_variables
!
!===============================================================================
!
! subroutine UPDATE_ERRORS_L2:
! ---------------------------
!
! Subroutine updates the errors with respect to the absolute and relative
! tolerances using the L2 norm. The errors are calculated per variable.
!
! Arguments:
!
! n - the index of the higher order solution
! m - the index of the lower order solution
!
!===============================================================================
!
subroutine update_errors_l2(n, m)
use blocks , only : block_data, list_data, get_nleafs
use coordinates, only : ncells, nb, ne
use equations , only : nf, errors
#ifdef MPI
use mpitools , only : reduce_sum
#endif /* MPI */
implicit none
integer, intent(in) :: n, m
integer :: l
real(kind=8) :: fnorm
type(block_data), pointer :: pdata
!-------------------------------------------------------------------------------
!
errors(:) = 0.0d+00
fnorm = 1.0d+00 / (get_nleafs() * ncells**NDIMS)
pdata => list_data
do while (associated(pdata))
do l = 1, nf
#if NDIMS == 3
errors(l) = errors(l) + &
sum((abs(pdata%uu(l,nb:ne,nb:ne,nb:ne,n) &
- pdata%uu(l,nb:ne,nb:ne,nb:ne,m)) / &
(atol + rtol * &
max(abs(pdata%uu(l,nb:ne,nb:ne,nb:ne,n)), &
abs(pdata%uu(l,nb:ne,nb:ne,nb:ne,m)))))**2)
#else /* NDIMS == 3 */
errors(l) = errors(l) + &
sum((abs(pdata%uu(l,nb:ne,nb:ne, : ,n) &
- pdata%uu(l,nb:ne,nb:ne, : ,m)) / &
(atol + rtol * &
max(abs(pdata%uu(l,nb:ne,nb:ne, : ,n)), &
abs(pdata%uu(l,nb:ne,nb:ne, : ,m)))))**2)
#endif /* NDIMS == 3 */
end do
pdata => pdata%next
end do
#ifdef MPI
call reduce_sum(errors)
#endif /* MPI */
errors(:) = sqrt(fnorm * errors(:))
!-------------------------------------------------------------------------------
!
end subroutine update_errors_l2
!
!===============================================================================
!
! subroutine UPDATE_ERRORS_MAX:
! ----------------------------
!
! Subroutine updates the errors with respect to the absolute and relative
! tolerances using the maximum norm. The errors are calculated per variable.
!
! Arguments:
!
! n - the index of the higher order solution
! m - the index of the lower order solution
!
!===============================================================================
!
subroutine update_errors_max(n, m)
use blocks , only : block_data, list_data
use coordinates, only : nb, ne
use equations , only : nf, errors
#ifdef MPI
use mpitools , only : reduce_maximum
#endif /* MPI */
implicit none
integer, intent(in) :: n, m
integer :: l
type(block_data), pointer :: pdata
!-------------------------------------------------------------------------------
!
errors(:) = 0.0d+00
pdata => list_data
do while (associated(pdata))
do l = 1, nf
#if NDIMS == 3
errors(l) = max(errors(l), &
maxval(abs(pdata%uu(l,nb:ne,nb:ne,nb:ne,n) &
- pdata%uu(l,nb:ne,nb:ne,nb:ne,m)) / &
(atol + rtol * &
max(abs(pdata%uu(l,nb:ne,nb:ne,nb:ne,n)), &
abs(pdata%uu(l,nb:ne,nb:ne,nb:ne,m))))))
#else /* NDIMS == 3 */
errors(l) = max(errors(l), &
maxval(abs(pdata%uu(l,nb:ne,nb:ne, : ,n) &
- pdata%uu(l,nb:ne,nb:ne, : ,m)) / &
(atol + rtol * &
max(abs(pdata%uu(l,nb:ne,nb:ne, : ,n)), &
abs(pdata%uu(l,nb:ne,nb:ne, : ,m))))))
#endif /* NDIMS == 3 */
end do
pdata => pdata%next
end do
#ifdef MPI
call reduce_maximum(errors)
#endif /* MPI */
!-------------------------------------------------------------------------------
!
end subroutine update_errors_max
#ifdef DEBUG
!
!===============================================================================
!
! subroutine CHECK_VARIABLES:
! --------------------------
!
! Subroutine iterates over all data blocks and converts the conservative
! variables to their primitive representation.
!
!
!===============================================================================
!
subroutine check_variables()
! include external procedures
!
use coordinates , only : nn => bcells
use equations , only : nv, pvars, cvars
use ieee_arithmetic, only : ieee_is_nan
! include external variables
!
use blocks , only : block_meta, list_meta
use blocks , only : block_data, list_data
! local variables are not implicit by default
!
implicit none
! local variables
!
integer :: i, j, k = 1, p
! local pointers
!
type(block_meta), pointer :: pmeta
type(block_data), pointer :: pdata
!
!-------------------------------------------------------------------------------
!
! associate the pointer with the first block on the data block list
!
pdata => list_data
! iterate over all data blocks
!
do while (associated(pdata))
! associate pmeta with the corresponding meta block
!
pmeta => pdata%meta
! check if there are NaNs in primitive variables
!
#if NDIMS == 3
do k = 1, nn
#endif /* NDIMS == 3 */
do j = 1, nn
do i = 1, nn
do p = 1, nv
if (ieee_is_nan(pdata%u(p,i,j,k))) then
print *, 'U NaN:', cvars(p), pdata%meta%id, i, j, k
end if
if (ieee_is_nan(pdata%q(p,i,j,k))) then
print *, 'Q NaN:', pvars(p), pdata%meta%id, i, j, k
end if
end do ! p = 1, nv
end do ! i = 1, nn
end do ! j = 1, nn
#if NDIMS == 3
end do ! k = 1, nn
#endif /* NDIMS == 3 */
! assign pointer to the next block
!
pdata => pdata%next
end do
!-------------------------------------------------------------------------------
!
end subroutine check_variables
#endif /* DEBUG */
!===============================================================================
!
end module evolution