2D Linear Advection - Sine Wave with Spatially-Varying Advection Speed
Location: hypar/Examples/2D/LinearAdvection/SineWave_NonConstantAdvection_SparseGrids (This directory contains all the input files needed to run this case.)
Governing equations: 2D Linear Advection Equation (linearadr.h)
Domain: \(0 \le x,y < 1\), "periodic" (_PERIODIC_) boundary conditions on all boundaries.
Initial solution: \(u\left(x,y,0\right) = u_0\left(x,y\right)= \cos\left(4\pi y\right)\)
Advection speed: \(a_x\left(x,y\right) = \sin\left(4\pi y\right)\), \(a_y\left(x,y\right) = -\cos\left(4\pi x\right)\)
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 yes
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 1
size 128 128
ghost 3
n_iter 1000
restart_iter 0
time_scheme rk
time_scheme_type 44
hyp_space_scheme upw5
hyp_interp_type components
dt 0.004
conservation_check yes
screen_op_iter 50
file_op_iter 50
op_file_format tecplot2d
ip_file_type binary
input_mode serial
output_mode serial
op_overwrite no
model linear-advection-diffusion-reaction
end
boundary.inp
4
periodic 0 1 0 0 0 1
periodic 0 -1 0 0 0 1
periodic 1 1 0 1 0 0
periodic 1 -1 0 1 0 0
physics.inp (specifies filename for advection field)
begin
advection_filename advection
end
Note: Do not include the ".inp" extension in the filename above.
To generate initial.inp and advection.inp, compile and run the following code in the run directory.
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <string.h>
int main()
{
const double pi = 4.0*atan(1.0);
int NI, NJ, ndims;
char ip_file_type[50];
strcpy(ip_file_type,"ascii");
FILE *in;
printf("Reading file \"solver.inp\"...\n");
in = fopen("solver.inp","r");
if (!in) {
fprintf(stderr,"Error: Input file \"solver.inp\" not found.\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, "ip_file_type")) {
fscanf(in,"%s",ip_file_type);
}
}
} else {
fprintf(stderr,"Error: Illegal format in solver.inp. Crash and burn!\n");
return(0);
}
}
fclose(in);
char adv_filename[500] = "none";
int flag = 0;
printf("Reading file \"physics.inp\"...\n");
in = fopen("physics.inp","r");
if (!in) {
printf("Error: Input file \"physics.inp\" not found.\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, "advection_filename")) {
fscanf(in,"%s", adv_filename);
flag = 1;
}
}
} else {
printf("Error: Illegal format in solver.inp. Crash and burn!\n");
}
}
fclose(in);
if (flag == 0) {
printf("Error: Input \"advection_filename\" not found in physics.inp file.\n");
return 0;
}
strcat(adv_filename, ".inp");
if (ndims != 2) {
fprintf(stderr,"ndims is not 2 in solver.inp. this code is to generate 2D initial conditions\n");
return(0);
}
printf("Grid: %d, %d\n",NI,NJ);
int i,j;
double dx = 1.0 / ((double)NI);
double dy = 1.0 / ((double)NJ);
double *x, *y, *u, *a;
x = (double*) calloc (NI , sizeof(double));
y = (double*) calloc (NJ , sizeof(double));
u = (double*) calloc (NI*NJ, sizeof(double));
a = (double*) calloc (ndims*NI*NJ, sizeof(double));
for (i = 0; i < NI; i++){
for (j = 0; j < NJ; j++){
x[i] = i*dx;
y[j] = j*dy;
int p = NI*j + i;
u[p] = cos(4*pi*y[j]);
a[ndims*p+0] = sin(4*pi*y[j]);
a[ndims*p+1] = -cos(4*pi*x[i]);
}
}
FILE *out;
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 ",u[p]);
}
}
fprintf(out,"\n");
fclose(out);
printf("Writing ASCII advection field file %s\n", adv_filename);
out = fopen(adv_filename,"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 ",a[ndims*p+0]);
}
}
fprintf(out,"\n");
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = NJ*i + j;
fprintf(out,"%lf ",a[ndims*p+1]);
}
}
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);
fwrite(u,sizeof(double),NI*NJ,out);
fclose(out);
printf("Writing binary advection field file %s\n", adv_filename);
out = fopen(adv_filename,"wb");
fwrite(x,sizeof(double),NI,out);
fwrite(y,sizeof(double),NJ,out);
fwrite(a,sizeof(double),ndims*NI*NJ,out);
fclose(out);
}
free(x);
free(y);
free(u);
free(a);
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_00020.dat: these contain the full grid solution at \(t=0, ..., 4\).
- op_sg_<n>_00000.dat, ..., op_sg_<n>_00020.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).
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 final column is the solution. 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 was generated from the full grid solution files that were computed by using the combination technique on the sparse grids:
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.
In addition to the usual output files, the linear advection physics module writes out the following files:
- advection_field_<n>_00000.dat, ..., advection_field_<n>_00020.dat: These files share the same format as the solution output files op_*.dat and contains the advection field \(a_x\left(x,y\right), a_y\left(x,y\right)\).
The following figure shows the vector plot of the velocity field on some of the sparse grids:
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, as well as the files conservation_<n>.dat:
128 8 1 1 4.0000000000000001E-03 9.5911234683498936E-17
32 32 1 1 4.0000000000000001E-03 2.7018318138338770E-16
8 128 1 1 4.0000000000000001E-03 2.6969529040576923E-17
32 16 1 1 4.0000000000000001E-03 5.3776427755281020E-17
8 64 1 1 4.0000000000000001E-03 1.8702487475374951E-17
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), and conservation error (HyPar::ConservationError).
Expected screen output:
HyPar - Parallel (MPI) version with 1 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 : 1
Domain size : 128 128
Processes along each dimension : 1 1
No. of ghosts pts : 3
No. of iter. : 1000
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-1stage
Spatial discretization scheme (parabolic ) : 2
Time Step : 4.000000E-03
Check for conservation : yes
Screen output iterations : 50
File output iterations : 50
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 : linear-advection-diffusion-reaction
Processor distribution for full grid object: 1 1
Initializing sparse grids...
Number of spatial dimensions: 2
Computing sparse grids dimensions...
Number of sparse grid domains in combination technique: 9
Computing processor decompositions...
Sparse Grids: Combination technique grids sizes and coefficients are:-
0: dim = ( 128 8 ), coeff = +1.00e+00, iproc = ( 1 1 )
1: dim = ( 64 16 ), coeff = +1.00e+00, iproc = ( 1 1 )
2: dim = ( 32 32 ), coeff = +1.00e+00, iproc = ( 1 1 )
3: dim = ( 16 64 ), coeff = +1.00e+00, iproc = ( 1 1 )
4: dim = ( 8 128 ), coeff = +1.00e+00, iproc = ( 1 1 )
5: dim = ( 64 8 ), coeff = -1.00e+00, iproc = ( 1 1 )
6: dim = ( 32 16 ), coeff = -1.00e+00, iproc = ( 1 1 )
7: dim = ( 16 32 ), coeff = -1.00e+00, iproc = ( 1 1 )
8: dim = ( 8 64 ), coeff = -1.00e+00, iproc = ( 1 1 )
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: -1.7631837830045516E-17
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.
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.
Initializing solvers.
Initializing physics. Model = "linear-advection-diffusion-reaction"
Reading physical model inputs from file "physics.inp".
Domain -1: Initializing physics data for model linear-advection-diffusion-reaction
Reading array from binary file advection.inp (Serial mode).
Domain -1: Initializing physics data for model linear-advection-diffusion-reaction
Reading array from binary file advection.inp (Serial mode).
Domain -1: Initializing physics data for model linear-advection-diffusion-reaction
Reading array from binary file advection.inp (Serial mode).
Domain -1: Initializing physics data for model linear-advection-diffusion-reaction
Reading array from binary file advection.inp (Serial mode).
Domain -1: Initializing physics data for model linear-advection-diffusion-reaction
Reading array from binary file advection.inp (Serial mode).
Domain -1: Initializing physics data for model linear-advection-diffusion-reaction
Reading array from binary file advection.inp (Serial mode).
Domain -1: Initializing physics data for model linear-advection-diffusion-reaction
Reading array from binary file advection.inp (Serial mode).
Domain -1: Initializing physics data for model linear-advection-diffusion-reaction
Reading array from binary file advection.inp (Serial mode).
Domain -1: Initializing physics data for model linear-advection-diffusion-reaction
Reading array from binary file advection.inp (Serial mode).
Interpolating initial solution to sparse grids domain 0.
Volume integral of the initial solution on sparse grids domain 0:
0: -6.7003694259604174E-17
Interpolating initial solution to sparse grids domain 1.
Volume integral of the initial solution on sparse grids domain 1:
0: 9.3740119833096713E-16
Interpolating initial solution to sparse grids domain 2.
Volume integral of the initial solution on sparse grids domain 2:
0: 2.0057740190981832E-16
Interpolating initial solution to sparse grids domain 3.
Volume integral of the initial solution on sparse grids domain 3:
0: -1.0386656812411132E-16
Interpolating initial solution to sparse grids domain 4.
Volume integral of the initial solution on sparse grids domain 4:
0: -7.5894152073985310E-18
Interpolating initial solution to sparse grids domain 5.
Volume integral of the initial solution on sparse grids domain 5:
0: 1.0191500421363742E-16
Interpolating initial solution to sparse grids domain 6.
Volume integral of the initial solution on sparse grids domain 6:
0: -1.2359904766334751E-16
Interpolating initial solution to sparse grids domain 7.
Volume integral of the initial solution on sparse grids domain 7:
0: -1.3010426069826053E-18
Interpolating initial solution to sparse grids domain 8.
Volume integral of the initial solution on sparse grids domain 8:
0: -2.7321894746634712E-17
Setting up time integration.
Solving in time (from 0 to 1000 iterations)
Writing solution file advection_field_0_00000.dat.
Writing solution file advection_field_1_00000.dat.
Writing solution file advection_field_2_00000.dat.
Writing solution file advection_field_3_00000.dat.
Writing solution file advection_field_4_00000.dat.
Writing solution file advection_field_5_00000.dat.
Writing solution file advection_field_6_00000.dat.
Writing solution file advection_field_7_00000.dat.
Writing solution file advection_field_8_00000.dat.
Writing solution file op_sg_0_00000.dat.
Writing solution file op_sg_1_00000.dat.
Writing solution file op_sg_2_00000.dat.
Writing solution file op_sg_3_00000.dat.
Writing solution file op_sg_4_00000.dat.
Writing solution file op_sg_5_00000.dat.
Writing solution file op_sg_6_00000.dat.
Writing solution file op_sg_7_00000.dat.
Writing solution file op_sg_8_00000.dat.
Writing solution file op_fg_00000.dat.
--
Iteration: 50, Time: 2.000e-01
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 2.2241E-02
Conservation loss: 1.1032E-15
--
Writing solution file advection_field_0_00001.dat.
Writing solution file advection_field_1_00001.dat.
Writing solution file advection_field_2_00001.dat.
Writing solution file advection_field_3_00001.dat.
Writing solution file advection_field_4_00001.dat.
Writing solution file advection_field_5_00001.dat.
Writing solution file advection_field_6_00001.dat.
Writing solution file advection_field_7_00001.dat.
Writing solution file advection_field_8_00001.dat.
Writing solution file op_sg_0_00001.dat.
Writing solution file op_sg_1_00001.dat.
Writing solution file op_sg_2_00001.dat.
Writing solution file op_sg_3_00001.dat.
Writing solution file op_sg_4_00001.dat.
Writing solution file op_sg_5_00001.dat.
Writing solution file op_sg_6_00001.dat.
Writing solution file op_sg_7_00001.dat.
Writing solution file op_sg_8_00001.dat.
Writing solution file op_fg_00001.dat.
--
Iteration: 100, Time: 4.000e-01
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 1.5906E-02
Conservation loss: 1.0999E-15
--
Writing solution file advection_field_0_00002.dat.
Writing solution file advection_field_1_00002.dat.
Writing solution file advection_field_2_00002.dat.
Writing solution file advection_field_3_00002.dat.
Writing solution file advection_field_4_00002.dat.
Writing solution file advection_field_5_00002.dat.
Writing solution file advection_field_6_00002.dat.
Writing solution file advection_field_7_00002.dat.
Writing solution file advection_field_8_00002.dat.
Writing solution file op_sg_0_00002.dat.
Writing solution file op_sg_1_00002.dat.
Writing solution file op_sg_2_00002.dat.
Writing solution file op_sg_3_00002.dat.
Writing solution file op_sg_4_00002.dat.
Writing solution file op_sg_5_00002.dat.
Writing solution file op_sg_6_00002.dat.
Writing solution file op_sg_7_00002.dat.
Writing solution file op_sg_8_00002.dat.
Writing solution file op_fg_00002.dat.
--
Iteration: 150, Time: 6.000e-01
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 1.0442E-02
Conservation loss: 9.5038E-16
--
Writing solution file advection_field_0_00003.dat.
Writing solution file advection_field_1_00003.dat.
Writing solution file advection_field_2_00003.dat.
Writing solution file advection_field_3_00003.dat.
Writing solution file advection_field_4_00003.dat.
Writing solution file advection_field_5_00003.dat.
Writing solution file advection_field_6_00003.dat.
Writing solution file advection_field_7_00003.dat.
Writing solution file advection_field_8_00003.dat.
Writing solution file op_sg_0_00003.dat.
Writing solution file op_sg_1_00003.dat.
Writing solution file op_sg_2_00003.dat.
Writing solution file op_sg_3_00003.dat.
Writing solution file op_sg_4_00003.dat.
Writing solution file op_sg_5_00003.dat.
Writing solution file op_sg_6_00003.dat.
Writing solution file op_sg_7_00003.dat.
Writing solution file op_sg_8_00003.dat.
Writing solution file op_fg_00003.dat.
--
Iteration: 200, Time: 8.000e-01
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 7.7374E-03
Conservation loss: 9.7977E-16
--
Writing solution file advection_field_0_00004.dat.
Writing solution file advection_field_1_00004.dat.
Writing solution file advection_field_2_00004.dat.
Writing solution file advection_field_3_00004.dat.
Writing solution file advection_field_4_00004.dat.
Writing solution file advection_field_5_00004.dat.
Writing solution file advection_field_6_00004.dat.
Writing solution file advection_field_7_00004.dat.
Writing solution file advection_field_8_00004.dat.
Writing solution file op_sg_0_00004.dat.
Writing solution file op_sg_1_00004.dat.
Writing solution file op_sg_2_00004.dat.
Writing solution file op_sg_3_00004.dat.
Writing solution file op_sg_4_00004.dat.
Writing solution file op_sg_5_00004.dat.
Writing solution file op_sg_6_00004.dat.
Writing solution file op_sg_7_00004.dat.
Writing solution file op_sg_8_00004.dat.
Writing solution file op_fg_00004.dat.
--
Iteration: 250, Time: 1.000e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 6.3623E-03
Conservation loss: 1.1215E-15
--
Writing solution file advection_field_0_00005.dat.
Writing solution file advection_field_1_00005.dat.
Writing solution file advection_field_2_00005.dat.
Writing solution file advection_field_3_00005.dat.
Writing solution file advection_field_4_00005.dat.
Writing solution file advection_field_5_00005.dat.
Writing solution file advection_field_6_00005.dat.
Writing solution file advection_field_7_00005.dat.
Writing solution file advection_field_8_00005.dat.
Writing solution file op_sg_0_00005.dat.
Writing solution file op_sg_1_00005.dat.
Writing solution file op_sg_2_00005.dat.
Writing solution file op_sg_3_00005.dat.
Writing solution file op_sg_4_00005.dat.
Writing solution file op_sg_5_00005.dat.
Writing solution file op_sg_6_00005.dat.
Writing solution file op_sg_7_00005.dat.
Writing solution file op_sg_8_00005.dat.
Writing solution file op_fg_00005.dat.
--
Iteration: 300, Time: 1.200e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 4.6742E-03
Conservation loss: 1.0754E-15
--
Writing solution file advection_field_0_00006.dat.
Writing solution file advection_field_1_00006.dat.
Writing solution file advection_field_2_00006.dat.
Writing solution file advection_field_3_00006.dat.
Writing solution file advection_field_4_00006.dat.
Writing solution file advection_field_5_00006.dat.
Writing solution file advection_field_6_00006.dat.
Writing solution file advection_field_7_00006.dat.
Writing solution file advection_field_8_00006.dat.
Writing solution file op_sg_0_00006.dat.
Writing solution file op_sg_1_00006.dat.
Writing solution file op_sg_2_00006.dat.
Writing solution file op_sg_3_00006.dat.
Writing solution file op_sg_4_00006.dat.
Writing solution file op_sg_5_00006.dat.
Writing solution file op_sg_6_00006.dat.
Writing solution file op_sg_7_00006.dat.
Writing solution file op_sg_8_00006.dat.
Writing solution file op_fg_00006.dat.
--
Iteration: 350, Time: 1.400e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 3.3827E-03
Conservation loss: 1.0024E-15
--
Writing solution file advection_field_0_00007.dat.
Writing solution file advection_field_1_00007.dat.
Writing solution file advection_field_2_00007.dat.
Writing solution file advection_field_3_00007.dat.
Writing solution file advection_field_4_00007.dat.
Writing solution file advection_field_5_00007.dat.
Writing solution file advection_field_6_00007.dat.
Writing solution file advection_field_7_00007.dat.
Writing solution file advection_field_8_00007.dat.
Writing solution file op_sg_0_00007.dat.
Writing solution file op_sg_1_00007.dat.
Writing solution file op_sg_2_00007.dat.
Writing solution file op_sg_3_00007.dat.
Writing solution file op_sg_4_00007.dat.
Writing solution file op_sg_5_00007.dat.
Writing solution file op_sg_6_00007.dat.
Writing solution file op_sg_7_00007.dat.
Writing solution file op_sg_8_00007.dat.
Writing solution file op_fg_00007.dat.
--
Iteration: 400, Time: 1.600e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 2.7570E-03
Conservation loss: 1.0048E-15
--
Writing solution file advection_field_0_00008.dat.
Writing solution file advection_field_1_00008.dat.
Writing solution file advection_field_2_00008.dat.
Writing solution file advection_field_3_00008.dat.
Writing solution file advection_field_4_00008.dat.
Writing solution file advection_field_5_00008.dat.
Writing solution file advection_field_6_00008.dat.
Writing solution file advection_field_7_00008.dat.
Writing solution file advection_field_8_00008.dat.
Writing solution file op_sg_0_00008.dat.
Writing solution file op_sg_1_00008.dat.
Writing solution file op_sg_2_00008.dat.
Writing solution file op_sg_3_00008.dat.
Writing solution file op_sg_4_00008.dat.
Writing solution file op_sg_5_00008.dat.
Writing solution file op_sg_6_00008.dat.
Writing solution file op_sg_7_00008.dat.
Writing solution file op_sg_8_00008.dat.
Writing solution file op_fg_00008.dat.
--
Iteration: 450, Time: 1.800e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 2.1329E-03
Conservation loss: 9.1568E-16
--
Writing solution file advection_field_0_00009.dat.
Writing solution file advection_field_1_00009.dat.
Writing solution file advection_field_2_00009.dat.
Writing solution file advection_field_3_00009.dat.
Writing solution file advection_field_4_00009.dat.
Writing solution file advection_field_5_00009.dat.
Writing solution file advection_field_6_00009.dat.
Writing solution file advection_field_7_00009.dat.
Writing solution file advection_field_8_00009.dat.
Writing solution file op_sg_0_00009.dat.
Writing solution file op_sg_1_00009.dat.
Writing solution file op_sg_2_00009.dat.
Writing solution file op_sg_3_00009.dat.
Writing solution file op_sg_4_00009.dat.
Writing solution file op_sg_5_00009.dat.
Writing solution file op_sg_6_00009.dat.
Writing solution file op_sg_7_00009.dat.
Writing solution file op_sg_8_00009.dat.
Writing solution file op_fg_00009.dat.
--
Iteration: 500, Time: 2.000e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 1.6837E-03
Conservation loss: 1.0634E-15
--
Writing solution file advection_field_0_00010.dat.
Writing solution file advection_field_1_00010.dat.
Writing solution file advection_field_2_00010.dat.
Writing solution file advection_field_3_00010.dat.
Writing solution file advection_field_4_00010.dat.
Writing solution file advection_field_5_00010.dat.
Writing solution file advection_field_6_00010.dat.
Writing solution file advection_field_7_00010.dat.
Writing solution file advection_field_8_00010.dat.
Writing solution file op_sg_0_00010.dat.
Writing solution file op_sg_1_00010.dat.
Writing solution file op_sg_2_00010.dat.
Writing solution file op_sg_3_00010.dat.
Writing solution file op_sg_4_00010.dat.
Writing solution file op_sg_5_00010.dat.
Writing solution file op_sg_6_00010.dat.
Writing solution file op_sg_7_00010.dat.
Writing solution file op_sg_8_00010.dat.
Writing solution file op_fg_00010.dat.
--
Iteration: 550, Time: 2.200e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 1.3778E-03
Conservation loss: 1.0364E-15
--
Writing solution file advection_field_0_00011.dat.
Writing solution file advection_field_1_00011.dat.
Writing solution file advection_field_2_00011.dat.
Writing solution file advection_field_3_00011.dat.
Writing solution file advection_field_4_00011.dat.
Writing solution file advection_field_5_00011.dat.
Writing solution file advection_field_6_00011.dat.
Writing solution file advection_field_7_00011.dat.
Writing solution file advection_field_8_00011.dat.
Writing solution file op_sg_0_00011.dat.
Writing solution file op_sg_1_00011.dat.
Writing solution file op_sg_2_00011.dat.
Writing solution file op_sg_3_00011.dat.
Writing solution file op_sg_4_00011.dat.
Writing solution file op_sg_5_00011.dat.
Writing solution file op_sg_6_00011.dat.
Writing solution file op_sg_7_00011.dat.
Writing solution file op_sg_8_00011.dat.
Writing solution file op_fg_00011.dat.
--
Iteration: 600, Time: 2.400e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 1.0731E-03
Conservation loss: 1.1319E-15
--
Writing solution file advection_field_0_00012.dat.
Writing solution file advection_field_1_00012.dat.
Writing solution file advection_field_2_00012.dat.
Writing solution file advection_field_3_00012.dat.
Writing solution file advection_field_4_00012.dat.
Writing solution file advection_field_5_00012.dat.
Writing solution file advection_field_6_00012.dat.
Writing solution file advection_field_7_00012.dat.
Writing solution file advection_field_8_00012.dat.
Writing solution file op_sg_0_00012.dat.
Writing solution file op_sg_1_00012.dat.
Writing solution file op_sg_2_00012.dat.
Writing solution file op_sg_3_00012.dat.
Writing solution file op_sg_4_00012.dat.
Writing solution file op_sg_5_00012.dat.
Writing solution file op_sg_6_00012.dat.
Writing solution file op_sg_7_00012.dat.
Writing solution file op_sg_8_00012.dat.
Writing solution file op_fg_00012.dat.
--
Iteration: 650, Time: 2.600e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 8.2563E-04
Conservation loss: 9.8764E-16
--
Writing solution file advection_field_0_00013.dat.
Writing solution file advection_field_1_00013.dat.
Writing solution file advection_field_2_00013.dat.
Writing solution file advection_field_3_00013.dat.
Writing solution file advection_field_4_00013.dat.
Writing solution file advection_field_5_00013.dat.
Writing solution file advection_field_6_00013.dat.
Writing solution file advection_field_7_00013.dat.
Writing solution file advection_field_8_00013.dat.
Writing solution file op_sg_0_00013.dat.
Writing solution file op_sg_1_00013.dat.
Writing solution file op_sg_2_00013.dat.
Writing solution file op_sg_3_00013.dat.
Writing solution file op_sg_4_00013.dat.
Writing solution file op_sg_5_00013.dat.
Writing solution file op_sg_6_00013.dat.
Writing solution file op_sg_7_00013.dat.
Writing solution file op_sg_8_00013.dat.
Writing solution file op_fg_00013.dat.
--
Iteration: 700, Time: 2.800e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 6.7015E-04
Conservation loss: 1.0517E-15
--
Writing solution file advection_field_0_00014.dat.
Writing solution file advection_field_1_00014.dat.
Writing solution file advection_field_2_00014.dat.
Writing solution file advection_field_3_00014.dat.
Writing solution file advection_field_4_00014.dat.
Writing solution file advection_field_5_00014.dat.
Writing solution file advection_field_6_00014.dat.
Writing solution file advection_field_7_00014.dat.
Writing solution file advection_field_8_00014.dat.
Writing solution file op_sg_0_00014.dat.
Writing solution file op_sg_1_00014.dat.
Writing solution file op_sg_2_00014.dat.
Writing solution file op_sg_3_00014.dat.
Writing solution file op_sg_4_00014.dat.
Writing solution file op_sg_5_00014.dat.
Writing solution file op_sg_6_00014.dat.
Writing solution file op_sg_7_00014.dat.
Writing solution file op_sg_8_00014.dat.
Writing solution file op_fg_00014.dat.
--
Iteration: 750, Time: 3.000e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 5.5489E-04
Conservation loss: 9.8038E-16
--
Writing solution file advection_field_0_00015.dat.
Writing solution file advection_field_1_00015.dat.
Writing solution file advection_field_2_00015.dat.
Writing solution file advection_field_3_00015.dat.
Writing solution file advection_field_4_00015.dat.
Writing solution file advection_field_5_00015.dat.
Writing solution file advection_field_6_00015.dat.
Writing solution file advection_field_7_00015.dat.
Writing solution file advection_field_8_00015.dat.
Writing solution file op_sg_0_00015.dat.
Writing solution file op_sg_1_00015.dat.
Writing solution file op_sg_2_00015.dat.
Writing solution file op_sg_3_00015.dat.
Writing solution file op_sg_4_00015.dat.
Writing solution file op_sg_5_00015.dat.
Writing solution file op_sg_6_00015.dat.
Writing solution file op_sg_7_00015.dat.
Writing solution file op_sg_8_00015.dat.
Writing solution file op_fg_00015.dat.
--
Iteration: 800, Time: 3.200e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 4.3893E-04
Conservation loss: 1.0627E-15
--
Writing solution file advection_field_0_00016.dat.
Writing solution file advection_field_1_00016.dat.
Writing solution file advection_field_2_00016.dat.
Writing solution file advection_field_3_00016.dat.
Writing solution file advection_field_4_00016.dat.
Writing solution file advection_field_5_00016.dat.
Writing solution file advection_field_6_00016.dat.
Writing solution file advection_field_7_00016.dat.
Writing solution file advection_field_8_00016.dat.
Writing solution file op_sg_0_00016.dat.
Writing solution file op_sg_1_00016.dat.
Writing solution file op_sg_2_00016.dat.
Writing solution file op_sg_3_00016.dat.
Writing solution file op_sg_4_00016.dat.
Writing solution file op_sg_5_00016.dat.
Writing solution file op_sg_6_00016.dat.
Writing solution file op_sg_7_00016.dat.
Writing solution file op_sg_8_00016.dat.
Writing solution file op_fg_00016.dat.
--
Iteration: 850, Time: 3.400e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 3.5386E-04
Conservation loss: 9.9935E-16
--
Writing solution file advection_field_0_00017.dat.
Writing solution file advection_field_1_00017.dat.
Writing solution file advection_field_2_00017.dat.
Writing solution file advection_field_3_00017.dat.
Writing solution file advection_field_4_00017.dat.
Writing solution file advection_field_5_00017.dat.
Writing solution file advection_field_6_00017.dat.
Writing solution file advection_field_7_00017.dat.
Writing solution file advection_field_8_00017.dat.
Writing solution file op_sg_0_00017.dat.
Writing solution file op_sg_1_00017.dat.
Writing solution file op_sg_2_00017.dat.
Writing solution file op_sg_3_00017.dat.
Writing solution file op_sg_4_00017.dat.
Writing solution file op_sg_5_00017.dat.
Writing solution file op_sg_6_00017.dat.
Writing solution file op_sg_7_00017.dat.
Writing solution file op_sg_8_00017.dat.
Writing solution file op_fg_00017.dat.
--
Iteration: 900, Time: 3.600e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 2.9575E-04
Conservation loss: 1.1153E-15
--
Writing solution file advection_field_0_00018.dat.
Writing solution file advection_field_1_00018.dat.
Writing solution file advection_field_2_00018.dat.
Writing solution file advection_field_3_00018.dat.
Writing solution file advection_field_4_00018.dat.
Writing solution file advection_field_5_00018.dat.
Writing solution file advection_field_6_00018.dat.
Writing solution file advection_field_7_00018.dat.
Writing solution file advection_field_8_00018.dat.
Writing solution file op_sg_0_00018.dat.
Writing solution file op_sg_1_00018.dat.
Writing solution file op_sg_2_00018.dat.
Writing solution file op_sg_3_00018.dat.
Writing solution file op_sg_4_00018.dat.
Writing solution file op_sg_5_00018.dat.
Writing solution file op_sg_6_00018.dat.
Writing solution file op_sg_7_00018.dat.
Writing solution file op_sg_8_00018.dat.
Writing solution file op_fg_00018.dat.
--
Iteration: 950, Time: 3.800e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 2.4308E-04
Conservation loss: 9.8256E-16
--
Writing solution file advection_field_0_00019.dat.
Writing solution file advection_field_1_00019.dat.
Writing solution file advection_field_2_00019.dat.
Writing solution file advection_field_3_00019.dat.
Writing solution file advection_field_4_00019.dat.
Writing solution file advection_field_5_00019.dat.
Writing solution file advection_field_6_00019.dat.
Writing solution file advection_field_7_00019.dat.
Writing solution file advection_field_8_00019.dat.
Writing solution file op_sg_0_00019.dat.
Writing solution file op_sg_1_00019.dat.
Writing solution file op_sg_2_00019.dat.
Writing solution file op_sg_3_00019.dat.
Writing solution file op_sg_4_00019.dat.
Writing solution file op_sg_5_00019.dat.
Writing solution file op_sg_6_00019.dat.
Writing solution file op_sg_7_00019.dat.
Writing solution file op_sg_8_00019.dat.
Writing solution file op_fg_00019.dat.
--
Iteration: 1000, Time: 4.000e+00
Max CFL: 3.196E-02, Max Diff. No.: 0.000E+00, Norm: 2.0336E-04
Conservation loss: 1.0652E-15
--
Writing solution file advection_field_0_00020.dat.
Writing solution file advection_field_1_00020.dat.
Writing solution file advection_field_2_00020.dat.
Writing solution file advection_field_3_00020.dat.
Writing solution file advection_field_4_00020.dat.
Writing solution file advection_field_5_00020.dat.
Writing solution file advection_field_6_00020.dat.
Writing solution file advection_field_7_00020.dat.
Writing solution file advection_field_8_00020.dat.
Writing solution file op_sg_0_00020.dat.
Writing solution file op_sg_1_00020.dat.
Writing solution file op_sg_2_00020.dat.
Writing solution file op_sg_3_00020.dat.
Writing solution file op_sg_4_00020.dat.
Writing solution file op_sg_5_00020.dat.
Writing solution file op_sg_6_00020.dat.
Writing solution file op_sg_7_00020.dat.
Writing solution file op_sg_8_00020.dat.
Writing solution file op_fg_00020.dat.
Completed time integration (Final time: 4.000000).
Conservation Errors:
9.5911234683498936E-17
Conservation Errors:
9.6971042307103517E-16
Conservation Errors:
2.7018318138338770E-16
Conservation Errors:
5.5402731014009277E-17
Conservation Errors:
2.6969529040576923E-17
Conservation Errors:
2.9438799488412659E-16
Conservation Errors:
5.3776427755281020E-17
Conservation Errors:
1.3595895242968226E-16
Conservation Errors:
1.8702487475374951E-17
Solver runtime (in seconds): 2.6698212999999999E+01
Total runtime (in seconds): 2.6852226999999999E+01
Deallocating arrays.
Finished.