SCHEMES: Add symmetry preserving optimizations to isothermal KEPES Riemann solver

- Update riemann_mhd_iso_kepes() subroutine to include missing symmetry
  preserving optimizations.
- Ensure consistency with the adiabatic version by adding and utilizing
  the ch variable.
- Correct the initialization and usage of transformation matrices (rm and tm).
- Adjust calculations to ensure symmetric properties, particularly in the handling
  of variables cs, ca, and cglm.
- Include appropriate sign functions for consistent behavior.
- Make changes to the calculation of intermediate variables to maintain symmetry
  and consistency.
- Adjusted flux calculation to correctly apply symmetry preserving
  terms.

Signed-off-by: Grzegorz Kowal <grzegorz@amuncode.org>

sources/schemes.F90

@@ -2155,7 +2155,7 @@ module schemes
   subroutine riemann_mhd_iso_kepes(ql, qr, fi)
 
     use coordinates, only : nn => bcells
-    use equations  , only : nf, csnd, csnd2, ch => cglm
+    use equations  , only : nf, csnd, csnd2, cglm
     use equations  , only : idn, imx, imy, imz, ivx, ivy, ivz, &
                             ibx, iby, ibz, ibp
 
@@ -2167,7 +2167,7 @@ module schemes
     integer      :: i
     real(kind=8) :: dna, pta, vxa, vya, vza, bxa, bya, bza, bpa
     real(kind=8) :: dnl, v2l, v2r, b2l, b2r, bp2, das, daf, csq
-    real(kind=8) :: ca, cs, cf, ca2, cs2, cf2, x2, x3
+    real(kind=8) :: ch, ca, cs, cf, ca2, cs2, cf2, x2, x3
     real(kind=8) :: alf, als, f1, f2, f3, f4
 
     real(kind=8), dimension(nf) :: v, lm, tm
@@ -2181,9 +2181,7 @@ module schemes
     if (first) then
       rm(:,:) =   0.0d+00
       rm(5,4) =   1.0d+00
-      rm(8,4) =   1.0d+00
       rm(5,5) =   1.0d+00
-      rm(8,5) = - 1.0d+00
 
       first = .false.
     end if
@@ -2219,6 +2217,7 @@ module schemes
       v(iby) = qr(iby,i) - ql(iby,i)
       v(ibz) = qr(ibz,i) - ql(ibz,i)
       v(ibp) = qr(ibp,i) - ql(ibp,i)
+      v( : ) = v( : ) / csnd2
 
       bp2 = bya * bya + bza * bza
       if (bp2 > 0.0d+00) then
@@ -2251,8 +2250,9 @@ module schemes
       end if
 
       cf = sign(sqrt(cf2), vxa)
-      ca =      sqrt(ca2)
       cs = sign(sqrt(cs2), vxa)
+      ca = sign(sqrt(ca2), vxa)
+      ch = sign(     cglm, vxa)
 
       lm(1) = vxa + cf
       lm(2) = vxa + ca
@@ -2263,10 +2263,10 @@ module schemes
       lm(7) = vxa - ca
       lm(8) = vxa - cf
 
-      tm(1) = 5.0d-01 / (dnl * csnd2)
+      tm(1) = 5.0d-01 / dnl
-      tm(2) = 5.0d-01 / dnl**2
+      tm(2) = 5.0d-01 * csnd2 / dnl**2
       tm(3) = tm(1)
-      tm(4) = 5.0d-01
+      tm(4) = 5.0d-01 * csnd2
       tm(5) = tm(4)
       tm(6) = tm(1)
       tm(7) = tm(2)
@@ -2276,24 +2276,24 @@ module schemes
       daf = dnl * alf
       csq = csnd * sqrt(dnl)
 
-      f1 = daf
+      f1 =   daf
-      f2 = sign(das, bxa) * cs
+      f2 =   sign(das, bxa) * cs
-      f3 = als * csq
+      f3 =   als * csq
       rm(1,1) = f1
-      rm(2,1) = f1 * (vxa + cf)
+      rm(2,1) = f1 * lm(1)
       rm(3,1) = f1 * vya - f2 * x2
       rm(4,1) = f1 * vza - f2 * x3
       rm(6,1) = f3 * x2
       rm(7,1) = f3 * x3
 
       rm(1,8) = f1
-      rm(2,8) = f1 * (vxa - cf)
+      rm(2,8) = f1 * lm(8)
       rm(3,8) = f1 * vya + f2 * x2
       rm(4,8) = f1 * vza + f2 * x3
       rm(6,8) = rm(6,1)
       rm(7,8) = rm(7,1)
 
-      f1 = dnl * sqrt(dna)
+      f1 = sign(dnl * sqrt(dna), vxa)
       rm(3,2) =    f1 * x3
       rm(4,2) = -  f1 * x2
       rm(6,2) = - dnl * x3
@@ -2304,31 +2304,35 @@ module schemes
       rm(6,7) =   rm(6,2)
       rm(7,7) =   rm(7,2)
 
-      f1 = das
+      f1 =   das
-      f2 = sign(daf, bxa) * cf
+      f2 =   sign(daf, bxa) * cf
       f3 = - alf * csq
       rm(1,3) = f1
-      rm(2,3) = f1 * (vxa + cs)
+      rm(2,3) = f1 * lm(3)
       rm(3,3) = f1 * vya + f2 * x2
       rm(4,3) = f1 * vza + f2 * x3
       rm(6,3) = f3 * x2
       rm(7,3) = f3 * x3
 
       rm(1,6) = f1
-      rm(2,6) = f1 * (vxa - cs)
+      rm(2,6) = f1 * lm(6)
       rm(3,6) = f1 * vya - f2 * x2
       rm(4,6) = f1 * vza - f2 * x3
       rm(6,6) = rm(6,3)
       rm(7,6) = rm(7,3)
 
+      f1 = sign(1.0d+00, vxa)
+      rm(8,4) =   f1
+      rm(8,5) = - f1
+
       fi(idn,i) = dnl * vxa
       fi(imx,i) = fi(idn,i) * vxa - bxa * bxa + pta
       fi(imy,i) = fi(idn,i) * vya - bxa * bya
       fi(imz,i) = fi(idn,i) * vza - bxa * bza
-      fi(ibx,i) = ch  * bpa
+      fi(ibx,i) = cglm * bpa
       fi(iby,i) = vxa * bya - bxa * vya
       fi(ibz,i) = vxa * bza - bxa * vza
-      fi(ibp,i) = ch  * bxa
+      fi(ibp,i) = cglm * bxa
 
       fi(:,i) = fi(:,i) - 5.0d-01 * matmul(rm, (abs(lm) * tm) * matmul(v, rm))