--- /dev/null
+
+c---------------------------------------------------------------------
+c---------------------------------------------------------------------
+
+ subroutine exact_rhs
+
+c---------------------------------------------------------------------
+c---------------------------------------------------------------------
+
+c---------------------------------------------------------------------
+c compute the right hand side based on exact solution
+c---------------------------------------------------------------------
+
+ include 'header.h'
+
+ double precision dtemp(5), xi, eta, zeta, dtpp
+ integer c, m, i, j, k, ip1, im1, jp1,
+ > jm1, km1, kp1
+
+
+c---------------------------------------------------------------------
+c loop over all cells owned by this node
+c---------------------------------------------------------------------
+ do c = 1, ncells
+
+c---------------------------------------------------------------------
+c initialize
+c---------------------------------------------------------------------
+ do k= 0, cell_size(3,c)-1
+ do j = 0, cell_size(2,c)-1
+ do i = 0, cell_size(1,c)-1
+ do m = 1, 5
+ forcing(m,i,j,k,c) = 0.0d0
+ enddo
+ enddo
+ enddo
+ enddo
+
+c---------------------------------------------------------------------
+c xi-direction flux differences
+c---------------------------------------------------------------------
+ do k = start(3,c), cell_size(3,c)-end(3,c)-1
+ zeta = dble(k+cell_low(3,c)) * dnzm1
+ do j = start(2,c), cell_size(2,c)-end(2,c)-1
+ eta = dble(j+cell_low(2,c)) * dnym1
+
+ do i=-2*(1-start(1,c)), cell_size(1,c)+1-2*end(1,c)
+ xi = dble(i+cell_low(1,c)) * dnxm1
+
+ call exact_solution(xi, eta, zeta, dtemp)
+ do m = 1, 5
+ ue(i,m) = dtemp(m)
+ enddo
+
+ dtpp = 1.0d0 / dtemp(1)
+
+ do m = 2, 5
+ buf(i,m) = dtpp * dtemp(m)
+ enddo
+
+ cuf(i) = buf(i,2) * buf(i,2)
+ buf(i,1) = cuf(i) + buf(i,3) * buf(i,3) +
+ > buf(i,4) * buf(i,4)
+ q(i) = 0.5d0*(buf(i,2)*ue(i,2) + buf(i,3)*ue(i,3) +
+ > buf(i,4)*ue(i,4))
+
+ enddo
+
+ do i = start(1,c), cell_size(1,c)-end(1,c)-1
+ im1 = i-1
+ ip1 = i+1
+
+ forcing(1,i,j,k,c) = forcing(1,i,j,k,c) -
+ > tx2*( ue(ip1,2)-ue(im1,2) )+
+ > dx1tx1*(ue(ip1,1)-2.0d0*ue(i,1)+ue(im1,1))
+
+ forcing(2,i,j,k,c) = forcing(2,i,j,k,c) - tx2 * (
+ > (ue(ip1,2)*buf(ip1,2)+c2*(ue(ip1,5)-q(ip1)))-
+ > (ue(im1,2)*buf(im1,2)+c2*(ue(im1,5)-q(im1))))+
+ > xxcon1*(buf(ip1,2)-2.0d0*buf(i,2)+buf(im1,2))+
+ > dx2tx1*( ue(ip1,2)-2.0d0* ue(i,2)+ue(im1,2))
+
+ forcing(3,i,j,k,c) = forcing(3,i,j,k,c) - tx2 * (
+ > ue(ip1,3)*buf(ip1,2)-ue(im1,3)*buf(im1,2))+
+ > xxcon2*(buf(ip1,3)-2.0d0*buf(i,3)+buf(im1,3))+
+ > dx3tx1*( ue(ip1,3)-2.0d0*ue(i,3) +ue(im1,3))
+
+ forcing(4,i,j,k,c) = forcing(4,i,j,k,c) - tx2*(
+ > ue(ip1,4)*buf(ip1,2)-ue(im1,4)*buf(im1,2))+
+ > xxcon2*(buf(ip1,4)-2.0d0*buf(i,4)+buf(im1,4))+
+ > dx4tx1*( ue(ip1,4)-2.0d0* ue(i,4)+ ue(im1,4))
+
+ forcing(5,i,j,k,c) = forcing(5,i,j,k,c) - tx2*(
+ > buf(ip1,2)*(c1*ue(ip1,5)-c2*q(ip1))-
+ > buf(im1,2)*(c1*ue(im1,5)-c2*q(im1)))+
+ > 0.5d0*xxcon3*(buf(ip1,1)-2.0d0*buf(i,1)+
+ > buf(im1,1))+
+ > xxcon4*(cuf(ip1)-2.0d0*cuf(i)+cuf(im1))+
+ > xxcon5*(buf(ip1,5)-2.0d0*buf(i,5)+buf(im1,5))+
+ > dx5tx1*( ue(ip1,5)-2.0d0* ue(i,5)+ ue(im1,5))
+ enddo
+
+c---------------------------------------------------------------------
+c Fourth-order dissipation
+c---------------------------------------------------------------------
+ if (start(1,c) .gt. 0) then
+ do m = 1, 5
+ i = 1
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp *
+ > (5.0d0*ue(i,m) - 4.0d0*ue(i+1,m) +ue(i+2,m))
+ i = 2
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp *
+ > (-4.0d0*ue(i-1,m) + 6.0d0*ue(i,m) -
+ > 4.0d0*ue(i+1,m) + ue(i+2,m))
+ enddo
+ endif
+
+ do i = start(1,c)*3, cell_size(1,c)-3*end(1,c)-1
+ do m = 1, 5
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp*
+ > (ue(i-2,m) - 4.0d0*ue(i-1,m) +
+ > 6.0d0*ue(i,m) - 4.0d0*ue(i+1,m) + ue(i+2,m))
+ enddo
+ enddo
+
+ if (end(1,c) .gt. 0) then
+ do m = 1, 5
+ i = cell_size(1,c)-3
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp *
+ > (ue(i-2,m) - 4.0d0*ue(i-1,m) +
+ > 6.0d0*ue(i,m) - 4.0d0*ue(i+1,m))
+ i = cell_size(1,c)-2
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp *
+ > (ue(i-2,m) - 4.0d0*ue(i-1,m) + 5.0d0*ue(i,m))
+ enddo
+ endif
+
+ enddo
+ enddo
+
+c---------------------------------------------------------------------
+c eta-direction flux differences
+c---------------------------------------------------------------------
+ do k = start(3,c), cell_size(3,c)-end(3,c)-1
+ zeta = dble(k+cell_low(3,c)) * dnzm1
+ do i=start(1,c), cell_size(1,c)-end(1,c)-1
+ xi = dble(i+cell_low(1,c)) * dnxm1
+
+ do j=-2*(1-start(2,c)), cell_size(2,c)+1-2*end(2,c)
+ eta = dble(j+cell_low(2,c)) * dnym1
+
+ call exact_solution(xi, eta, zeta, dtemp)
+ do m = 1, 5
+ ue(j,m) = dtemp(m)
+ enddo
+
+ dtpp = 1.0d0/dtemp(1)
+
+ do m = 2, 5
+ buf(j,m) = dtpp * dtemp(m)
+ enddo
+
+ cuf(j) = buf(j,3) * buf(j,3)
+ buf(j,1) = cuf(j) + buf(j,2) * buf(j,2) +
+ > buf(j,4) * buf(j,4)
+ q(j) = 0.5d0*(buf(j,2)*ue(j,2) + buf(j,3)*ue(j,3) +
+ > buf(j,4)*ue(j,4))
+ enddo
+
+ do j = start(2,c), cell_size(2,c)-end(2,c)-1
+ jm1 = j-1
+ jp1 = j+1
+
+ forcing(1,i,j,k,c) = forcing(1,i,j,k,c) -
+ > ty2*( ue(jp1,3)-ue(jm1,3) )+
+ > dy1ty1*(ue(jp1,1)-2.0d0*ue(j,1)+ue(jm1,1))
+
+ forcing(2,i,j,k,c) = forcing(2,i,j,k,c) - ty2*(
+ > ue(jp1,2)*buf(jp1,3)-ue(jm1,2)*buf(jm1,3))+
+ > yycon2*(buf(jp1,2)-2.0d0*buf(j,2)+buf(jm1,2))+
+ > dy2ty1*( ue(jp1,2)-2.0* ue(j,2)+ ue(jm1,2))
+
+ forcing(3,i,j,k,c) = forcing(3,i,j,k,c) - ty2*(
+ > (ue(jp1,3)*buf(jp1,3)+c2*(ue(jp1,5)-q(jp1)))-
+ > (ue(jm1,3)*buf(jm1,3)+c2*(ue(jm1,5)-q(jm1))))+
+ > yycon1*(buf(jp1,3)-2.0d0*buf(j,3)+buf(jm1,3))+
+ > dy3ty1*( ue(jp1,3)-2.0d0*ue(j,3) +ue(jm1,3))
+
+ forcing(4,i,j,k,c) = forcing(4,i,j,k,c) - ty2*(
+ > ue(jp1,4)*buf(jp1,3)-ue(jm1,4)*buf(jm1,3))+
+ > yycon2*(buf(jp1,4)-2.0d0*buf(j,4)+buf(jm1,4))+
+ > dy4ty1*( ue(jp1,4)-2.0d0*ue(j,4)+ ue(jm1,4))
+
+ forcing(5,i,j,k,c) = forcing(5,i,j,k,c) - ty2*(
+ > buf(jp1,3)*(c1*ue(jp1,5)-c2*q(jp1))-
+ > buf(jm1,3)*(c1*ue(jm1,5)-c2*q(jm1)))+
+ > 0.5d0*yycon3*(buf(jp1,1)-2.0d0*buf(j,1)+
+ > buf(jm1,1))+
+ > yycon4*(cuf(jp1)-2.0d0*cuf(j)+cuf(jm1))+
+ > yycon5*(buf(jp1,5)-2.0d0*buf(j,5)+buf(jm1,5))+
+ > dy5ty1*(ue(jp1,5)-2.0d0*ue(j,5)+ue(jm1,5))
+ enddo
+
+c---------------------------------------------------------------------
+c Fourth-order dissipation
+c---------------------------------------------------------------------
+ if (start(2,c) .gt. 0) then
+ do m = 1, 5
+ j = 1
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp *
+ > (5.0d0*ue(j,m) - 4.0d0*ue(j+1,m) +ue(j+2,m))
+ j = 2
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp *
+ > (-4.0d0*ue(j-1,m) + 6.0d0*ue(j,m) -
+ > 4.0d0*ue(j+1,m) + ue(j+2,m))
+ enddo
+ endif
+
+ do j = start(2,c)*3, cell_size(2,c)-3*end(2,c)-1
+ do m = 1, 5
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp*
+ > (ue(j-2,m) - 4.0d0*ue(j-1,m) +
+ > 6.0d0*ue(j,m) - 4.0d0*ue(j+1,m) + ue(j+2,m))
+ enddo
+ enddo
+
+ if (end(2,c) .gt. 0) then
+ do m = 1, 5
+ j = cell_size(2,c)-3
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp *
+ > (ue(j-2,m) - 4.0d0*ue(j-1,m) +
+ > 6.0d0*ue(j,m) - 4.0d0*ue(j+1,m))
+ j = cell_size(2,c)-2
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp *
+ > (ue(j-2,m) - 4.0d0*ue(j-1,m) + 5.0d0*ue(j,m))
+
+ enddo
+ endif
+
+ enddo
+ enddo
+
+c---------------------------------------------------------------------
+c zeta-direction flux differences
+c---------------------------------------------------------------------
+ do j=start(2,c), cell_size(2,c)-end(2,c)-1
+ eta = dble(j+cell_low(2,c)) * dnym1
+ do i = start(1,c), cell_size(1,c)-end(1,c)-1
+ xi = dble(i+cell_low(1,c)) * dnxm1
+
+ do k=-2*(1-start(3,c)), cell_size(3,c)+1-2*end(3,c)
+ zeta = dble(k+cell_low(3,c)) * dnzm1
+
+ call exact_solution(xi, eta, zeta, dtemp)
+ do m = 1, 5
+ ue(k,m) = dtemp(m)
+ enddo
+
+ dtpp = 1.0d0/dtemp(1)
+
+ do m = 2, 5
+ buf(k,m) = dtpp * dtemp(m)
+ enddo
+
+ cuf(k) = buf(k,4) * buf(k,4)
+ buf(k,1) = cuf(k) + buf(k,2) * buf(k,2) +
+ > buf(k,3) * buf(k,3)
+ q(k) = 0.5d0*(buf(k,2)*ue(k,2) + buf(k,3)*ue(k,3) +
+ > buf(k,4)*ue(k,4))
+ enddo
+
+ do k=start(3,c), cell_size(3,c)-end(3,c)-1
+ km1 = k-1
+ kp1 = k+1
+
+ forcing(1,i,j,k,c) = forcing(1,i,j,k,c) -
+ > tz2*( ue(kp1,4)-ue(km1,4) )+
+ > dz1tz1*(ue(kp1,1)-2.0d0*ue(k,1)+ue(km1,1))
+
+ forcing(2,i,j,k,c) = forcing(2,i,j,k,c) - tz2 * (
+ > ue(kp1,2)*buf(kp1,4)-ue(km1,2)*buf(km1,4))+
+ > zzcon2*(buf(kp1,2)-2.0d0*buf(k,2)+buf(km1,2))+
+ > dz2tz1*( ue(kp1,2)-2.0d0* ue(k,2)+ ue(km1,2))
+
+ forcing(3,i,j,k,c) = forcing(3,i,j,k,c) - tz2 * (
+ > ue(kp1,3)*buf(kp1,4)-ue(km1,3)*buf(km1,4))+
+ > zzcon2*(buf(kp1,3)-2.0d0*buf(k,3)+buf(km1,3))+
+ > dz3tz1*(ue(kp1,3)-2.0d0*ue(k,3)+ue(km1,3))
+
+ forcing(4,i,j,k,c) = forcing(4,i,j,k,c) - tz2 * (
+ > (ue(kp1,4)*buf(kp1,4)+c2*(ue(kp1,5)-q(kp1)))-
+ > (ue(km1,4)*buf(km1,4)+c2*(ue(km1,5)-q(km1))))+
+ > zzcon1*(buf(kp1,4)-2.0d0*buf(k,4)+buf(km1,4))+
+ > dz4tz1*( ue(kp1,4)-2.0d0*ue(k,4) +ue(km1,4))
+
+ forcing(5,i,j,k,c) = forcing(5,i,j,k,c) - tz2 * (
+ > buf(kp1,4)*(c1*ue(kp1,5)-c2*q(kp1))-
+ > buf(km1,4)*(c1*ue(km1,5)-c2*q(km1)))+
+ > 0.5d0*zzcon3*(buf(kp1,1)-2.0d0*buf(k,1)
+ > +buf(km1,1))+
+ > zzcon4*(cuf(kp1)-2.0d0*cuf(k)+cuf(km1))+
+ > zzcon5*(buf(kp1,5)-2.0d0*buf(k,5)+buf(km1,5))+
+ > dz5tz1*( ue(kp1,5)-2.0d0*ue(k,5)+ ue(km1,5))
+ enddo
+
+c---------------------------------------------------------------------
+c Fourth-order dissipation
+c---------------------------------------------------------------------
+ if (start(3,c) .gt. 0) then
+ do m = 1, 5
+ k = 1
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp *
+ > (5.0d0*ue(k,m) - 4.0d0*ue(k+1,m) +ue(k+2,m))
+ k = 2
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp *
+ > (-4.0d0*ue(k-1,m) + 6.0d0*ue(k,m) -
+ > 4.0d0*ue(k+1,m) + ue(k+2,m))
+ enddo
+ endif
+
+ do k = start(3,c)*3, cell_size(3,c)-3*end(3,c)-1
+ do m = 1, 5
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp*
+ > (ue(k-2,m) - 4.0d0*ue(k-1,m) +
+ > 6.0d0*ue(k,m) - 4.0d0*ue(k+1,m) + ue(k+2,m))
+ enddo
+ enddo
+
+ if (end(3,c) .gt. 0) then
+ do m = 1, 5
+ k = cell_size(3,c)-3
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp *
+ > (ue(k-2,m) - 4.0d0*ue(k-1,m) +
+ > 6.0d0*ue(k,m) - 4.0d0*ue(k+1,m))
+ k = cell_size(3,c)-2
+ forcing(m,i,j,k,c) = forcing(m,i,j,k,c) - dssp *
+ > (ue(k-2,m) - 4.0d0*ue(k-1,m) + 5.0d0*ue(k,m))
+ enddo
+ endif
+
+ enddo
+ enddo
+
+c---------------------------------------------------------------------
+c now change the sign of the forcing function,
+c---------------------------------------------------------------------
+ do k = start(3,c), cell_size(3,c)-end(3,c)-1
+ do j = start(2,c), cell_size(2,c)-end(2,c)-1
+ do i = start(1,c), cell_size(1,c)-end(1,c)-1
+ do m = 1, 5
+ forcing(m,i,j,k,c) = -1.d0 * forcing(m,i,j,k,c)
+ enddo
+ enddo
+ enddo
+ enddo
+
+ enddo
+
+ return
+ end
