bool
AmesosBTFGlobal_LinearProblem::
rvs()
{
// cout << "AmesosBTFGlobal_LinearProblem: NewLHS_" << endl;
// cout << *NewLHS_ << endl;
OldLHS_->Import( *NewLHS_, *Importer2_, Insert );
int numrhs = OldLHS_->NumVectors();
std::vector<double> actual_resids( numrhs ), rhs_norm( numrhs );
Epetra_MultiVector resid( OldLHS_->Map(), numrhs );
OldMatrix_->Apply( *OldLHS_, resid );
resid.Update( -1.0, *OldRHS_, 1.0 );
resid.Norm2( &actual_resids[0] );
OldRHS_->Norm2( &rhs_norm[0] );
if (OldLHS_->Comm().MyPID() == 0 ) {
for (int i=0; i<numrhs; i++ ) {
std::cout << "Problem " << i << " (in AmesosBTFGlobal): \t" << actual_resids[i]/rhs_norm[i] << std::endl;
}
}
//cout << "AmesosBTFGlobal_LinearProblem: OldLHS_" << endl;
//cout << *OldLHS_ << endl;
return true;
}
//---------------------------------------------------------------------
// verification routine
//---------------------------------------------------------------------
void verify(int no_time_steps, char *Class, logical *verified)
{
double xcrref[5], xceref[5], xcrdif[5], xcedif[5];
double epsilon, xce[5], xcr[5], dtref = 0.0;
int m;
//---------------------------------------------------------------------
// tolerance level
//---------------------------------------------------------------------
epsilon = 1.0e-08;
//---------------------------------------------------------------------
// compute the error norm and the residual norm, and exit if not printing
//---------------------------------------------------------------------
printf("[%s:%d]\n", __FILE__, __LINE__);
error_norm(xce);
printf("[%s:%d]\n", __FILE__, __LINE__);
copy_faces();
printf("[%s:%d]\n", __FILE__, __LINE__);
rhs_norm(xcr);
printf("[%s:%d]\n", __FILE__, __LINE__);
for (m = 0; m < 5; m++) {
xcr[m] = xcr[m] / dt;
}
*Class = 'U';
*verified = true;
for (m = 0; m < 5; m++) {
xcrref[m] = 1.0;
xceref[m] = 1.0;
}
//---------------------------------------------------------------------
// reference data for 12X12X12 grids after 100 time steps,
// with DT = 1.50e-02
//---------------------------------------------------------------------
if ( (grid_points[0] == 12) && (grid_points[1] == 12) &&
(grid_points[2] == 12) && (no_time_steps == 100) ) {
*Class = 'S';
dtref = 1.5e-2;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref[0] = 2.7470315451339479e-02;
xcrref[1] = 1.0360746705285417e-02;
xcrref[2] = 1.6235745065095532e-02;
xcrref[3] = 1.5840557224455615e-02;
xcrref[4] = 3.4849040609362460e-02;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref[0] = 2.7289258557377227e-05;
xceref[1] = 1.0364446640837285e-05;
xceref[2] = 1.6154798287166471e-05;
xceref[3] = 1.5750704994480102e-05;
xceref[4] = 3.4177666183390531e-05;
//---------------------------------------------------------------------
// reference data for 36X36X36 grids after 400 time steps,
// with DT = 1.5e-03
//---------------------------------------------------------------------
} else if ( (grid_points[0] == 36) && (grid_points[1] == 36) &&
(grid_points[2] == 36) && (no_time_steps == 400) ) {
*Class = 'W';
dtref = 1.5e-3;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref[0] = 0.1893253733584e-02;
xcrref[1] = 0.1717075447775e-03;
xcrref[2] = 0.2778153350936e-03;
xcrref[3] = 0.2887475409984e-03;
xcrref[4] = 0.3143611161242e-02;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref[0] = 0.7542088599534e-04;
xceref[1] = 0.6512852253086e-05;
xceref[2] = 0.1049092285688e-04;
xceref[3] = 0.1128838671535e-04;
xceref[4] = 0.1212845639773e-03;
//---------------------------------------------------------------------
// reference data for 64X64X64 grids after 400 time steps,
// with DT = 1.5e-03
//---------------------------------------------------------------------
} else if ( (grid_points[0] == 64) && (grid_points[1] == 64) &&
(grid_points[2] == 64) && (no_time_steps == 400) ) {
*Class = 'A';
dtref = 1.5e-3;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref[0] = 2.4799822399300195;
xcrref[1] = 1.1276337964368832;
xcrref[2] = 1.5028977888770491;
xcrref[3] = 1.4217816211695179;
xcrref[4] = 2.1292113035138280;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref[0] = 1.0900140297820550e-04;
xceref[1] = 3.7343951769282091e-05;
xceref[2] = 5.0092785406541633e-05;
xceref[3] = 4.7671093939528255e-05;
xceref[4] = 1.3621613399213001e-04;
//---------------------------------------------------------------------
// reference data for 102X102X102 grids after 400 time steps,
// with DT = 1.0e-03
//---------------------------------------------------------------------
} else if ( (grid_points[0] == 102) && (grid_points[1] == 102) &&
(grid_points[2] == 102) && (no_time_steps == 400) ) {
*Class = 'B';
dtref = 1.0e-3;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref[0] = 0.6903293579998e+02;
xcrref[1] = 0.3095134488084e+02;
xcrref[2] = 0.4103336647017e+02;
xcrref[3] = 0.3864769009604e+02;
xcrref[4] = 0.5643482272596e+02;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref[0] = 0.9810006190188e-02;
xceref[1] = 0.1022827905670e-02;
xceref[2] = 0.1720597911692e-02;
xceref[3] = 0.1694479428231e-02;
xceref[4] = 0.1847456263981e-01;
//---------------------------------------------------------------------
// reference data for 162X162X162 grids after 400 time steps,
// with DT = 0.67e-03
//---------------------------------------------------------------------
} else if ( (grid_points[0] == 162) && (grid_points[1] == 162) &&
(grid_points[2] == 162) && (no_time_steps == 400) ) {
*Class = 'C';
dtref = 0.67e-3;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref[0] = 0.5881691581829e+03;
xcrref[1] = 0.2454417603569e+03;
xcrref[2] = 0.3293829191851e+03;
xcrref[3] = 0.3081924971891e+03;
xcrref[4] = 0.4597223799176e+03;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref[0] = 0.2598120500183e+00;
xceref[1] = 0.2590888922315e-01;
xceref[2] = 0.5132886416320e-01;
xceref[3] = 0.4806073419454e-01;
xceref[4] = 0.5483377491301e+00;
//---------------------------------------------------------------------
// reference data for 408X408X408 grids after 500 time steps,
// with DT = 0.3e-03
//---------------------------------------------------------------------
} else if ( (grid_points[0] == 408) && (grid_points[1] == 408) &&
(grid_points[2] == 408) && (no_time_steps == 500) ) {
*Class = 'D';
dtref = 0.30e-3;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref[0] = 0.1044696216887e+05;
xcrref[1] = 0.3204427762578e+04;
xcrref[2] = 0.4648680733032e+04;
xcrref[3] = 0.4238923283697e+04;
xcrref[4] = 0.7588412036136e+04;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref[0] = 0.5089471423669e+01;
xceref[1] = 0.5323514855894e+00;
xceref[2] = 0.1187051008971e+01;
xceref[3] = 0.1083734951938e+01;
xceref[4] = 0.1164108338568e+02;
//---------------------------------------------------------------------
// reference data for 1020X1020X1020 grids after 500 time steps,
// with DT = 0.1e-03
//---------------------------------------------------------------------
} else if ( (grid_points[0] == 1020) && (grid_points[1] == 1020) &&
(grid_points[2] == 1020) && (no_time_steps == 500) ) {
*Class = 'E';
dtref = 0.10e-3;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref[0] = 0.6255387422609e+05;
xcrref[1] = 0.1495317020012e+05;
xcrref[2] = 0.2347595750586e+05;
xcrref[3] = 0.2091099783534e+05;
xcrref[4] = 0.4770412841218e+05;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref[0] = 0.6742735164909e+02;
xceref[1] = 0.5390656036938e+01;
xceref[2] = 0.1680647196477e+02;
xceref[3] = 0.1536963126457e+02;
xceref[4] = 0.1575330146156e+03;
} else {
*verified = false;
}
//---------------------------------------------------------------------
// verification test for residuals if gridsize is one of
// the defined grid sizes above (class .ne. 'U')
//---------------------------------------------------------------------
//---------------------------------------------------------------------
// Compute the difference of solution values and the known reference values.
//---------------------------------------------------------------------
for (m = 0; m < 5; m++) {
xcrdif[m] = fabs((xcr[m]-xcrref[m])/xcrref[m]);
xcedif[m] = fabs((xce[m]-xceref[m])/xceref[m]);
}
//---------------------------------------------------------------------
// Output the comparison of computed results to known cases.
//---------------------------------------------------------------------
if (*Class != 'U') {
printf(" Verification being performed for class %c\n", *Class);
printf(" accuracy setting for epsilon = %20.13E\n", epsilon);
*verified = (fabs(dt-dtref) <= epsilon);
if (!(*verified)) {
*Class = 'U';
printf(" DT does not match the reference value of %15.8E\n", dtref);
}
} else {
printf(" Unknown class\n");
}
if (*Class != 'U') {
printf(" Comparison of RMS-norms of residual\n");
} else {
printf(" RMS-norms of residual\n");
}
for (m = 0; m < 5; m++) {
if (*Class == 'U') {
printf(" %2d%20.13E\n", m+1, xcr[m]);
} else if (xcrdif[m] <= epsilon) {
printf(" %2d%20.13E%20.13E%20.13E\n",
m+1, xcr[m], xcrref[m], xcrdif[m]);
} else {
*verified = false;
printf(" FAILURE: %2d%20.13E%20.13E%20.13E\n",
m+1, xcr[m], xcrref[m], xcrdif[m]);
}
}
if (*Class != 'U') {
printf(" Comparison of RMS-norms of solution error\n");
} else {
printf(" RMS-norms of solution error\n");
}
for (m = 0; m < 5; m++) {
if (*Class == 'U') {
printf(" %2d%20.13E\n", m+1, xce[m]);
} else if (xcedif[m] <= epsilon) {
printf(" %2d%20.13E%20.13E%20.13E\n",
m+1, xce[m], xceref[m], xcedif[m]);
} else {
*verified = false;
printf(" FAILURE: %2d%20.13E%20.13E%20.13E\n",
m+1, xce[m], xceref[m], xcedif[m]);
}
}
if (*Class == 'U') {
printf(" No reference values provided\n");
printf(" No verification performed\n");
} else if (*verified) {
printf(" Verification Successful\n");
} else {
printf(" Verification failed\n");
}
}
//---------------------------------------------------------------------
// verification routine
//---------------------------------------------------------------------
void verify(int no_time_steps, char *Class, logical *verified)
{
double xcrref[5], xceref[5], xcrdif[5], xcedif[5];
double epsilon, xce[5], xcr[5], dtref = 0.0;
int m;
//---------------------------------------------------------------------
// tolerance level
//---------------------------------------------------------------------
epsilon = 1.0e-08;
//---------------------------------------------------------------------
// compute the error norm and the residual norm, and exit if not printing
//---------------------------------------------------------------------
error_norm(xce);
compute_rhs();
rhs_norm(xcr);
for (m = 0; m < 5; m++) {
xcr[m] = xcr[m] / dt;
}
*Class = 'U';
*verified = true;
for (m = 0; m < 5; m++) {
xcrref[m] = 1.0;
xceref[m] = 1.0;
}
//---------------------------------------------------------------------
// reference data for 12X12X12 grids after 60 time steps, with DT = 1.0e-02
//---------------------------------------------------------------------
if ( (grid_points[0] == 12) && (grid_points[1] == 12) &&
(grid_points[2] == 12) && (no_time_steps == 60)) {
*Class = 'S';
dtref = 1.0e-2;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref[0] = 1.7034283709541311e-01;
xcrref[1] = 1.2975252070034097e-02;
xcrref[2] = 3.2527926989486055e-02;
xcrref[3] = 2.6436421275166801e-02;
xcrref[4] = 1.9211784131744430e-01;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref[0] = 4.9976913345811579e-04;
xceref[1] = 4.5195666782961927e-05;
xceref[2] = 7.3973765172921357e-05;
xceref[3] = 7.3821238632439731e-05;
xceref[4] = 8.9269630987491446e-04;
//---------------------------------------------------------------------
// reference data for 24X24X24 grids after 200 time steps,
// with DT = 0.8e-3
//---------------------------------------------------------------------
} else if ( (grid_points[0] == 24) && (grid_points[1] == 24) &&
(grid_points[2] == 24) && (no_time_steps == 200) ) {
*Class = 'W';
dtref = 0.8e-3;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref[0] = 0.1125590409344e+03;
xcrref[1] = 0.1180007595731e+02;
xcrref[2] = 0.2710329767846e+02;
xcrref[3] = 0.2469174937669e+02;
xcrref[4] = 0.2638427874317e+03;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref[0] = 0.4419655736008e+01;
xceref[1] = 0.4638531260002e+00;
xceref[2] = 0.1011551749967e+01;
xceref[3] = 0.9235878729944e+00;
xceref[4] = 0.1018045837718e+02;
//---------------------------------------------------------------------
// reference data for 64X64X64 grids after 200 time steps,
// with DT = 0.8e-3
//---------------------------------------------------------------------
} else if ( (grid_points[0] == 64) && (grid_points[1] == 64) &&
(grid_points[2] == 64) && (no_time_steps == 200) ) {
*Class = 'A';
dtref = 0.8e-3;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref[0] = 1.0806346714637264e+02;
xcrref[1] = 1.1319730901220813e+01;
xcrref[2] = 2.5974354511582465e+01;
xcrref[3] = 2.3665622544678910e+01;
xcrref[4] = 2.5278963211748344e+02;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref[0] = 4.2348416040525025e+00;
xceref[1] = 4.4390282496995698e-01;
xceref[2] = 9.6692480136345650e-01;
xceref[3] = 8.8302063039765474e-01;
xceref[4] = 9.7379901770829278e+00;
//---------------------------------------------------------------------
// reference data for 102X102X102 grids after 200 time steps,
// with DT = 3.0e-04
//---------------------------------------------------------------------
} else if ( (grid_points[0] == 102) && (grid_points[1] == 102) &&
(grid_points[2] == 102) && (no_time_steps == 200) ) {
*Class = 'B';
dtref = 3.0e-4;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref[0] = 1.4233597229287254e+03;
xcrref[1] = 9.9330522590150238e+01;
xcrref[2] = 3.5646025644535285e+02;
xcrref[3] = 3.2485447959084092e+02;
xcrref[4] = 3.2707541254659363e+03;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref[0] = 5.2969847140936856e+01;
xceref[1] = 4.4632896115670668e+00;
xceref[2] = 1.3122573342210174e+01;
xceref[3] = 1.2006925323559144e+01;
xceref[4] = 1.2459576151035986e+02;
//---------------------------------------------------------------------
// reference data for 162X162X162 grids after 200 time steps,
// with DT = 1.0e-04
//---------------------------------------------------------------------
} else if ( (grid_points[0] == 162) && (grid_points[1] == 162) &&
(grid_points[2] == 162) && (no_time_steps == 200) ) {
*Class = 'C';
dtref = 1.0e-4;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref[0] = 0.62398116551764615e+04;
xcrref[1] = 0.50793239190423964e+03;
xcrref[2] = 0.15423530093013596e+04;
xcrref[3] = 0.13302387929291190e+04;
xcrref[4] = 0.11604087428436455e+05;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref[0] = 0.16462008369091265e+03;
xceref[1] = 0.11497107903824313e+02;
xceref[2] = 0.41207446207461508e+02;
xceref[3] = 0.37087651059694167e+02;
xceref[4] = 0.36211053051841265e+03;
//---------------------------------------------------------------------
// reference data for 408x408x408 grids after 250 time steps,
// with DT = 0.2e-04
//---------------------------------------------------------------------
} else if ( (grid_points[0] == 408) && (grid_points[1] == 408) &&
(grid_points[2] == 408) && (no_time_steps == 250) ) {
*Class = 'D';
dtref = 0.2e-4;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref[0] = 0.2533188551738e+05;
xcrref[1] = 0.2346393716980e+04;
xcrref[2] = 0.6294554366904e+04;
xcrref[3] = 0.5352565376030e+04;
xcrref[4] = 0.3905864038618e+05;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref[0] = 0.3100009377557e+03;
xceref[1] = 0.2424086324913e+02;
xceref[2] = 0.7782212022645e+02;
xceref[3] = 0.6835623860116e+02;
xceref[4] = 0.6065737200368e+03;
//---------------------------------------------------------------------
// reference data for 1020x1020x1020 grids after 250 time steps,
// with DT = 0.4e-05
//---------------------------------------------------------------------
} else if ( (grid_points[0] == 1020) && (grid_points[1] == 1020) &&
(grid_points[2] == 1020) && (no_time_steps == 250) ) {
*Class = 'E';
dtref = 0.4e-5;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref[0] = 0.9795372484517e+05;
xcrref[1] = 0.9739814511521e+04;
xcrref[2] = 0.2467606342965e+05;
xcrref[3] = 0.2092419572860e+05;
xcrref[4] = 0.1392138856939e+06;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref[0] = 0.4327562208414e+03;
xceref[1] = 0.3699051964887e+02;
xceref[2] = 0.1089845040954e+03;
xceref[3] = 0.9462517622043e+02;
xceref[4] = 0.7765512765309e+03;
} else {
*verified = false;
}
//---------------------------------------------------------------------
// verification test for residuals if gridsize is one of
// the defined grid sizes above (*Class != 'U')
//---------------------------------------------------------------------
//---------------------------------------------------------------------
// Compute the difference of solution values and the known reference values.
//---------------------------------------------------------------------
for (m = 0; m < 5; m++) {
xcrdif[m] = fabs((xcr[m]-xcrref[m])/xcrref[m]);
xcedif[m] = fabs((xce[m]-xceref[m])/xceref[m]);
}
//---------------------------------------------------------------------
// Output the comparison of computed results to known cases.
//---------------------------------------------------------------------
if (*Class != 'U') {
printf(" Verification being performed for class %c\n", *Class);
printf(" accuracy setting for epsilon = %20.13E\n", epsilon);
*verified = (fabs(dt-dtref) <= epsilon);
if (!(*verified)) {
*Class = 'U';
printf(" DT does not match the reference value of %15.8E\n", dtref);
}
} else {
printf(" Unknown class\n");
}
if (*Class != 'U') {
printf(" Comparison of RMS-norms of residual\n");
} else {
printf(" RMS-norms of residual\n");
}
for (m = 0; m < 5; m++) {
if (*Class == 'U') {
printf(" %2d%20.13E\n", m+1, xcr[m]);
} else if (xcrdif[m] <= epsilon) {
printf(" %2d%20.13E%20.13E%20.13E\n",
m+1, xcr[m], xcrref[m], xcrdif[m]);
} else {
*verified = false;
printf(" FAILURE: %2d%20.13E%20.13E%20.13E\n",
m+1, xcr[m], xcrref[m], xcrdif[m]);
}
}
if (*Class != 'U') {
printf(" Comparison of RMS-norms of solution error\n");
} else {
printf(" RMS-norms of solution error\n");
}
for (m = 0; m < 5; m++) {
if (*Class == 'U') {
printf(" %2d%20.13E\n", m+1, xce[m]);
} else if (xcedif[m] <= epsilon) {
printf(" %2d%20.13E%20.13E%20.13E\n",
m+1, xce[m], xceref[m], xcedif[m]);
} else {
*verified = false;
printf(" FAILURE: %2d%20.13E%20.13E%20.13E\n",
m+1, xce[m], xceref[m], xcedif[m]);
}
}
if (*Class == 'U') {
printf(" No reference values provided\n");
printf(" No verification performed\n");
} else if (*verified) {
printf(" Verification Successful\n");
} else {
printf(" Verification failed\n");
}
}
void verify(int no_time_steps, char *class_r, int *verified_r) {
//---------------------------------------------------------------------
//---------------------------------------------------------------------
//---------------------------------------------------------------------
// verification routine
//---------------------------------------------------------------------
double xcrref[5],xceref[5],xcrdif[5],xcedif[5],
epsilon, xce[5], xcr[5], dtref;
int m;
char class;
int verified;
#define xcrref(m) xcrref[m-1]
#define xceref(m) xceref[m-1]
#define xcrdif(m) xcrdif[m-1]
#define xcedif(m) xcedif[m-1]
#define xce(m) xce[m-1]
#define xcr(m) xcr[m-1]
//---------------------------------------------------------------------
// tolerance level
//---------------------------------------------------------------------
epsilon = 1.0e-08;
verified = 1;
//---------------------------------------------------------------------
// compute the error norm and the residual norm, and exit if not printing
//---------------------------------------------------------------------
error_norm(xce);
copy_faces();
rhs_norm(xcr);
for (m = 1; m <= 5; m++) {
xcr(m) = xcr(m) / dt;
}
if (node != root) return;
class = 'U';
for (m = 1; m <= 5; m++) {
xcrref(m) = 1.0;
xceref(m) = 1.0;
}
//---------------------------------------------------------------------
// reference data for 12X12X12 grids after 60 time steps, with DT = 1.0e-02
//---------------------------------------------------------------------
if ( (grid_points(1) == 12 ) &&
(grid_points(2) == 12 ) &&
(grid_points(3) == 12 ) &&
(no_time_steps == 60 )) {
class = 'S';
dtref = 1.0e-2;
//---------------------------------------------------------------------
// Reference values of RMS-norms of residual.
//---------------------------------------------------------------------
xcrref(1) = 1.7034283709541311e-01;
xcrref(2) = 1.2975252070034097e-02;
xcrref(3) = 3.2527926989486055e-02;
xcrref(4) = 2.6436421275166801e-02;
xcrref(5) = 1.9211784131744430e-01;
//---------------------------------------------------------------------
// Reference values of RMS-norms of solution error.
//---------------------------------------------------------------------
xceref(1) = 4.9976913345811579e-04;
xceref(2) = 4.5195666782961927e-05;
xceref(3) = 7.3973765172921357e-05;
xceref(4) = 7.3821238632439731e-05;
xceref(5) = 8.9269630987491446e-04;
//---------------------------------------------------------------------
// reference data for 24X24X24 grids after 200 time steps, with DT = 0.8e-3
//---------------------------------------------------------------------
} else if ( (grid_points(1) == 24) &&
