2D Euler Equations - Isentropic Vortex Convection
Location: hypar/Examples/2D/NavierStokes2D/InviscidVortexConvection_SparseGrids (This directory contains all the input files needed to run this case.)
Governing equations: 2D Euler Equations (navierstokes2d.h - By default, NavierStokes2D::Re is set to -1 which makes the code skip the parabolic terms, i.e., the 2D Euler equations are solved.)
Reference: C.-W. Shu, "Essentially Non-oscillatory and Weighted Essentially Non-oscillatory Schemes for Hyperbolic Conservation Laws", ICASE Report 97-65, 1997
Domain: \(0 \le x,y \le 10\), "periodic" (_PERIODIC_) boundary conditions.
Initial solution: The freestream flow is given by
\begin{equation} \rho_\infty = 1,\ u_\infty = 0.1,\ v_\infty = 0,\ p_\infty = 1 \end{equation}
and a vortex is introduced, specified as
\begin{align} \rho &= \left[ 1 - \frac{\left(\gamma-1\right)b^2}{8\gamma\pi^2} e^{1-r^2} \right]^{\frac{1}{\gamma-1}},\ p = \rho^\gamma, \\ u &= u_\infty - \frac{b}{2\pi} e^{\frac{1}{2}\left(1-r^2\right)} \left(y-y_c\right),\ v = v_\infty + \frac{b}{2\pi} e^{\frac{1}{2}\left(1-r^2\right)} \left(x-x_c\right), \end{align}
where \(b=0.5\) is the vortex strength and \(r = \left[(x-x_c)^2 + (y-y_c)^2 \right]^{1/2}\) is the distance from the vortex center \(\left(x_c,y_c\right) = \left(5,5\right)\).
Numerical method:
- Spatial discretization (hyperbolic): 5th order upwind (Interp1PrimFifthOrderUpwind())
- Time integration: RK4 (TimeRK(), _RK_44_)
Input files required:
sparse_grids.inp
begin
log2_imin 3
interp_order 6
write_sg_solutions yes
write_sg_errors no
end
Note: The remaining files are the same as what would be required for a conventional (non-sparse-grids) simulation.
solver.inp
begin
ndims 2
nvars 4
size 256 256
ghost 3
n_iter 800
restart_iter 0
time_scheme rk
time_scheme_type 44
hyp_space_scheme upw5
hyp_flux_split no
hyp_interp_type components
par_space_type nonconservative-2stage
par_space_scheme 4
dt 0.025
conservation_check yes
screen_op_iter 10
file_op_iter 80
input_mode serial
ip_file_type binary
output_mode serial
op_file_format tecplot2d
op_overwrite no
model navierstokes2d
end
boundary.inp
4
periodic 0 1 0 0 0 10.0
periodic 0 -1 0 0 0 10.0
periodic 1 1 0 10.0 0 0
periodic 1 -1 0 10.0 0 0
physics.inp
begin
gamma 1.4
upwinding roe
end
To generate initial.inp (initial solution) and exact.inp (exact solution), compile and run the following code in the run directory. (It is the same file as the one used in running a conventional non-sparse-grids simulation).
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <string.h>
double power(double x,double a)
{
return(exp(a*log(x)));
}
int main(){
const double pi = 4.0*atan(1.0);
const double GAMMA = 1.4;
int NI,NJ,ndims,n_iter;
double tf, dt;
char ip_file_type[50]; strcpy(ip_file_type,"ascii");
FILE *in, *out;
printf("Reading file \"solver.inp\"...\n");
in = fopen("solver.inp","r");
if (!in) {
printf("Error: Input file \"solver.inp\" not found. Default values will be used.\n");
return(0);
} else {
char word[500];
fscanf(in,"%s",word);
if (!strcmp(word, "begin")) {
while (strcmp(word, "end")) {
fscanf(in,"%s",word);
if (!strcmp(word, "ndims")) fscanf(in,"%d",&ndims);
else if (!strcmp(word, "size")) {
fscanf(in,"%d",&NI);
fscanf(in,"%d",&NJ);
} else if (!strcmp(word, "n_iter")) fscanf(in,"%d",&n_iter);
else if (!strcmp(word, "dt")) fscanf(in,"%lf",&dt);
else if (!strcmp(word, "ip_file_type")) fscanf(in,"%s",ip_file_type);
}
} else printf("Error: Illegal format in solver.inp. Crash and burn!\n");
}
fclose(in);
if (ndims != 2) {
printf("ndims is not 2 in solver.inp. this code is to generate 2D exact solution\n");
return(0);
}
printf("Grid:\t\t\t%d X %d\n",NI,NJ);
int i,j;
double dx = 10.0 / ((double)NI);
double dy = 10.0 / ((double)NJ);
tf = (double)n_iter * dt; double tff = tf;
while (tf > 20) tf -= 20; // Time period
double *x, *y, *u0, *u1, *u2, *u3;
x = (double*) calloc (NI , sizeof(double));
y = (double*) calloc (NJ , sizeof(double));
u0 = (double*) calloc (NI*NJ, sizeof(double));
u1 = (double*) calloc (NI*NJ, sizeof(double));
u2 = (double*) calloc (NI*NJ, sizeof(double));
u3 = (double*) calloc (NI*NJ, sizeof(double));
double u_inf = 0.5;
double v_inf = 0.0;
double b = u_inf;
double x0, y0;
/* Initial solution */
x0 = 5.0, y0 = 5.0;
for (i = 0; i < NI; i++){
for (j = 0; j < NJ; j++){
x[i] = i*dx;
y[j] = j*dy;
int p = NJ*i + j;
double rx, ry;
rx = (x[i] - x0);
ry = (y[j] - y0);
if (rx < -5) { rx += 10; }
else if (rx > 5) { rx -= 10; }
double rsq = rx*rx + ry*ry;
double rho, u, v, P;
double du, dv;
rho = power(1.0 - ((GAMMA-1.0)*b*b)/(8.0*GAMMA*pi*pi) * exp(1.0-rsq), 1.0/(GAMMA-1.0));
P = power(rho,GAMMA);
du = - b/(2.0*pi) * exp(0.5*(1.0-rsq)) * ry;
dv = b/(2.0*pi) * exp(0.5*(1.0-rsq)) * rx;
u = u_inf + du;
v = v_inf + dv;
u0[p] = rho;
u1[p] = rho*u;
u2[p] = rho*v;
u3[p] = P/(GAMMA-1.0) + 0.5*rho*(u*u+v*v);
}
}
if (!strcmp(ip_file_type,"ascii")) {
printf("Writing ASCII initial solution file initial.inp\n");
out = fopen("initial.inp","w");
for (i = 0; i < NI; i++) fprintf(out,"%lf ",x[i]);
fprintf(out,"\n");
for (j = 0; j < NJ; j++) fprintf(out,"%lf ",y[j]);
fprintf(out,"\n");
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = NJ*i + j;
fprintf(out,"%lf ",u0[p]);
}
}
fprintf(out,"\n");
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = NJ*i + j;
fprintf(out,"%lf ",u1[p]);
}
}
fprintf(out,"\n");
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = NJ*i + j;
fprintf(out,"%lf ",u2[p]);
}
}
fprintf(out,"\n");
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = NJ*i + j;
fprintf(out,"%lf ",u3[p]);
}
}
fprintf(out,"\n");
fclose(out);
} else if ((!strcmp(ip_file_type,"binary")) || (!strcmp(ip_file_type,"bin"))) {
printf("Writing binary initial solution file initial.inp\n");
out = fopen("initial.inp","wb");
fwrite(x,sizeof(double),NI,out);
fwrite(y,sizeof(double),NJ,out);
double *U = (double*) calloc (4*NI*NJ,sizeof(double));
for (i=0; i < NI; i++) {
for (j = 0; j < NJ; j++) {
int p = NJ*i + j;
int q = NI*j + i;
U[4*q+0] = u0[p];
U[4*q+1] = u1[p];
U[4*q+2] = u2[p];
U[4*q+3] = u3[p];
}
}
fwrite(U,sizeof(double),4*NI*NJ,out);
free(U);
fclose(out);
}
/* Exact solution */
x0 = 5.0+tf*u_inf, y0 = 5.0;
if (x0 > 10) x0 -= 10; //periodic domain
printf("Final time: %lf, Vortex center: %lf, %lf\n",tff,x0,y0);
for (i = 0; i < NI; i++){
for (j = 0; j < NJ; j++){
x[i] = i*dx;
y[j] = j*dy;
int p = NJ*i + j;
double rx, ry;
rx = (x[i] - x0);
ry = (y[j] - y0);
if (rx < -5) { rx += 10; }
else if (rx > 5) { rx -= 10; }
double rsq = rx*rx + ry*ry;
double rho, u, v, P;
double du, dv;
rho = power(1.0 - ((GAMMA-1.0)*b*b)/(8.0*GAMMA*pi*pi) * exp(1.0-rsq), 1.0/(GAMMA-1.0));
P = power(rho,GAMMA);
du = - b/(2.0*pi) * exp(0.5*(1.0-rsq)) * ry;
dv = b/(2.0*pi) * exp(0.5*(1.0-rsq)) * rx;
u = u_inf + du;
v = v_inf + dv;
u0[p] = rho;
u1[p] = rho*u;
u2[p] = rho*v;
u3[p] = P/(GAMMA-1.0) + 0.5*rho*(u*u+v*v);
}
}
if (!strcmp(ip_file_type,"ascii")) {
printf("Writing ASCII exact solution file exact.inp\n");
out = fopen("exact.inp","w");
for (i = 0; i < NI; i++) fprintf(out,"%lf ",x[i]);
fprintf(out,"\n");
for (j = 0; j < NJ; j++) fprintf(out,"%lf ",y[j]);
fprintf(out,"\n");
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = NJ*i + j;
fprintf(out,"%lf ",u0[p]);
}
}
fprintf(out,"\n");
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = NJ*i + j;
fprintf(out,"%lf ",u1[p]);
}
}
fprintf(out,"\n");
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = NJ*i + j;
fprintf(out,"%lf ",u2[p]);
}
}
fprintf(out,"\n");
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = NJ*i + j;
fprintf(out,"%lf ",u3[p]);
}
}
fprintf(out,"\n");
fclose(out);
} else if ((!strcmp(ip_file_type,"binary")) || (!strcmp(ip_file_type,"bin"))) {
printf("Writing binary exact solution file exact.inp\n");
out = fopen("exact.inp","wb");
fwrite(x,sizeof(double),NI,out);
fwrite(y,sizeof(double),NJ,out);
double *U = (double*) calloc (4*NI*NJ,sizeof(double));
for (i=0; i < NI; i++) {
for (j = 0; j < NJ; j++) {
int p = NJ*i + j;
int q = NI*j + i;
U[4*q+0] = u0[p];
U[4*q+1] = u1[p];
U[4*q+2] = u2[p];
U[4*q+3] = u3[p];
}
}
fwrite(U,sizeof(double),4*NI*NJ,out);
free(U);
fclose(out);
}
free(x);
free(y);
free(u0);
free(u1);
free(u2);
free(u3);
return(0);
}
Output:
Note that iproc does not need to be set for simulations using sparse grids. HyPar will automatically calculate the load balanced processor distribution for each sparse grid. If too many processors are specified, then it will return an error.
After running the code, there should be the following output files:
- op_fg_00000.dat, ..., op_fg_00010.dat: these contain the full grid solution at \(t=0, ..., 20\).
- op_sg_<n>_00000.dat, ..., op_sg_<n>_00010.dat: these contain the solution on each of the sparse grids in the combination technique. These are written out because write_sg_solutions is set to yes in sparse_grids.inp (SparseGridsSimulation::m_write_sg_solutions).
Since HyPar::op_overwrite is set to no in solver.inp, separate files are written for solutions at each output time.
HyPar::op_file_format is set to tecplot2d in solver.inp, and thus, all the files are in a format that Tecplot (http://www.tecplot.com/) or other visualization software supporting the Tecplot format (e.g. VisIt - https://wci.llnl.gov/simulation/computer-codes/visit/) can read. In these files, the first two lines are the Tecplot headers, after which the data is written out as: the first two columns are grid indices, the next two columns are x and y coordinates, and the remaining columns are the solution components. HyPar::op_file_format can be set to text to get the solution files in plain text format (which can be read in and visualized in MATLAB for example).
The following animation shows the density contours as the vortex convects over the domain:,
The following animations show the solution on some of the sparse grids in the combination technique. The simulation was actually carried out on these grids. Note the different grid sizes.
Note: Some of the coarse grid solutions look like nonsense, but they are still important constituents of the combination technique and contribute towards the full grid solution.
Since the exact solution is available at the final time, the numerical errors are calculated for the recombined full grid solution and reported on screen (see below) as well as errors_fg.dat:
256 256 8 4 2.5000000000000001E-02 2.8601039537110856E-06 6.9470161546834434E-06 5.2476165544926512E-05 1.6932818000000001E+01 1.7046122000000000E+01
The numbers are: number of grid points in each dimension (HyPar::dim_global), number of processors in each dimension (MPIVariables::iproc), time step size (HyPar::dt), L1, L2, and L-infinity errors (HyPar::error), solver wall time (seconds) (i.e., not accounting for initialization, and cleaning up), and total wall time.
Since write_sg_errors is set to to no in sparse_grids.inp (SparseGridsSimulation::m_print_sg_errors), the errors for each of the sparse grids are not computed or reported.
Since HyPar::ConservationCheck is set to yes in solver.inp, the code checks for conservation errors for each of the sparse grids and prints it to screen, but does not write it to files because write_sg_errors is set to to no in sparse_grids.inp.
Expected screen output:
HyPar - Parallel (MPI) version with 32 processes
Compiled with PETSc time integration.
-- Sparse Grids Simulation --
Sparse grids inputs:
log2 of minimum grid size: 3
interpolation order: 6
write sparse grids solutions? yes
Allocated full grid simulation object(s).
Reading solver inputs from file "solver.inp".
No. of dimensions : 2
No. of variables : 4
Domain size : 256 256
Processes along each dimension : 8 4
No. of ghosts pts : 3
No. of iter. : 800
Restart iteration : 0
Time integration scheme : rk (44)
Spatial discretization scheme (hyperbolic) : upw5
Split hyperbolic flux term? : no
Interpolation type for hyperbolic term : components
Spatial discretization type (parabolic ) : nonconservative-2stage
Spatial discretization scheme (parabolic ) : 4
Time Step : 2.500000E-02
Check for conservation : yes
Screen output iterations : 10
File output iterations : 80
Initial solution file type : binary
Initial solution read mode : serial
Solution file write mode : serial
Solution file format : tecplot2d
Overwrite solution file : no
Physical model : navierstokes2d
Processor distribution for full grid object: 8 4
Initializing sparse grids...
Number of spatial dimensions: 2
Computing sparse grids dimensions...
Number of sparse grid domains in combination technique: 11
Computing processor decompositions...
Sparse Grids: Combination technique grids sizes and coefficients are:-
0: dim = ( 256 8 ), coeff = +1.00e+00, iproc = ( 32 1 )
1: dim = ( 128 16 ), coeff = +1.00e+00, iproc = ( 16 2 )
2: dim = ( 64 32 ), coeff = +1.00e+00, iproc = ( 8 4 )
3: dim = ( 32 64 ), coeff = +1.00e+00, iproc = ( 4 8 )
4: dim = ( 16 128 ), coeff = +1.00e+00, iproc = ( 2 16 )
5: dim = ( 8 256 ), coeff = +1.00e+00, iproc = ( 1 32 )
6: dim = ( 128 8 ), coeff = -1.00e+00, iproc = ( 32 1 )
7: dim = ( 64 16 ), coeff = -1.00e+00, iproc = ( 16 2 )
8: dim = ( 32 32 ), coeff = -1.00e+00, iproc = ( 8 4 )
9: dim = ( 16 64 ), coeff = -1.00e+00, iproc = ( 2 16 )
10: dim = ( 8 128 ), coeff = -1.00e+00, iproc = ( 1 32 )
Allocating data arrays for full grid.
Partitioning domain and allocating data arrays.
Reading array from binary file initial.inp (Serial mode).
Volume integral of the initial solution:
0: 9.9980704056028827E+01
1: 4.9990352267387813E+01
2: -2.3937350376260907E-07
3: 2.6245709153809656E+02
Interpolating grid coordinates to sparse grids domain 0.
Interpolating grid coordinates to sparse grids domain 1.
Interpolating grid coordinates to sparse grids domain 2.
Interpolating grid coordinates to sparse grids domain 3.
Interpolating grid coordinates to sparse grids domain 4.
Interpolating grid coordinates to sparse grids domain 5.
Interpolating grid coordinates to sparse grids domain 6.
Interpolating grid coordinates to sparse grids domain 7.
Interpolating grid coordinates to sparse grids domain 8.
Interpolating grid coordinates to sparse grids domain 9.
Interpolating grid coordinates to sparse grids domain 10.
Domain 0: Reading boundary conditions from boundary.inp.
Boundary periodic: Along dimension 0 and face +1
Boundary periodic: Along dimension 0 and face -1
Boundary periodic: Along dimension 1 and face +1
Boundary periodic: Along dimension 1 and face -1
4 boundary condition(s) read.
Domain 1: Reading boundary conditions from boundary.inp.
Boundary periodic: Along dimension 0 and face +1
Boundary periodic: Along dimension 0 and face -1
Boundary periodic: Along dimension 1 and face +1
Boundary periodic: Along dimension 1 and face -1
4 boundary condition(s) read.
Domain 2: Reading boundary conditions from boundary.inp.
Boundary periodic: Along dimension 0 and face +1
Boundary periodic: Along dimension 0 and face -1
Boundary periodic: Along dimension 1 and face +1
Boundary periodic: Along dimension 1 and face -1
4 boundary condition(s) read.
Domain 3: Reading boundary conditions from boundary.inp.
Boundary periodic: Along dimension 0 and face +1
Boundary periodic: Along dimension 0 and face -1
Boundary periodic: Along dimension 1 and face +1
Boundary periodic: Along dimension 1 and face -1
4 boundary condition(s) read.
Domain 4: Reading boundary conditions from boundary.inp.
Boundary periodic: Along dimension 0 and face +1
Boundary periodic: Along dimension 0 and face -1
Boundary periodic: Along dimension 1 and face +1
Boundary periodic: Along dimension 1 and face -1
4 boundary condition(s) read.
Domain 5: Reading boundary conditions from boundary.inp.
Boundary periodic: Along dimension 0 and face +1
Boundary periodic: Along dimension 0 and face -1
Boundary periodic: Along dimension 1 and face +1
Boundary periodic: Along dimension 1 and face -1
4 boundary condition(s) read.
Domain 6: Reading boundary conditions from boundary.inp.
Boundary periodic: Along dimension 0 and face +1
Boundary periodic: Along dimension 0 and face -1
Boundary periodic: Along dimension 1 and face +1
Boundary periodic: Along dimension 1 and face -1
4 boundary condition(s) read.
Domain 7: Reading boundary conditions from boundary.inp.
Boundary periodic: Along dimension 0 and face +1
Boundary periodic: Along dimension 0 and face -1
Boundary periodic: Along dimension 1 and face +1
Boundary periodic: Along dimension 1 and face -1
4 boundary condition(s) read.
Domain 8: Reading boundary conditions from boundary.inp.
Boundary periodic: Along dimension 0 and face +1
Boundary periodic: Along dimension 0 and face -1
Boundary periodic: Along dimension 1 and face +1
Boundary periodic: Along dimension 1 and face -1
4 boundary condition(s) read.
Domain 9: Reading boundary conditions from boundary.inp.
Boundary periodic: Along dimension 0 and face +1
Boundary periodic: Along dimension 0 and face -1
Boundary periodic: Along dimension 1 and face +1
Boundary periodic: Along dimension 1 and face -1
4 boundary condition(s) read.
Domain 10: Reading boundary conditions from boundary.inp.
Boundary periodic: Along dimension 0 and face +1
Boundary periodic: Along dimension 0 and face -1
Boundary periodic: Along dimension 1 and face +1
Boundary periodic: Along dimension 1 and face -1
4 boundary condition(s) read.
Initializing solvers.
Initializing physics. Model = "navierstokes2d"
Reading physical model inputs from file "physics.inp".
Interpolating initial solution to sparse grids domain 0.
Volume integral of the initial solution on sparse grids domain 0:
0: 9.9980771982607934E+01
1: 4.9990391607657912E+01
2: -2.3937205473334444E-07
3: 2.6245784617971401E+02
Interpolating initial solution to sparse grids domain 1.
Volume integral of the initial solution on sparse grids domain 1:
0: 9.9980704055917187E+01
1: 4.9990352197094573E+01
2: -2.3937354948211365E-07
3: 2.6245709149936698E+02
Interpolating initial solution to sparse grids domain 2.
Volume integral of the initial solution on sparse grids domain 2:
0: 9.9980704056028671E+01
1: 4.9990352247317951E+01
2: -2.3403495727448639E-07
3: 2.6245709152806205E+02
Interpolating initial solution to sparse grids domain 3.
Volume integral of the initial solution on sparse grids domain 3:
0: 9.9980704056028685E+01
1: 4.9990352262049285E+01
2: -2.1930360996768816E-07
3: 2.6245709153542776E+02
Interpolating initial solution to sparse grids domain 4.
Volume integral of the initial solution on sparse grids domain 4:
0: 9.9980704055917187E+01
1: 4.9990352267332149E+01
2: -1.6913597602997063E-07
3: 2.6245709153448581E+02
Interpolating initial solution to sparse grids domain 5.
Volume integral of the initial solution on sparse grids domain 5:
0: 9.9980771982607934E+01
1: 4.9990386230676023E+01
2: -5.6163539423498036E-06
3: 2.6245784349122300E+02
Interpolating initial solution to sparse grids domain 6.
Volume integral of the initial solution on sparse grids domain 6:
0: 9.9980771982607948E+01
1: 4.9990391607657912E+01
2: -2.3937205473334444E-07
3: 2.6245784617971401E+02
Interpolating initial solution to sparse grids domain 7.
Volume integral of the initial solution on sparse grids domain 7:
0: 9.9980704055917187E+01
1: 4.9990352197094573E+01
2: -2.3403498880525397E-07
3: 2.6245709149936704E+02
Interpolating initial solution to sparse grids domain 8.
Volume integral of the initial solution on sparse grids domain 8:
0: 9.9980704056028657E+01
1: 4.9990352247317944E+01
2: -2.1930361898095559E-07
3: 2.6245709152806228E+02
Interpolating initial solution to sparse grids domain 9.
Volume integral of the initial solution on sparse grids domain 9:
0: 9.9980704055917201E+01
1: 4.9990352261993586E+01
2: -1.6913597870430817E-07
3: 2.6245709153181656E+02
Interpolating initial solution to sparse grids domain 10.
Volume integral of the initial solution on sparse grids domain 10:
0: 9.9980771982607948E+01
1: 4.9990386230676023E+01
2: -5.6163539423644200E-06
3: 2.6245784349122300E+02
Setting up time integration.
Solving in time (from 0 to 800 iterations)
Writing solution file op_sg_00_00000.dat.
Writing solution file op_sg_01_00000.dat.
Writing solution file op_sg_02_00000.dat.
Writing solution file op_sg_03_00000.dat.
Writing solution file op_sg_04_00000.dat.
Writing solution file op_sg_05_00000.dat.
Writing solution file op_sg_06_00000.dat.
Writing solution file op_sg_07_00000.dat.
Writing solution file op_sg_08_00000.dat.
Writing solution file op_sg_09_00000.dat.
Writing solution file op_sg_10_00000.dat.
Writing solution file op_fg_00000.dat.
--
Iteration: 10, Time: 2.500e-01
Max CFL: 4.014E-01, Max Diff. No.: -1.000E+00, Norm: 3.3049E-04
Conservation loss: 7.5746E-16
--
--
Iteration: 20, Time: 5.000e-01
Max CFL: 4.018E-01, Max Diff. No.: -1.000E+00, Norm: 3.1356E-04
Conservation loss: 6.0432E-16
--
--
Iteration: 30, Time: 7.500e-01
Max CFL: 4.018E-01, Max Diff. No.: -1.000E+00, Norm: 3.0382E-04
Conservation loss: 5.6672E-16
--
--
Iteration: 40, Time: 1.000e+00
Max CFL: 4.014E-01, Max Diff. No.: -1.000E+00, Norm: 3.0078E-04
Conservation loss: 7.5893E-16
--
--
Iteration: 50, Time: 1.250e+00
Max CFL: 4.009E-01, Max Diff. No.: -1.000E+00, Norm: 2.9996E-04
Conservation loss: 8.2964E-16
--
--
Iteration: 60, Time: 1.500e+00
Max CFL: 4.003E-01, Max Diff. No.: -1.000E+00, Norm: 2.9815E-04
Conservation loss: 7.0913E-16
--
--
Iteration: 70, Time: 1.750e+00
Max CFL: 3.997E-01, Max Diff. No.: -1.000E+00, Norm: 2.9500E-04
Conservation loss: 9.0264E-16
--
--
Iteration: 80, Time: 2.000e+00
Max CFL: 3.990E-01, Max Diff. No.: -1.000E+00, Norm: 2.9147E-04
Conservation loss: 9.8112E-16
--
Writing solution file op_sg_00_00001.dat.
Writing solution file op_sg_01_00001.dat.
Writing solution file op_sg_02_00001.dat.
Writing solution file op_sg_03_00001.dat.
Writing solution file op_sg_04_00001.dat.
Writing solution file op_sg_05_00001.dat.
Writing solution file op_sg_06_00001.dat.
Writing solution file op_sg_07_00001.dat.
Writing solution file op_sg_08_00001.dat.
Writing solution file op_sg_09_00001.dat.
Writing solution file op_sg_10_00001.dat.
Writing solution file op_fg_00001.dat.
--
Iteration: 90, Time: 2.250e+00
Max CFL: 3.982E-01, Max Diff. No.: -1.000E+00, Norm: 2.8833E-04
Conservation loss: 9.7630E-16
--
--
Iteration: 100, Time: 2.500e+00
Max CFL: 3.972E-01, Max Diff. No.: -1.000E+00, Norm: 2.8586E-04
Conservation loss: 1.1037E-15
--
--
Iteration: 110, Time: 2.750e+00
Max CFL: 3.960E-01, Max Diff. No.: -1.000E+00, Norm: 2.8406E-04
Conservation loss: 1.0997E-15
--
--
Iteration: 120, Time: 3.000e+00
Max CFL: 3.947E-01, Max Diff. No.: -1.000E+00, Norm: 2.8280E-04
Conservation loss: 1.1201E-15
--
--
Iteration: 130, Time: 3.250e+00
Max CFL: 3.947E-01, Max Diff. No.: -1.000E+00, Norm: 2.8187E-04
Conservation loss: 9.5860E-16
--
--
Iteration: 140, Time: 3.500e+00
Max CFL: 3.957E-01, Max Diff. No.: -1.000E+00, Norm: 2.8100E-04
Conservation loss: 9.7995E-16
--
--
Iteration: 150, Time: 3.750e+00
Max CFL: 3.965E-01, Max Diff. No.: -1.000E+00, Norm: 2.8006E-04
Conservation loss: 1.0788E-15
--
--
Iteration: 160, Time: 4.000e+00
Max CFL: 3.969E-01, Max Diff. No.: -1.000E+00, Norm: 2.7914E-04
Conservation loss: 1.0612E-15
--
Writing solution file op_sg_00_00002.dat.
Writing solution file op_sg_01_00002.dat.
Writing solution file op_sg_02_00002.dat.
Writing solution file op_sg_03_00002.dat.
Writing solution file op_sg_04_00002.dat.
Writing solution file op_sg_05_00002.dat.
Writing solution file op_sg_06_00002.dat.
Writing solution file op_sg_07_00002.dat.
Writing solution file op_sg_08_00002.dat.
Writing solution file op_sg_09_00002.dat.
Writing solution file op_sg_10_00002.dat.
Writing solution file op_fg_00002.dat.
--
Iteration: 170, Time: 4.250e+00
Max CFL: 3.971E-01, Max Diff. No.: -1.000E+00, Norm: 2.7836E-04
Conservation loss: 1.1094E-15
--
--
Iteration: 180, Time: 4.500e+00
Max CFL: 3.970E-01, Max Diff. No.: -1.000E+00, Norm: 2.7767E-04
Conservation loss: 1.1035E-15
--
--
Iteration: 190, Time: 4.750e+00
Max CFL: 3.967E-01, Max Diff. No.: -1.000E+00, Norm: 2.7697E-04
Conservation loss: 1.0396E-15
--
--
Iteration: 200, Time: 5.000e+00
Max CFL: 3.962E-01, Max Diff. No.: -1.000E+00, Norm: 2.7613E-04
Conservation loss: 9.3420E-16
--
--
Iteration: 210, Time: 5.250e+00
Max CFL: 3.956E-01, Max Diff. No.: -1.000E+00, Norm: 2.7511E-04
Conservation loss: 1.1196E-15
--
--
Iteration: 220, Time: 5.500e+00
Max CFL: 3.947E-01, Max Diff. No.: -1.000E+00, Norm: 2.7413E-04
Conservation loss: 1.1269E-15
--
--
Iteration: 230, Time: 5.750e+00
Max CFL: 3.937E-01, Max Diff. No.: -1.000E+00, Norm: 2.7348E-04
Conservation loss: 1.1007E-15
--
--
Iteration: 240, Time: 6.000e+00
Max CFL: 3.929E-01, Max Diff. No.: -1.000E+00, Norm: 2.7316E-04
Conservation loss: 1.2379E-15
--
Writing solution file op_sg_00_00003.dat.
Writing solution file op_sg_01_00003.dat.
Writing solution file op_sg_02_00003.dat.
Writing solution file op_sg_03_00003.dat.
Writing solution file op_sg_04_00003.dat.
Writing solution file op_sg_05_00003.dat.
Writing solution file op_sg_06_00003.dat.
Writing solution file op_sg_07_00003.dat.
Writing solution file op_sg_08_00003.dat.
Writing solution file op_sg_09_00003.dat.
Writing solution file op_sg_10_00003.dat.
Writing solution file op_fg_00003.dat.
--
Iteration: 250, Time: 6.250e+00
Max CFL: 3.935E-01, Max Diff. No.: -1.000E+00, Norm: 2.7284E-04
Conservation loss: 1.2991E-15
--
--
Iteration: 260, Time: 6.500e+00
Max CFL: 3.942E-01, Max Diff. No.: -1.000E+00, Norm: 2.7230E-04
Conservation loss: 1.3243E-15
--
--
Iteration: 270, Time: 6.750e+00
Max CFL: 3.947E-01, Max Diff. No.: -1.000E+00, Norm: 2.7160E-04
Conservation loss: 1.2739E-15
--
--
Iteration: 280, Time: 7.000e+00
Max CFL: 3.949E-01, Max Diff. No.: -1.000E+00, Norm: 2.7091E-04
Conservation loss: 1.1281E-15
--
--
Iteration: 290, Time: 7.250e+00
Max CFL: 3.949E-01, Max Diff. No.: -1.000E+00, Norm: 2.7035E-04
Conservation loss: 1.1390E-15
--
--
Iteration: 300, Time: 7.500e+00
Max CFL: 3.948E-01, Max Diff. No.: -1.000E+00, Norm: 2.6990E-04
Conservation loss: 1.2292E-15
--
--
Iteration: 310, Time: 7.750e+00
Max CFL: 3.944E-01, Max Diff. No.: -1.000E+00, Norm: 2.6950E-04
Conservation loss: 1.1823E-15
--
--
Iteration: 320, Time: 8.000e+00
Max CFL: 3.939E-01, Max Diff. No.: -1.000E+00, Norm: 2.6913E-04
Conservation loss: 9.2646E-16
--
Writing solution file op_sg_00_00004.dat.
Writing solution file op_sg_01_00004.dat.
Writing solution file op_sg_02_00004.dat.
Writing solution file op_sg_03_00004.dat.
Writing solution file op_sg_04_00004.dat.
Writing solution file op_sg_05_00004.dat.
Writing solution file op_sg_06_00004.dat.
Writing solution file op_sg_07_00004.dat.
Writing solution file op_sg_08_00004.dat.
Writing solution file op_sg_09_00004.dat.
Writing solution file op_sg_10_00004.dat.
Writing solution file op_fg_00004.dat.
--
Iteration: 330, Time: 8.250e+00
Max CFL: 3.933E-01, Max Diff. No.: -1.000E+00, Norm: 2.6876E-04
Conservation loss: 9.5619E-16
--
--
Iteration: 340, Time: 8.500e+00
Max CFL: 3.925E-01, Max Diff. No.: -1.000E+00, Norm: 2.6836E-04
Conservation loss: 9.3920E-16
--
--
Iteration: 350, Time: 8.750e+00
Max CFL: 3.918E-01, Max Diff. No.: -1.000E+00, Norm: 2.6793E-04
Conservation loss: 1.0439E-15
--
--
Iteration: 360, Time: 9.000e+00
Max CFL: 3.925E-01, Max Diff. No.: -1.000E+00, Norm: 2.6753E-04
Conservation loss: 1.0907E-15
--
--
Iteration: 370, Time: 9.250e+00
Max CFL: 3.931E-01, Max Diff. No.: -1.000E+00, Norm: 2.6722E-04
Conservation loss: 1.0355E-15
--
--
Iteration: 380, Time: 9.500e+00
Max CFL: 3.935E-01, Max Diff. No.: -1.000E+00, Norm: 2.6695E-04
Conservation loss: 1.1399E-15
--
--
Iteration: 390, Time: 9.750e+00
Max CFL: 3.936E-01, Max Diff. No.: -1.000E+00, Norm: 2.6657E-04
Conservation loss: 1.0894E-15
--
--
Iteration: 400, Time: 1.000e+01
Max CFL: 3.936E-01, Max Diff. No.: -1.000E+00, Norm: 2.6605E-04
Conservation loss: 1.1089E-15
--
Writing solution file op_sg_00_00005.dat.
Writing solution file op_sg_01_00005.dat.
Writing solution file op_sg_02_00005.dat.
Writing solution file op_sg_03_00005.dat.
Writing solution file op_sg_04_00005.dat.
Writing solution file op_sg_05_00005.dat.
Writing solution file op_sg_06_00005.dat.
Writing solution file op_sg_07_00005.dat.
Writing solution file op_sg_08_00005.dat.
Writing solution file op_sg_09_00005.dat.
Writing solution file op_sg_10_00005.dat.
Writing solution file op_fg_00005.dat.
--
Iteration: 410, Time: 1.025e+01
Max CFL: 3.934E-01, Max Diff. No.: -1.000E+00, Norm: 2.6550E-04
Conservation loss: 1.1351E-15
--
--
Iteration: 420, Time: 1.050e+01
Max CFL: 3.931E-01, Max Diff. No.: -1.000E+00, Norm: 2.6506E-04
Conservation loss: 1.1216E-15
--
--
Iteration: 430, Time: 1.075e+01
Max CFL: 3.926E-01, Max Diff. No.: -1.000E+00, Norm: 2.6473E-04
Conservation loss: 1.2011E-15
--
--
Iteration: 440, Time: 1.100e+01
Max CFL: 3.921E-01, Max Diff. No.: -1.000E+00, Norm: 2.6440E-04
Conservation loss: 1.1521E-15
--
--
Iteration: 450, Time: 1.125e+01
Max CFL: 3.914E-01, Max Diff. No.: -1.000E+00, Norm: 2.6403E-04
Conservation loss: 1.3241E-15
--
--
Iteration: 460, Time: 1.150e+01
Max CFL: 3.915E-01, Max Diff. No.: -1.000E+00, Norm: 2.6368E-04
Conservation loss: 1.4016E-15
--
--
Iteration: 470, Time: 1.175e+01
Max CFL: 3.921E-01, Max Diff. No.: -1.000E+00, Norm: 2.6344E-04
Conservation loss: 1.3749E-15
--
--
Iteration: 480, Time: 1.200e+01
Max CFL: 3.925E-01, Max Diff. No.: -1.000E+00, Norm: 2.6330E-04
Conservation loss: 1.3038E-15
--
Writing solution file op_sg_00_00006.dat.
Writing solution file op_sg_01_00006.dat.
Writing solution file op_sg_02_00006.dat.
Writing solution file op_sg_03_00006.dat.
Writing solution file op_sg_04_00006.dat.
Writing solution file op_sg_05_00006.dat.
Writing solution file op_sg_06_00006.dat.
Writing solution file op_sg_07_00006.dat.
Writing solution file op_sg_08_00006.dat.
Writing solution file op_sg_09_00006.dat.
Writing solution file op_sg_10_00006.dat.
Writing solution file op_fg_00006.dat.
--
Iteration: 490, Time: 1.225e+01
Max CFL: 3.928E-01, Max Diff. No.: -1.000E+00, Norm: 2.6318E-04
Conservation loss: 1.3768E-15
--
--
Iteration: 500, Time: 1.250e+01
Max CFL: 3.929E-01, Max Diff. No.: -1.000E+00, Norm: 2.6299E-04
Conservation loss: 1.2515E-15
--
--
Iteration: 510, Time: 1.275e+01
Max CFL: 3.929E-01, Max Diff. No.: -1.000E+00, Norm: 2.6268E-04
Conservation loss: 1.2171E-15
--
--
Iteration: 520, Time: 1.300e+01
Max CFL: 3.927E-01, Max Diff. No.: -1.000E+00, Norm: 2.6229E-04
Conservation loss: 1.1241E-15
--
--
Iteration: 530, Time: 1.325e+01
Max CFL: 3.923E-01, Max Diff. No.: -1.000E+00, Norm: 2.6191E-04
Conservation loss: 1.4518E-15
--
--
Iteration: 540, Time: 1.350e+01
Max CFL: 3.919E-01, Max Diff. No.: -1.000E+00, Norm: 2.6157E-04
Conservation loss: 1.5080E-15
--
--
Iteration: 550, Time: 1.375e+01
Max CFL: 3.913E-01, Max Diff. No.: -1.000E+00, Norm: 2.6124E-04
Conservation loss: 1.3999E-15
--
--
Iteration: 560, Time: 1.400e+01
Max CFL: 3.907E-01, Max Diff. No.: -1.000E+00, Norm: 2.6092E-04
Conservation loss: 1.5722E-15
--
Writing solution file op_sg_00_00007.dat.
Writing solution file op_sg_01_00007.dat.
Writing solution file op_sg_02_00007.dat.
Writing solution file op_sg_03_00007.dat.
Writing solution file op_sg_04_00007.dat.
Writing solution file op_sg_05_00007.dat.
Writing solution file op_sg_06_00007.dat.
Writing solution file op_sg_07_00007.dat.
Writing solution file op_sg_08_00007.dat.
Writing solution file op_sg_09_00007.dat.
Writing solution file op_sg_10_00007.dat.
Writing solution file op_fg_00007.dat.
--
Iteration: 570, Time: 1.425e+01
Max CFL: 3.912E-01, Max Diff. No.: -1.000E+00, Norm: 2.6065E-04
Conservation loss: 1.5222E-15
--
--
Iteration: 580, Time: 1.450e+01
Max CFL: 3.917E-01, Max Diff. No.: -1.000E+00, Norm: 2.6041E-04
Conservation loss: 1.4880E-15
--
--
Iteration: 590, Time: 1.475e+01
Max CFL: 3.920E-01, Max Diff. No.: -1.000E+00, Norm: 2.6015E-04
Conservation loss: 1.4763E-15
--
--
Iteration: 600, Time: 1.500e+01
Max CFL: 3.922E-01, Max Diff. No.: -1.000E+00, Norm: 2.5989E-04
Conservation loss: 1.3194E-15
--
--
Iteration: 610, Time: 1.525e+01
Max CFL: 3.923E-01, Max Diff. No.: -1.000E+00, Norm: 2.5968E-04
Conservation loss: 1.2840E-15
--
--
Iteration: 620, Time: 1.550e+01
Max CFL: 3.922E-01, Max Diff. No.: -1.000E+00, Norm: 2.5952E-04
Conservation loss: 1.2997E-15
--
--
Iteration: 630, Time: 1.575e+01
Max CFL: 3.920E-01, Max Diff. No.: -1.000E+00, Norm: 2.5934E-04
Conservation loss: 1.3699E-15
--
--
Iteration: 640, Time: 1.600e+01
Max CFL: 3.917E-01, Max Diff. No.: -1.000E+00, Norm: 2.5913E-04
Conservation loss: 1.3629E-15
--
Writing solution file op_sg_00_00008.dat.
Writing solution file op_sg_01_00008.dat.
Writing solution file op_sg_02_00008.dat.
Writing solution file op_sg_03_00008.dat.
Writing solution file op_sg_04_00008.dat.
Writing solution file op_sg_05_00008.dat.
Writing solution file op_sg_06_00008.dat.
Writing solution file op_sg_07_00008.dat.
Writing solution file op_sg_08_00008.dat.
Writing solution file op_sg_09_00008.dat.
Writing solution file op_sg_10_00008.dat.
Writing solution file op_fg_00008.dat.
--
Iteration: 650, Time: 1.625e+01
Max CFL: 3.913E-01, Max Diff. No.: -1.000E+00, Norm: 2.5890E-04
Conservation loss: 1.4159E-15
--
--
Iteration: 660, Time: 1.650e+01
Max CFL: 3.908E-01, Max Diff. No.: -1.000E+00, Norm: 2.5869E-04
Conservation loss: 1.3844E-15
--
--
Iteration: 670, Time: 1.675e+01
Max CFL: 3.905E-01, Max Diff. No.: -1.000E+00, Norm: 2.5851E-04
Conservation loss: 1.4539E-15
--
--
Iteration: 680, Time: 1.700e+01
Max CFL: 3.910E-01, Max Diff. No.: -1.000E+00, Norm: 2.5830E-04
Conservation loss: 1.5666E-15
--
--
Iteration: 690, Time: 1.725e+01
Max CFL: 3.914E-01, Max Diff. No.: -1.000E+00, Norm: 2.5806E-04
Conservation loss: 1.4202E-15
--
--
Iteration: 700, Time: 1.750e+01
Max CFL: 3.916E-01, Max Diff. No.: -1.000E+00, Norm: 2.5780E-04
Conservation loss: 1.3192E-15
--
--
Iteration: 710, Time: 1.775e+01
Max CFL: 3.918E-01, Max Diff. No.: -1.000E+00, Norm: 2.5751E-04
Conservation loss: 1.2006E-15
--
--
Iteration: 720, Time: 1.800e+01
Max CFL: 3.918E-01, Max Diff. No.: -1.000E+00, Norm: 2.5724E-04
Conservation loss: 1.1660E-15
--
Writing solution file op_sg_00_00009.dat.
Writing solution file op_sg_01_00009.dat.
Writing solution file op_sg_02_00009.dat.
Writing solution file op_sg_03_00009.dat.
Writing solution file op_sg_04_00009.dat.
Writing solution file op_sg_05_00009.dat.
Writing solution file op_sg_06_00009.dat.
Writing solution file op_sg_07_00009.dat.
Writing solution file op_sg_08_00009.dat.
Writing solution file op_sg_09_00009.dat.
Writing solution file op_sg_10_00009.dat.
Writing solution file op_fg_00009.dat.
--
Iteration: 730, Time: 1.825e+01
Max CFL: 3.917E-01, Max Diff. No.: -1.000E+00, Norm: 2.5702E-04
Conservation loss: 1.1348E-15
--
--
Iteration: 740, Time: 1.850e+01
Max CFL: 3.915E-01, Max Diff. No.: -1.000E+00, Norm: 2.5687E-04
Conservation loss: 1.1627E-15
--
--
Iteration: 750, Time: 1.875e+01
Max CFL: 3.911E-01, Max Diff. No.: -1.000E+00, Norm: 2.5673E-04
Conservation loss: 1.0387E-15
--
--
Iteration: 760, Time: 1.900e+01
Max CFL: 3.907E-01, Max Diff. No.: -1.000E+00, Norm: 2.5657E-04
Conservation loss: 1.1611E-15
--
--
Iteration: 770, Time: 1.925e+01
Max CFL: 3.902E-01, Max Diff. No.: -1.000E+00, Norm: 2.5639E-04
Conservation loss: 1.1230E-15
--
--
Iteration: 780, Time: 1.950e+01
Max CFL: 3.904E-01, Max Diff. No.: -1.000E+00, Norm: 2.5619E-04
Conservation loss: 1.1702E-15
--
--
Iteration: 790, Time: 1.975e+01
Max CFL: 3.908E-01, Max Diff. No.: -1.000E+00, Norm: 2.5599E-04
Conservation loss: 1.2247E-15
--
--
Iteration: 800, Time: 2.000e+01
Max CFL: 3.911E-01, Max Diff. No.: -1.000E+00, Norm: 2.5581E-04
Conservation loss: 1.2456E-15
--
Writing solution file op_sg_00_00010.dat.
Writing solution file op_sg_01_00010.dat.
Writing solution file op_sg_02_00010.dat.
Writing solution file op_sg_03_00010.dat.
Writing solution file op_sg_04_00010.dat.
Writing solution file op_sg_05_00010.dat.
Writing solution file op_sg_06_00010.dat.
Writing solution file op_sg_07_00010.dat.
Writing solution file op_sg_08_00010.dat.
Writing solution file op_sg_09_00010.dat.
Writing solution file op_sg_10_00010.dat.
Writing solution file op_fg_00010.dat.
Completed time integration (Final time: 20.000000).
Reading array from binary file exact.inp (Serial mode).
Computed errors for full grid solution:
L1 Error: 2.8601039537110856E-06
L2 Error: 6.9470161546834434E-06
Linf Error: 5.2476165544926512E-05
Solver runtime (in seconds): 1.6932818000000001E+01
Total runtime (in seconds): 1.7046122000000000E+01
Deallocating arrays.
Finished.