HyPar  1.0
Finite-Difference Hyperbolic-Parabolic PDE Solver on Cartesian Grids
3D NUMA (Nonhydrostatic Unified Model of the Atmosphere) Equations - Rising Thermal Bubble

Location: hypar/Examples/3D/NUMA/RisingThermalBubble (This directory contains all the input files needed to run this case. If there is a Run.m, run it in MATLAB to quickly set up, run, and visualize the example).

Governing equations: 3D NUMA (Nonhydrostatic Unified Model of the Atmosphere) Equations (numa3d.h)

Domain: \(0 \le x,y,z < 1000\,{\rm m}\), "numa-nfbc" (_NO_FLUX_BC_) boundaries everywhere.

Reference:

  • Kelly, J. F., Giraldo, F. X., "Continuous and discontinuous Galerkin methods for a scalable three-dimensional nonhydrostatic atmospheric model: Limited-area mode", J. Comput. Phys., 231, 2012, pp. 7988-8008 (see section 5.1.2).
  • Giraldo, F. X., Kelly, J. F., Constantinescu, E. M., "Implicit-Explicit Formulations of a Three-Dimensional Nonhydrostatic Unified Model of the Atmosphere (NUMA)", SIAM J. Sci. Comput., 35 (5), 2013, pp. B1162-B1194 (see section 4.1).

Initial solution: A warm bubble in cool ambient atmosphere.

Other parameters (all dimensional quantities are in SI units):

  • Specific heat ratio \(\gamma = 1.4\) (Numa3D::gamma)
  • Universal gas constant \(R = 287.058\) (Numa3D::R)
  • Gravitational force per unit mass \(g = 9.8\) (Numa3D::g)
  • Angular speed of earth \(\Omega = 0\) (Numa3D::Omega)
  • Reference pressure (at zero altitude) \(P_{ref} = 100000\) (Numa3D::Pref)
  • Reference temperature (at zero altitude) \(T_{ref} = 300\) (Numa3D::Tref)

Numerical Method:

Input files required:

solver.inp

begin
ndims 3
nvars 5
size 101 101 101
iproc 2 2 2
ghost 3
n_iter 10000
time_scheme rk
time_scheme_type 44
hyp_space_scheme weno5
hyp_interp_type components
dt 0.02
screen_op_iter 50
file_op_iter 250
input_mode serial
ip_file_type binary
op_file_format binary
op_overwrite no
model numa3d
end

boundary.inp

6
numa-nfbc 0 1 0 0 0 1000.0 0.0 1000.0
numa-nfbc 0 -1 0 0 0 1000.0 0.0 1000.0
numa-nfbc 1 1 0.0 1000.0 0 0 0.0 1000.0
numa-nfbc 1 -1 0.0 1000.0 0 0 0.0 1000.0
numa-nfbc 2 1 0.0 1000.0 0 1000.0 0.0 0.0
numa-nfbc 2 -1 0.0 1000.0 0 1000.0 0.0 0.0

physics.inp

begin
gamma 1.4
R 287.058
g 9.8
Omega 0.0
Pref 100000.0
Tref 300.0
end

weno.inp (optional)

begin
mapped 0
borges 0
yc 0
no_limiting 1
epsilon 0.000001
p 2.0
rc 0.3
xi 0.001
end

To generate initial.inp (initial solution), compile and run the following code in the run directory.

/*
This code generates the initial solution for the
3D rising thermal bubble case for the non-hydrostatic
unified model of the atmosphere (NUMA).
*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <string.h>
double raiseto(double x, double a)
{
return(exp(a*log(x)));
}
int main()
{
int NI,NJ,NK,ndims;
char ip_file_type[50];
FILE *in;
strcpy (ip_file_type,"ascii");
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");
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);
fscanf(in,"%d",&NK);
} 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 != 3) {
printf("ndims is not 3 in solver.inp. this code is to generate 3D exact solution\n");
return(0);
}
printf("Grid:\t\t\t%d x %d x %d\n", NI, NJ, NK);
/* read in zero-altitude pressure and temperature */
double P0 = 100000.0;
double T0 = 300.0;
double g = 9.8;
double R = 287.058;
double gamma = 1.4;
printf("Reading file \"physics.inp\"...\n");
in = fopen("physics.inp","r");
if (!in) printf ("Error: Input file \"physics.inp\" not found. Default values will be used.\n");
else {
char word[500];
fscanf(in,"%s",word);
if (!strcmp(word, "begin")){
while (strcmp(word, "end")){
fscanf(in,"%s",word);
if (!strcmp(word, "Pref" )) fscanf(in,"%lf",&P0);
else if (!strcmp(word, "Tref" )) fscanf(in,"%lf",&T0);
else if (!strcmp(word, "gamma")) fscanf(in,"%lf",&gamma);
else if (!strcmp(word, "g" )) fscanf(in,"%lf",&g);
else if (!strcmp(word, "R" )) fscanf(in,"%lf",&R);
}
} else printf("Error: Illegal format in physics.inp. Crash and burn!\n");
}
fclose(in);
double inv_gamma_m1 = 1.0 / (gamma-1.0);
double Cp = gamma * inv_gamma_m1 * R;
/* Define the domain */
double xmin, xmax, ymin, ymax, zmin, zmax;
xmin = 0.0;
xmax = 1000.0;
ymin = 0.0;
ymax = 1000.0;
zmin = 0.0;
zmax = 1000.0 ;
double Lx = xmax - xmin;
double Ly = ymax - ymin;
double Lz = zmax - zmin;
int i,j,k;
double dx = Lx / ((double)NI-1);
double dy = Ly / ((double)NJ-1);
double dz = Lz / ((double)NK-1);
/* Initial perturbation center */
double xc = 500;
double yc = 500;
double zc = 260;
/* initial perturbation parameters */
double pi = 4.0*atan(1.0);
double rc = 250.0;
double *x, *y, *z, *U;
FILE *out;
x = (double*) calloc (NI , sizeof(double));
y = (double*) calloc (NJ , sizeof(double));
z = (double*) calloc (NK , sizeof(double));
U = (double*) calloc (5*NI*NJ*NK, sizeof(double));
for (i = 0; i < NI; i++){
for (j = 0; j < NJ; j++){
for (k = 0; k < NK; k++){
x[i] = xmin + i*dx;
y[j] = ymin + j*dy;
z[k] = zmin + k*dz;
int p = i + NI*j + NI*NJ*k;
/* temperature peturbation */
double r = sqrt((x[i]-xc)*(x[i]-xc)+(y[j]-yc)*(y[j]-yc)+(z[k]-zc)*(z[k]-zc));
double dtheta = (r>rc ? 0.0 : (0.5*(1.0+cos(pi*r/rc))) );
double theta = T0 + dtheta; /* temperature potential */
double Pexner = 1.0 - (g/(Cp*T0))*z[k]; /* Exner pressure */
double rho = (P0/(R*theta)) * raiseto(Pexner,inv_gamma_m1); /* density */
double theta_ref = T0; /* reference temperature potential */
double Pexner_ref = 1.0 - (g/(Cp*T0))*z[k]; /* reference Exner pressure */
double rho_ref = (P0/(R*theta_ref)) * raiseto(Pexner_ref,inv_gamma_m1); /* reference density */
U[5*p+0] = rho - rho_ref;
U[5*p+1] = 0.0;
U[5*p+2] = 0.0;
U[5*p+3] = 0.0;
U[5*p+4] = (rho*theta) - (rho_ref*theta_ref);
}
}
}
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,"%1.16E ",x[i]);
fprintf(out,"\n");
for (j = 0; j < NJ; j++) fprintf(out,"%1.16E ",y[j]);
fprintf(out,"\n");
for (k = 0; k < NK; k++) fprintf(out,"%1.16E ",z[k]);
fprintf(out,"\n");
for (k = 0; k < NK; k++) {
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = i + NI*j + NI*NJ*k;
fprintf(out,"%1.16E ",U[5*p+0]);
}
}
}
fprintf(out,"\n");
for (k = 0; k < NK; k++) {
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = i + NI*j + NI*NJ*k;
fprintf(out,"%1.16E ",U[5*p+1]);
}
}
}
fprintf(out,"\n");
for (k = 0; k < NK; k++) {
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = i + NI*j + NI*NJ*k;
fprintf(out,"%1.16E ",U[5*p+2]);
}
}
}
fprintf(out,"\n");
for (k = 0; k < NK; k++) {
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = i + NI*j + NI*NJ*k;
fprintf(out,"%1.16E ",U[5*p+3]);
}
}
}
fprintf(out,"\n");
for (k = 0; k < NK; k++) {
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = i + NI*j + NI*NJ*k;
fprintf(out,"%1.16E ",U[5*p+4]);
}
}
}
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(z,sizeof(double),NK,out);
fwrite(U,sizeof(double),5*NI*NJ*NK,out);
fclose(out);
}
free(x);
free(y);
free(z);
free(U);
return(0);
}

Output:

Note that iproc is set to 

  2 2 2

in solver.inp (i.e., 2 processors along x, 2 processors along y, and 2 processor along z). Thus, this example should be run with 8 MPI ranks (or change iproc).

After running the code, there should be 41 output files op_00000.bin, op_00001.bin, ... op_00040.bin; the first one is the solution at \(t=0\) and the final one is the solution at \(t=200\,{\rm s}\). Since HyPar::op_overwrite is set to no in solver.inp, separate files are written for solutions at each output time. All the files are binary (HyPar::op_file_format is set to binary in solver.inp).

The code hypar/Extras/BinaryToTecplot.c can be used to convert the binary solution files to 3D Tecplot files that can be visualized in any software supporting the Tecplot format. Similarly, the code hypar/Extras/BinaryToText.c can be used to convert the binary solution files to ASCII text files with the following data layout: the first three columns are grid indices, the next three columns are x, y, and z coordinates, and the remaining columns are the solution components ( \(\rho{'}, \rho u, \rho v, \rho w, \Theta{'}\)).

The following figure shows the density pertubation iso-surface for the initial and final solutions (plotted in VisIt):

Solution_3DNUMA_Bubble3D.png

The code hypar/Extras/ExtractSlice.c can be used to extract a slice along at a specified location along any of the dimensions (Note: make a subdirectory slices before using this). It will write out 2D slice solutions in binary format in the slices subdirectory with the same names as the original solution files. The codes hypar/Extras/BinaryToTecplot.c and hypar/Extras/BinaryToText.c can then be used to convert these to text or Tecplot formats for visualization. (Note: A copy of the original solver.inp is required in the slices subdirectory with ndims reduced by 1, and size and iproc with the appropriate component removed such that they have 2 components.)

The following figure shows the density pertubation along a slice at \(y=500\,{\rm m}\) (plotted in VisIt):

Solution_3DNUMA_Bubble.gif

Expected screen output:

HyPar - Parallel (MPI) version with 8 processes
Reading solver inputs from file "solver.inp".
No. of dimensions : 3
No. of variables : 5
Domain size : 101 101 101
Processes along each dimension : 2 2 2
No. of ghosts pts : 3
No. of iter. : 10000
Restart iteration : 0
Time integration scheme : rk (44)
Spatial discretization scheme (hyperbolic) : weno5
Split hyperbolic flux term? : no
Interpolation type for hyperbolic term : components
Spatial discretization type (parabolic ) : nonconservative-1stage
Spatial discretization scheme (parabolic ) : 2
Time Step : 2.000000E-02
Check for conservation : no
Screen output iterations : 50
File output iterations : 250
Initial solution file type : binary
Initial solution read mode : serial
Solution file write mode : serial
Solution file format : binary
Overwrite solution file : no
Physical model : numa3d
Partitioning domain.
Allocating data arrays.
Reading array from binary file initial.inp (Serial mode).
Volume integral of the initial solution:
0: -4.8546502532699233E+04
1: 0.0000000000000000E+00
2: 0.0000000000000000E+00
3: 0.0000000000000000E+00
4: 9.0358298621140420E-07
Reading boundary conditions from "boundary.inp".
Boundary numa-nfbc: Along dimension 0 and face +1
Boundary numa-nfbc: Along dimension 0 and face -1
Boundary numa-nfbc: Along dimension 1 and face +1
Boundary numa-nfbc: Along dimension 1 and face -1
Boundary numa-nfbc: Along dimension 2 and face +1
Boundary numa-nfbc: Along dimension 2 and face -1
6 boundary condition(s) read.
Initializing solvers.
Reading WENO parameters from weno.inp.
Initializing physics. Model = "numa3d"
Reading physical model inputs from file "physics.inp".
Setting up time integration.
Solving in time (from 0 to 10000 iterations)
Writing solution file op_00000.bin.
Iteration: 50 Time: 1.000E+00 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4953E-05
Iteration: 100 Time: 2.000E+00 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4495E-05
Iteration: 150 Time: 3.000E+00 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4759E-05
Iteration: 200 Time: 4.000E+00 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4678E-05
Iteration: 250 Time: 5.000E+00 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4703E-05
Writing solution file op_00001.bin.
Iteration: 300 Time: 6.000E+00 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.5033E-05
Iteration: 350 Time: 7.000E+00 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4666E-05
Iteration: 400 Time: 8.000E+00 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4306E-05
Iteration: 450 Time: 9.000E+00 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4542E-05
Iteration: 500 Time: 1.000E+01 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4360E-05
Writing solution file op_00002.bin.
Iteration: 550 Time: 1.100E+01 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4262E-05
Iteration: 600 Time: 1.200E+01 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4773E-05
Iteration: 650 Time: 1.300E+01 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4519E-05
Iteration: 700 Time: 1.400E+01 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.3980E-05
Iteration: 750 Time: 1.500E+01 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4245E-05
Writing solution file op_00003.bin.
Iteration: 800 Time: 1.600E+01 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4249E-05
Iteration: 850 Time: 1.700E+01 Max CFL: 6.945E-01 Max Diff. No.: -1.000E+00 Norm: 5.4642E-05
Iteration: 900 Time: 1.800E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4647E-05
Iteration: 950 Time: 1.900E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4413E-05
Iteration: 1000 Time: 2.000E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4371E-05
Writing solution file op_00004.bin.
Iteration: 1050 Time: 2.100E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4682E-05
Iteration: 1100 Time: 2.200E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4832E-05
Iteration: 1150 Time: 2.300E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4748E-05
Iteration: 1200 Time: 2.400E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4710E-05
Iteration: 1250 Time: 2.500E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4875E-05
Writing solution file op_00005.bin.
Iteration: 1300 Time: 2.600E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4741E-05
Iteration: 1350 Time: 2.700E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4929E-05
Iteration: 1400 Time: 2.800E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4767E-05
Iteration: 1450 Time: 2.900E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4848E-05
Iteration: 1500 Time: 3.000E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4885E-05
Writing solution file op_00006.bin.
Iteration: 1550 Time: 3.100E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4911E-05
Iteration: 1600 Time: 3.200E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4797E-05
Iteration: 1650 Time: 3.300E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4526E-05
Iteration: 1700 Time: 3.400E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4808E-05
Iteration: 1750 Time: 3.500E+01 Max CFL: 6.946E-01 Max Diff. No.: -1.000E+00 Norm: 5.4788E-05
Writing solution file op_00007.bin.
Iteration: 1800 Time: 3.600E+01 Max CFL: 6.947E-01 Max Diff. No.: -1.000E+00 Norm: 5.4370E-05
Iteration: 1850 Time: 3.700E+01 Max CFL: 6.947E-01 Max Diff. No.: -1.000E+00 Norm: 5.4635E-05
Iteration: 1900 Time: 3.800E+01 Max CFL: 6.947E-01 Max Diff. No.: -1.000E+00 Norm: 5.5018E-05
Iteration: 1950 Time: 3.900E+01 Max CFL: 6.947E-01 Max Diff. No.: -1.000E+00 Norm: 5.4628E-05
Iteration: 2000 Time: 4.000E+01 Max CFL: 6.947E-01 Max Diff. No.: -1.000E+00 Norm: 5.4511E-05
Writing solution file op_00008.bin.
Iteration: 2050 Time: 4.100E+01 Max CFL: 6.947E-01 Max Diff. No.: -1.000E+00 Norm: 5.4561E-05
Iteration: 2100 Time: 4.200E+01 Max CFL: 6.947E-01 Max Diff. No.: -1.000E+00 Norm: 5.4580E-05
Iteration: 2150 Time: 4.300E+01 Max CFL: 6.947E-01 Max Diff. No.: -1.000E+00 Norm: 5.4623E-05
Iteration: 2200 Time: 4.400E+01 Max CFL: 6.947E-01 Max Diff. No.: -1.000E+00 Norm: 5.5212E-05
Iteration: 2250 Time: 4.500E+01 Max CFL: 6.947E-01 Max Diff. No.: -1.000E+00 Norm: 5.4662E-05
Writing solution file op_00009.bin.
Iteration: 2300 Time: 4.600E+01 Max CFL: 6.948E-01 Max Diff. No.: -1.000E+00 Norm: 5.4624E-05
Iteration: 2350 Time: 4.700E+01 Max CFL: 6.948E-01 Max Diff. No.: -1.000E+00 Norm: 5.5041E-05
Iteration: 2400 Time: 4.800E+01 Max CFL: 6.948E-01 Max Diff. No.: -1.000E+00 Norm: 5.4923E-05
Iteration: 2450 Time: 4.900E+01 Max CFL: 6.948E-01 Max Diff. No.: -1.000E+00 Norm: 5.5057E-05
Iteration: 2500 Time: 5.000E+01 Max CFL: 6.949E-01 Max Diff. No.: -1.000E+00 Norm: 5.5360E-05
Writing solution file op_00010.bin.
Iteration: 2550 Time: 5.100E+01 Max CFL: 6.949E-01 Max Diff. No.: -1.000E+00 Norm: 5.5153E-05
Iteration: 2600 Time: 5.200E+01 Max CFL: 6.949E-01 Max Diff. No.: -1.000E+00 Norm: 5.5086E-05
Iteration: 2650 Time: 5.300E+01 Max CFL: 6.950E-01 Max Diff. No.: -1.000E+00 Norm: 5.5272E-05
Iteration: 2700 Time: 5.400E+01 Max CFL: 6.950E-01 Max Diff. No.: -1.000E+00 Norm: 5.5442E-05
Iteration: 2750 Time: 5.500E+01 Max CFL: 6.950E-01 Max Diff. No.: -1.000E+00 Norm: 5.5286E-05
Writing solution file op_00011.bin.
Iteration: 2800 Time: 5.600E+01 Max CFL: 6.950E-01 Max Diff. No.: -1.000E+00 Norm: 5.5432E-05
Iteration: 2850 Time: 5.700E+01 Max CFL: 6.951E-01 Max Diff. No.: -1.000E+00 Norm: 5.5426E-05
Iteration: 2900 Time: 5.800E+01 Max CFL: 6.951E-01 Max Diff. No.: -1.000E+00 Norm: 5.5098E-05
Iteration: 2950 Time: 5.900E+01 Max CFL: 6.951E-01 Max Diff. No.: -1.000E+00 Norm: 5.5474E-05
Iteration: 3000 Time: 6.000E+01 Max CFL: 6.952E-01 Max Diff. No.: -1.000E+00 Norm: 5.5585E-05
Writing solution file op_00012.bin.
Iteration: 3050 Time: 6.100E+01 Max CFL: 6.952E-01 Max Diff. No.: -1.000E+00 Norm: 5.5692E-05
Iteration: 3100 Time: 6.200E+01 Max CFL: 6.952E-01 Max Diff. No.: -1.000E+00 Norm: 5.5517E-05
Iteration: 3150 Time: 6.300E+01 Max CFL: 6.952E-01 Max Diff. No.: -1.000E+00 Norm: 5.5282E-05
Iteration: 3200 Time: 6.400E+01 Max CFL: 6.953E-01 Max Diff. No.: -1.000E+00 Norm: 5.5364E-05
Iteration: 3250 Time: 6.500E+01 Max CFL: 6.953E-01 Max Diff. No.: -1.000E+00 Norm: 5.5511E-05
Writing solution file op_00013.bin.
Iteration: 3300 Time: 6.600E+01 Max CFL: 6.953E-01 Max Diff. No.: -1.000E+00 Norm: 5.5704E-05
Iteration: 3350 Time: 6.700E+01 Max CFL: 6.954E-01 Max Diff. No.: -1.000E+00 Norm: 5.5741E-05
Iteration: 3400 Time: 6.800E+01 Max CFL: 6.954E-01 Max Diff. No.: -1.000E+00 Norm: 5.5420E-05
Iteration: 3450 Time: 6.900E+01 Max CFL: 6.954E-01 Max Diff. No.: -1.000E+00 Norm: 5.5698E-05
Iteration: 3500 Time: 7.000E+01 Max CFL: 6.954E-01 Max Diff. No.: -1.000E+00 Norm: 5.5820E-05
Writing solution file op_00014.bin.
Iteration: 3550 Time: 7.100E+01 Max CFL: 6.955E-01 Max Diff. No.: -1.000E+00 Norm: 5.5885E-05
Iteration: 3600 Time: 7.200E+01 Max CFL: 6.955E-01 Max Diff. No.: -1.000E+00 Norm: 5.6114E-05
Iteration: 3650 Time: 7.300E+01 Max CFL: 6.955E-01 Max Diff. No.: -1.000E+00 Norm: 5.6167E-05
Iteration: 3700 Time: 7.400E+01 Max CFL: 6.955E-01 Max Diff. No.: -1.000E+00 Norm: 5.6050E-05
Iteration: 3750 Time: 7.500E+01 Max CFL: 6.956E-01 Max Diff. No.: -1.000E+00 Norm: 5.6226E-05
Writing solution file op_00015.bin.
Iteration: 3800 Time: 7.600E+01 Max CFL: 6.956E-01 Max Diff. No.: -1.000E+00 Norm: 5.6689E-05
Iteration: 3850 Time: 7.700E+01 Max CFL: 6.956E-01 Max Diff. No.: -1.000E+00 Norm: 5.6495E-05
Iteration: 3900 Time: 7.800E+01 Max CFL: 6.956E-01 Max Diff. No.: -1.000E+00 Norm: 5.6878E-05
Iteration: 3950 Time: 7.900E+01 Max CFL: 6.957E-01 Max Diff. No.: -1.000E+00 Norm: 5.7137E-05
Iteration: 4000 Time: 8.000E+01 Max CFL: 6.957E-01 Max Diff. No.: -1.000E+00 Norm: 5.6831E-05
Writing solution file op_00016.bin.
Iteration: 4050 Time: 8.100E+01 Max CFL: 6.957E-01 Max Diff. No.: -1.000E+00 Norm: 5.7122E-05
Iteration: 4100 Time: 8.200E+01 Max CFL: 6.957E-01 Max Diff. No.: -1.000E+00 Norm: 5.7636E-05
Iteration: 4150 Time: 8.300E+01 Max CFL: 6.958E-01 Max Diff. No.: -1.000E+00 Norm: 5.7509E-05
Iteration: 4200 Time: 8.400E+01 Max CFL: 6.958E-01 Max Diff. No.: -1.000E+00 Norm: 5.7724E-05
Iteration: 4250 Time: 8.500E+01 Max CFL: 6.958E-01 Max Diff. No.: -1.000E+00 Norm: 5.7896E-05
Writing solution file op_00017.bin.
Iteration: 4300 Time: 8.600E+01 Max CFL: 6.958E-01 Max Diff. No.: -1.000E+00 Norm: 5.7588E-05
Iteration: 4350 Time: 8.700E+01 Max CFL: 6.959E-01 Max Diff. No.: -1.000E+00 Norm: 5.7981E-05
Iteration: 4400 Time: 8.800E+01 Max CFL: 6.959E-01 Max Diff. No.: -1.000E+00 Norm: 5.8715E-05
Iteration: 4450 Time: 8.900E+01 Max CFL: 6.959E-01 Max Diff. No.: -1.000E+00 Norm: 5.8283E-05
Iteration: 4500 Time: 9.000E+01 Max CFL: 6.959E-01 Max Diff. No.: -1.000E+00 Norm: 5.8084E-05
Writing solution file op_00018.bin.
Iteration: 4550 Time: 9.100E+01 Max CFL: 6.959E-01 Max Diff. No.: -1.000E+00 Norm: 5.8909E-05
Iteration: 4600 Time: 9.200E+01 Max CFL: 6.960E-01 Max Diff. No.: -1.000E+00 Norm: 5.8603E-05
Iteration: 4650 Time: 9.300E+01 Max CFL: 6.960E-01 Max Diff. No.: -1.000E+00 Norm: 5.8613E-05
Iteration: 4700 Time: 9.400E+01 Max CFL: 6.960E-01 Max Diff. No.: -1.000E+00 Norm: 5.9079E-05
Iteration: 4750 Time: 9.500E+01 Max CFL: 6.960E-01 Max Diff. No.: -1.000E+00 Norm: 5.8911E-05
Writing solution file op_00019.bin.
Iteration: 4800 Time: 9.600E+01 Max CFL: 6.960E-01 Max Diff. No.: -1.000E+00 Norm: 5.9094E-05
Iteration: 4850 Time: 9.700E+01 Max CFL: 6.961E-01 Max Diff. No.: -1.000E+00 Norm: 5.9537E-05
Iteration: 4900 Time: 9.800E+01 Max CFL: 6.961E-01 Max Diff. No.: -1.000E+00 Norm: 5.9603E-05
Iteration: 4950 Time: 9.900E+01 Max CFL: 6.961E-01 Max Diff. No.: -1.000E+00 Norm: 5.9735E-05
Iteration: 5000 Time: 1.000E+02 Max CFL: 6.961E-01 Max Diff. No.: -1.000E+00 Norm: 6.0152E-05
Writing solution file op_00020.bin.
Iteration: 5050 Time: 1.010E+02 Max CFL: 6.961E-01 Max Diff. No.: -1.000E+00 Norm: 6.0315E-05
Iteration: 5100 Time: 1.020E+02 Max CFL: 6.962E-01 Max Diff. No.: -1.000E+00 Norm: 6.0030E-05
Iteration: 5150 Time: 1.030E+02 Max CFL: 6.962E-01 Max Diff. No.: -1.000E+00 Norm: 6.0672E-05
Iteration: 5200 Time: 1.040E+02 Max CFL: 6.962E-01 Max Diff. No.: -1.000E+00 Norm: 6.1104E-05
Iteration: 5250 Time: 1.050E+02 Max CFL: 6.962E-01 Max Diff. No.: -1.000E+00 Norm: 6.1356E-05
Writing solution file op_00021.bin.
Iteration: 5300 Time: 1.060E+02 Max CFL: 6.962E-01 Max Diff. No.: -1.000E+00 Norm: 6.1546E-05
Iteration: 5350 Time: 1.070E+02 Max CFL: 6.963E-01 Max Diff. No.: -1.000E+00 Norm: 6.1557E-05
Iteration: 5400 Time: 1.080E+02 Max CFL: 6.963E-01 Max Diff. No.: -1.000E+00 Norm: 6.1961E-05
Iteration: 5450 Time: 1.090E+02 Max CFL: 6.963E-01 Max Diff. No.: -1.000E+00 Norm: 6.2611E-05
Iteration: 5500 Time: 1.100E+02 Max CFL: 6.963E-01 Max Diff. No.: -1.000E+00 Norm: 6.2876E-05
Writing solution file op_00022.bin.
Iteration: 5550 Time: 1.110E+02 Max CFL: 6.963E-01 Max Diff. No.: -1.000E+00 Norm: 6.3017E-05
Iteration: 5600 Time: 1.120E+02 Max CFL: 6.963E-01 Max Diff. No.: -1.000E+00 Norm: 6.3186E-05
Iteration: 5650 Time: 1.130E+02 Max CFL: 6.964E-01 Max Diff. No.: -1.000E+00 Norm: 6.3484E-05
Iteration: 5700 Time: 1.140E+02 Max CFL: 6.964E-01 Max Diff. No.: -1.000E+00 Norm: 6.3950E-05
Iteration: 5750 Time: 1.150E+02 Max CFL: 6.964E-01 Max Diff. No.: -1.000E+00 Norm: 6.4172E-05
Writing solution file op_00023.bin.
Iteration: 5800 Time: 1.160E+02 Max CFL: 6.964E-01 Max Diff. No.: -1.000E+00 Norm: 6.4638E-05
Iteration: 5850 Time: 1.170E+02 Max CFL: 6.964E-01 Max Diff. No.: -1.000E+00 Norm: 6.4794E-05
Iteration: 5900 Time: 1.180E+02 Max CFL: 6.964E-01 Max Diff. No.: -1.000E+00 Norm: 6.5130E-05
Iteration: 5950 Time: 1.190E+02 Max CFL: 6.965E-01 Max Diff. No.: -1.000E+00 Norm: 6.5188E-05
Iteration: 6000 Time: 1.200E+02 Max CFL: 6.965E-01 Max Diff. No.: -1.000E+00 Norm: 6.5597E-05
Writing solution file op_00024.bin.
Iteration: 6050 Time: 1.210E+02 Max CFL: 6.965E-01 Max Diff. No.: -1.000E+00 Norm: 6.6009E-05
Iteration: 6100 Time: 1.220E+02 Max CFL: 6.965E-01 Max Diff. No.: -1.000E+00 Norm: 6.6752E-05
Iteration: 6150 Time: 1.230E+02 Max CFL: 6.965E-01 Max Diff. No.: -1.000E+00 Norm: 6.6837E-05
Iteration: 6200 Time: 1.240E+02 Max CFL: 6.965E-01 Max Diff. No.: -1.000E+00 Norm: 6.6672E-05
Iteration: 6250 Time: 1.250E+02 Max CFL: 6.965E-01 Max Diff. No.: -1.000E+00 Norm: 6.7220E-05
Writing solution file op_00025.bin.
Iteration: 6300 Time: 1.260E+02 Max CFL: 6.965E-01 Max Diff. No.: -1.000E+00 Norm: 6.8083E-05
Iteration: 6350 Time: 1.270E+02 Max CFL: 6.966E-01 Max Diff. No.: -1.000E+00 Norm: 6.8253E-05
Iteration: 6400 Time: 1.280E+02 Max CFL: 6.966E-01 Max Diff. No.: -1.000E+00 Norm: 6.8632E-05
Iteration: 6450 Time: 1.290E+02 Max CFL: 6.966E-01 Max Diff. No.: -1.000E+00 Norm: 6.9186E-05
Iteration: 6500 Time: 1.300E+02 Max CFL: 6.966E-01 Max Diff. No.: -1.000E+00 Norm: 6.9390E-05
Writing solution file op_00026.bin.
Iteration: 6550 Time: 1.310E+02 Max CFL: 6.966E-01 Max Diff. No.: -1.000E+00 Norm: 6.9888E-05
Iteration: 6600 Time: 1.320E+02 Max CFL: 6.966E-01 Max Diff. No.: -1.000E+00 Norm: 7.0484E-05
Iteration: 6650 Time: 1.330E+02 Max CFL: 6.966E-01 Max Diff. No.: -1.000E+00 Norm: 7.0875E-05
Iteration: 6700 Time: 1.340E+02 Max CFL: 6.966E-01 Max Diff. No.: -1.000E+00 Norm: 7.1308E-05
Iteration: 6750 Time: 1.350E+02 Max CFL: 6.966E-01 Max Diff. No.: -1.000E+00 Norm: 7.2240E-05
Writing solution file op_00027.bin.
Iteration: 6800 Time: 1.360E+02 Max CFL: 6.966E-01 Max Diff. No.: -1.000E+00 Norm: 7.2301E-05
Iteration: 6850 Time: 1.370E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 7.2669E-05
Iteration: 6900 Time: 1.380E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 7.3460E-05
Iteration: 6950 Time: 1.390E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 7.4033E-05
Iteration: 7000 Time: 1.400E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 7.4315E-05
Writing solution file op_00028.bin.
Iteration: 7050 Time: 1.410E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 7.4963E-05
Iteration: 7100 Time: 1.420E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 7.5325E-05
Iteration: 7150 Time: 1.430E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 7.6003E-05
Iteration: 7200 Time: 1.440E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 7.6399E-05
Iteration: 7250 Time: 1.450E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 7.6972E-05
Writing solution file op_00029.bin.
Iteration: 7300 Time: 1.460E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 7.7347E-05
Iteration: 7350 Time: 1.470E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 7.8046E-05
Iteration: 7400 Time: 1.480E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 7.8666E-05
Iteration: 7450 Time: 1.490E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 7.9052E-05
Iteration: 7500 Time: 1.500E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 7.9451E-05
Writing solution file op_00030.bin.
Iteration: 7550 Time: 1.510E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 8.0081E-05
Iteration: 7600 Time: 1.520E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 8.0821E-05
Iteration: 7650 Time: 1.530E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 8.1456E-05
Iteration: 7700 Time: 1.540E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 8.1905E-05
Iteration: 7750 Time: 1.550E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 8.2437E-05
Writing solution file op_00031.bin.
Iteration: 7800 Time: 1.560E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 8.3077E-05
Iteration: 7850 Time: 1.570E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 8.3719E-05
Iteration: 7900 Time: 1.580E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 8.4564E-05
Iteration: 7950 Time: 1.590E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 8.5104E-05
Iteration: 8000 Time: 1.600E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 8.6071E-05
Writing solution file op_00032.bin.
Iteration: 8050 Time: 1.610E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 8.6568E-05
Iteration: 8100 Time: 1.620E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 8.6958E-05
Iteration: 8150 Time: 1.630E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 8.7743E-05
Iteration: 8200 Time: 1.640E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 8.8827E-05
Iteration: 8250 Time: 1.650E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 8.9405E-05
Writing solution file op_00033.bin.
Iteration: 8300 Time: 1.660E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.0245E-05
Iteration: 8350 Time: 1.670E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.0852E-05
Iteration: 8400 Time: 1.680E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.1315E-05
Iteration: 8450 Time: 1.690E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.2100E-05
Iteration: 8500 Time: 1.700E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.3237E-05
Writing solution file op_00034.bin.
Iteration: 8550 Time: 1.710E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.3599E-05
Iteration: 8600 Time: 1.720E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.4381E-05
Iteration: 8650 Time: 1.730E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.5145E-05
Iteration: 8700 Time: 1.740E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.5562E-05
Iteration: 8750 Time: 1.750E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.6308E-05
Writing solution file op_00035.bin.
Iteration: 8800 Time: 1.760E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.7419E-05
Iteration: 8850 Time: 1.770E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.7910E-05
Iteration: 8900 Time: 1.780E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.8511E-05
Iteration: 8950 Time: 1.790E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.9374E-05
Iteration: 9000 Time: 1.800E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 9.9868E-05
Writing solution file op_00036.bin.
Iteration: 9050 Time: 1.810E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 1.0059E-04
Iteration: 9100 Time: 1.820E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 1.0164E-04
Iteration: 9150 Time: 1.830E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 1.0232E-04
Iteration: 9200 Time: 1.840E+02 Max CFL: 6.968E-01 Max Diff. No.: -1.000E+00 Norm: 1.0282E-04
Iteration: 9250 Time: 1.850E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.0378E-04
Writing solution file op_00037.bin.
Iteration: 9300 Time: 1.860E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.0438E-04
Iteration: 9350 Time: 1.870E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.0531E-04
Iteration: 9400 Time: 1.880E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.0624E-04
Iteration: 9450 Time: 1.890E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.0691E-04
Iteration: 9500 Time: 1.900E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.0756E-04
Writing solution file op_00038.bin.
Iteration: 9550 Time: 1.910E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.0836E-04
Iteration: 9600 Time: 1.920E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.0916E-04
Iteration: 9650 Time: 1.930E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.0993E-04
Iteration: 9700 Time: 1.940E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.1076E-04
Iteration: 9750 Time: 1.950E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.1155E-04
Writing solution file op_00039.bin.
Iteration: 9800 Time: 1.960E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.1218E-04
Iteration: 9850 Time: 1.970E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.1301E-04
Iteration: 9900 Time: 1.980E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.1373E-04
Iteration: 9950 Time: 1.990E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.1446E-04
Iteration: 10000 Time: 2.000E+02 Max CFL: 6.967E-01 Max Diff. No.: -1.000E+00 Norm: 1.1519E-04
Writing solution file op_00040.bin.
Completed time integration (Final time: 200.000000).
Computed errors:
L1 Error : 0.0000000000000000E+00
L2 Error : 0.0000000000000000E+00
Linfinity Error : 0.0000000000000000E+00
Conservation Errors:
0.0000000000000000E+00
0.0000000000000000E+00
0.0000000000000000E+00
0.0000000000000000E+00
0.0000000000000000E+00
Solver runtime (in seconds): 2.3506969013999998E+04
Total runtime (in seconds): 2.3507416699000001E+04
Deallocating arrays.
Finished.