2D Euler Equations (with gravitational force) - Rising Thermal Bubble

Location: hypar/Examples/2D/NavierStokes2D/RisingThermalBubble_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:

• Giraldo, F.X., Restelli, M., "A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases", J. Comput. Phys., 227, 2008, 3849–3877, (Section 3.2).

Domain: \(0 \le x,y \le 1000\,m\), "slip-wall" (_SLIP_WALL_) boundary conditions on all sides.

Initial solution: See references above.

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

• Specific heat ratio \(\gamma = 1.4\) (NavierStokes2D::gamma)
• Universal gas constant \(R = 287.058\) (NavierStokes2D::R)
• Gravitational force per unit mass \(g = 9.8\) along y-axis (NavierStokes2D::grav_y)
• Reference density (at zero altitude) \(\rho_{ref} = 1.1612055171196529\) (NavierStokes2D::rho0)
• Reference pressure (at zero altitude) \(P_{ref} = 100000\) (NavierStokes2D::p0)
• Hydrostatic balance type 2 (NavierStokes2D::HB)

Numerical method:

• Spatial discretization (hyperbolic): 5th order upwind (Interp1PrimFifthOrderUpwind())
• Time integration: SSPRK3 (TimeRK(), _RK_SSP3_)

Input files required:

sparse_grids.inp

begin
  log2_imin           3
  interp_order        6
  write_sg_solutions  no
  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              30000
  restart_iter        0
  time_scheme         rk
  time_scheme_type    ssprk3
  hyp_space_scheme    upw5
  hyp_flux_split      no
  hyp_interp_type     components
  par_space_type      nonconservative-2stage
  par_space_scheme    4
  dt                  0.01
  conservation_check  no
  screen_op_iter      500
  file_op_iter        1000
  input_mode          serial
  ip_file_type        binary
  output_mode         serial
  op_file_format      binary
  op_overwrite        no
  model               navierstokes2d
end

boundary.inp

4
slip-wall   0     1     0        0      0   1000.0
0.0 0.0
slip-wall   0    -1     0        0      0   1000.0
0.0 0.0
slip-wall   1     1     0   1000.0      0        0
0.0 0.0
slip-wall   1    -1     0   1000.0      0        0
0.0 0.0

physics.inp

begin
  gamma     1.4
  upwinding rusanov
  gravity   0.0 9.8
  rho_ref   1.1612055171196529
  p_ref     100000.0
  R         287.058
  HB        2
end

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

#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()
{
  double gamma    = 1.4;
  double R        = 287.058;
  double rho_ref  = 1.1612055171196529;
  double p_ref    = 100000.0;
  double grav_x   = 0.0;
  double grav_y   = 9.8;
  int    HB       = 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) {
    printf("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 printf("Error: Illegal format in solver.inp. Crash and burn!\n");
  }
  fclose(in);

  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, "rho_ref")) fscanf(in,"%lf",&rho_ref);
        else if (!strcmp(word, "p_ref"  )) fscanf(in,"%lf",&p_ref  );
        else if (!strcmp(word, "gamma"  )) fscanf(in,"%lf",&gamma  );
        else if (!strcmp(word, "R"      )) fscanf(in,"%lf",&R      );
        else if (!strcmp(word, "HB"     )) fscanf(in,"%d" ,&HB     );
        else if (!strcmp(word, "gravity")) {
          fscanf(in,"%lf",&grav_x );
          fscanf(in,"%lf",&grav_y );
        }
      }
    } else printf("Error: Illegal format in physics.inp. Crash and burn!\n");
  }
  fclose(in);

  if (ndims != 2) {
    printf("ndims is not 2 in solver.inp. this code is to generate 2D initial conditions\n");
    return(0);
  }
  if (HB != 2) {
    printf("Error: Specify \"HB\" as 2 in physics.inp.\n");
  }
    printf("Grid:\t\t\t%d X %d\n",NI,NJ);
  printf("Reference density and pressure: %lf, %lf.\n",rho_ref,p_ref);

    int i,j;
    double dx = 1000.0  / ((double)(NI-1));
    double dy = 1000.0  / ((double)(NJ-1));

    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));

  /* Initial perturbation center */
  double xc = 500;
  double yc = 350;
  double Cp = gamma * R / (gamma-1.0);

  /* initial perturbation parameters */
  double tc = 0.5;
  double pi = 4.0*atan(1.0);
  double rc = 250.0;
  double T_ref = p_ref / (R * rho_ref);

    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;

      /* temperature peturbation */
      double r      = sqrt((x[i]-xc)*(x[i]-xc)+(y[j]-yc)*(y[j]-yc));
      double dtheta = (r>rc ? 0.0 : (0.5*tc*(1.0+cos(pi*r/rc))) );
      double theta  = T_ref + dtheta;
      double Pexner = 1.0 - (grav_y*y[j])/(Cp*T_ref);

      double rho    = (p_ref/(R*theta)) * raiseto(Pexner, (1.0/(gamma-1.0)));
      double E      = rho * (R/(gamma-1.0)) * theta*Pexner;

      u0[p] = rho;
      u1[p] = 0.0;
      u2[p] = 0.0;
      u3[p] = E;
      }
    }

  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 ",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);
  }

    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_00030.dat: these contain the full grid solution at \(t=0, ..., 300\).

Since write_sg_solutions is set to no in sparse_grids.inp (SparseGridsSimulation::m_write_sg_solutions), sparse grid solution files are not written out.

HyPar::op_file_format is set to binary in solver.inp, and thus, all the files are written out in the binary format, see WriteBinary(). The binary file contains the conserved variables \(\left(\rho, \rho u, \rho v, e\right)\). The following code converts these variables to the primitive variables of interest to atmospheric flows \(\left(\rho, u, v, p, \theta\right)\). It also writes out the hydrostatically balanced quantities \(\left(\rho_0,\pi_0, \theta_0\right)\) for this case that can be used to compute and plot the temperature and density perturbations. These variables are then written to either a tecplot2d or text file. (compile and run it in the run directory):

/*
  Rising Thermal Bubble:-
  The code takes a binary solution file (that contains the 
  conserved variable (rho, rho*u, rho*v, e) as its input 
  and calculates the primitive atmospheric flow variables:
  rho, u, v, P, theta, pi, rho0, P0, theta0, pi0
  and writes them to a tecplot file.
*/

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>

typedef struct _parameters_{
  double grav_x, grav_y, R, gamma, P_ref, rho_ref;
  int HB;
} Parameters;

void IncrementFilename(char *f)
{
  if (f[10] == '9') {
    f[10] = '0';
    if (f[9] == '9') {
      f[9] = '0';
      if (f[8] == '9') {
        f[8] = '0';
        if (f[7] == '9') {
          f[7] = '0';
          if (f[6] == '9') {
            f[6] = '0';
            fprintf(stderr,"Warning: file increment hit max limit. Resetting to zero.\n");
          } else {
            f[6]++;
          }
        } else {
          f[7]++;
        }
      } else {
        f[8]++;
      }
    } else {
      f[9]++;
    }
  } else {
    f[10]++;
  }
}

double raiseto(double x, double a)
{
  return(exp(a*log(x)));
}

void WriteTecplot2D(int nvars,int imax, int jmax,double *x,double *u,char *f)
{
  printf("\tWriting tecplot solution file %s.\n",f);
  FILE *out;
  out = fopen(f,"w");
  if (!out) {
    fprintf(stderr,"Error: could not open %s for writing.\n",f);
    return;
  }

  double *X = x;
  double *Y = x+imax;

  /* writing tecplot data file headers */
  fprintf(out,"VARIABLES=\"I\",\"J\",\"X\",\"Y\",");
  fprintf(out,"\"RHO\",\"U\",\"V\",\"P\",");
  fprintf(out,"\"THETA\",\"RHO0\",\"P0\",");
  fprintf(out,"\"PI0\",\"THETA0\",\n");
  fprintf(out,"ZONE I=%d,J=%d,F=POINT\n",imax,jmax);

  /* writing the data */
  int i,j;
  for (j=0; j<jmax; j++) {
    for (i=0; i<imax; i++) {
      int v, p = i + imax*j;
      fprintf(out,"%4d %4d ",i,j);
      fprintf(out,"%1.16E %1.16E ",X[i],Y[j]);
      for (v=0; v<nvars; v++) fprintf(out,"%1.16E ",u[nvars*p+v]);
      fprintf(out,"\n");
    }
  }
  fclose(out);
  return;
}

void WriteText2D(int nvars,int imax, int jmax,double *x,double *u,char *f)
{
  printf("\tWriting text solution file %s.\n",f);
  FILE *out;
  out = fopen(f,"w");
  if (!out) {
    fprintf(stderr,"Error: could not open %s for writing.\n",f);
    return;
  }

  double *X = x;
  double *Y = x+imax;

  /* writing the data */
  int i,j;
  for (j=0; j<jmax; j++) {
    for (i=0; i<imax; i++) {
      int v, p = i + imax*j;
      fprintf(out,"%4d %4d ",i,j);
      fprintf(out,"%1.16E %1.16E ",X[i],Y[j]);
      for (v=0; v<nvars; v++) fprintf(out,"%1.16E ",u[nvars*p+v]);
      fprintf(out,"\n");
    }
  }
  fclose(out);
  return;
}

int PostProcess(char *fname, char *oname, void *p, int flag)
{
  Parameters *params = (Parameters*) p;
  FILE *in; in = fopen(fname,"rb");

  if (!in) return(-1);

  printf("Reading file %s.\n",fname);
  int ndims, nvars;
  double *U,*x;

  /* read the file headers */
  fread(&ndims,sizeof(int),1,in);
  fread(&nvars,sizeof(int),1,in);

  /* some checks */
  if (ndims != 2) {
    printf("Error: ndims in %s not equal to 2!\n",fname);
    return(1);
  }
  if (nvars != 4) {
    printf("Error: nvars in %s not equal to 4!\n",fname);
    return(1);
  }

  /* read dimensions */
  int dims[ndims];
  fread(dims,sizeof(int),ndims,in);
  printf("Dimensions: %d x %d\n",dims[0],dims[1]);
  printf("Nvars     : %d\n",nvars);

  /* allocate grid and solution arrays */
  x = (double*) calloc (dims[0]+dims[1]       ,sizeof(double));
  U = (double*) calloc (dims[0]*dims[1]*nvars ,sizeof(double));

  /* read grid and solution */
  fread(x,sizeof(double),dims[0]+dims[1]      ,in);
  fread(U,sizeof(double),dims[0]*dims[1]*nvars,in);
  /* done reading */
  fclose(in);

  int imax = dims[0];
  int jmax = dims[1];

  /* allocate primitive variable array (rho, u, v, P, theta, rho0, P0, pi0, theta0) */
  int evars = 5;
  double *Q = (double*) calloc ((nvars+evars)*imax*jmax,sizeof(double));
    
  /* calculate primitive variables */
  int i, j;
  double *X           = x;
  double *Y           = x+imax;
  double grav_y       = params->grav_y;
  double R            = params->R;
  double gamma        = params->gamma;
  double P_ref        = params->P_ref;
  double rho_ref      = params->rho_ref;
  double T_ref        = P_ref / (R*rho_ref);
  double inv_gamma_m1 = 1.0 / (gamma-1.0);
  double Cp           = gamma * inv_gamma_m1 * R;

  for (i=0; i<imax; i++) {
    for (j=0; j<jmax; j++) {
      int p = i + imax*j;

      double rho0, theta0, Pexner, P0;
      theta0  = T_ref;
      Pexner  = 1.0 - (grav_y*Y[j])/(Cp*T_ref);
      rho0    = (P_ref/(R*theta0)) * raiseto(Pexner, inv_gamma_m1);
      P0      = P_ref   * raiseto(Pexner, gamma*inv_gamma_m1);

      double rho, uvel, vvel, E, P, theta;
      rho   = U[nvars*p+0];
      uvel  = U[nvars*p+1] / rho;
      vvel  = U[nvars*p+2] / rho;
      E     = U[nvars*p+3];
      P     = (gamma-1.0) * (E - 0.5*rho*(uvel*uvel+vvel*vvel));
      theta = (E-0.5*rho*(uvel*uvel+vvel*vvel))/(Pexner*rho) * ((gamma-1.0)/R);

      Q[(nvars+evars)*p+0] = rho;
      Q[(nvars+evars)*p+1] = uvel;
      Q[(nvars+evars)*p+2] = vvel;
      Q[(nvars+evars)*p+3] = P;
      Q[(nvars+evars)*p+4] = theta;
      Q[(nvars+evars)*p+5] = rho0;
      Q[(nvars+evars)*p+6] = P0;
      Q[(nvars+evars)*p+7] = Pexner;
      Q[(nvars+evars)*p+8] = theta0;
    }
  }

  /* write Tecplot/Text file */
  if (flag) WriteTecplot2D(nvars+evars,imax,jmax,x,Q,oname);
  else      WriteText2D   (nvars+evars,imax,jmax,x,Q,oname);

  /* clean up */
  free(U);
  free(Q);
  free(x);
}

int main()
{
  FILE *out1, *out2, *in, *inputs;
  char filename[50], op_file_format[50], tecfile[50], overwrite[50];

  int flag;
  printf("Write tecplot file (1) or plain text file (0): ");
  scanf("%d",&flag);

  if ((flag != 1) && (flag != 0)) {
    printf("Error: Invalid input. Should be 1 or 0.\n");
    return(0);
  }

  printf("Reading solver.inp.\n");
  inputs = fopen("solver.inp","r");
  if (!inputs) {
    fprintf(stderr,"Error: File \"solver.inp\" not found.\n");
    return(1);
  } else {
      char word[100];
    fscanf(inputs,"%s",word);
    if (!strcmp(word, "begin")){
        while (strcmp(word, "end")){
            fscanf(inputs,"%s",word);
            if      (!strcmp(word, "op_file_format"   ))  fscanf(inputs,"%s" ,op_file_format);
            else if (!strcmp(word, "op_overwrite"     ))  fscanf(inputs,"%s" ,overwrite      );
      }
    }
    fclose(inputs);
  }
  if (strcmp(op_file_format,"binary") && strcmp(op_file_format,"bin")) {
    printf("Error: solution output needs to be in binary files.\n");
    return(0);
  }

  Parameters params;
  /* default values */
  params.grav_x   = 0.0;
  params.grav_y   = 9.8;
  params.R        = 287.058;
  params.gamma    = 1.4;
  params.P_ref    = 100000.0;
  params.rho_ref  = 100000.0 / (params.R * 300.0);
  params.HB       = 0;
  /* read these parameters from file */
  printf("Reading physics.inp.\n");
  inputs = fopen("physics.inp","r");
  if (!inputs) {
    fprintf(stderr,"Error: File \"physics.inp\" not found.\n");
    return(1);
  } else {
      char word[100];
    fscanf(inputs,"%s",word);
    if (!strcmp(word, "begin")){
        while (strcmp(word, "end")){
            fscanf(inputs,"%s",word);
            if      (!strcmp(word, "gamma"))    fscanf(inputs,"%lf",&params.gamma);
            else if (!strcmp(word, "gravity")) {
          fscanf(inputs,"%lf",&params.grav_x);
          fscanf(inputs,"%lf",&params.grav_y);
        } else if (!strcmp(word,"p_ref"))   fscanf(inputs,"%lf",&params.P_ref);
        else if   (!strcmp(word,"rho_ref")) fscanf(inputs,"%lf",&params.rho_ref);
        else if   (!strcmp(word,"HB"))      fscanf(inputs,"%d",&params.HB);
      }
    }
    fclose(inputs);
  }
  if (params.HB != 2) {
    printf("Error: \"HB\" must be specified as 2 in physics.inp.\n");
    return(0);
  }

  if (!strcmp(overwrite,"no")) {
    strcpy(filename,"op_fg_00000.bin");
    while(1) {
      /* set filename */
      strcpy(tecfile,filename);
      tecfile[12] = 'd';
      tecfile[13] = 'a';
      tecfile[14] = 't';
      int err = PostProcess(filename, tecfile, &params, flag);
      if (err == -1) {
        printf("No more files found. Exiting.\n");
        break;
      }
      IncrementFilename(filename);
    }
  } else if (!strcmp(overwrite,"yes")) {
    strcpy(filename,"op_fg.bin");
    /* set filename */
    strcpy(tecfile,filename);
    tecfile[6] = 'd';
    tecfile[7] = 'a';
    tecfile[8] = 't';
    int err = PostProcess(filename, tecfile, &params, flag);
    if (err == -1) {
      printf("Error: op.bin not found.\n");
      return(0);
    }
  }

  return(0);
}

The following animation shows the potential temperature perturbation. It was plotted using VisIt (https://wci.llnl.gov/simulation/computer-codes/visit/) with tecplot2d format chosen in the above postprocessing code.

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?  no
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.                               : 30000
  Restart iteration                          : 0
  Time integration scheme                    : rk (ssprk3)
  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                                  : 1.000000E-02
  Check for conservation                     : no
  Screen output iterations                   : 500
  File output iterations                     : 1000
  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                             : 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:  1.1234279568688401E+06
 1:  0.0000000000000000E+00
 2:  0.0000000000000000E+00
 3:  2.3801254424407306E+11
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                      slip-wall:  Along dimension  0 and face +1
  Boundary                      slip-wall:  Along dimension  0 and face -1
  Boundary                      slip-wall:  Along dimension  1 and face +1
  Boundary                      slip-wall:  Along dimension  1 and face -1
4 boundary condition(s) read.
Domain 1: Reading boundary conditions from boundary.inp.
  Boundary                      slip-wall:  Along dimension  0 and face +1
  Boundary                      slip-wall:  Along dimension  0 and face -1
  Boundary                      slip-wall:  Along dimension  1 and face +1
  Boundary                      slip-wall:  Along dimension  1 and face -1
4 boundary condition(s) read.
Domain 2: Reading boundary conditions from boundary.inp.
  Boundary                      slip-wall:  Along dimension  0 and face +1
  Boundary                      slip-wall:  Along dimension  0 and face -1
  Boundary                      slip-wall:  Along dimension  1 and face +1
  Boundary                      slip-wall:  Along dimension  1 and face -1
4 boundary condition(s) read.
Domain 3: Reading boundary conditions from boundary.inp.
  Boundary                      slip-wall:  Along dimension  0 and face +1
  Boundary                      slip-wall:  Along dimension  0 and face -1
  Boundary                      slip-wall:  Along dimension  1 and face +1
  Boundary                      slip-wall:  Along dimension  1 and face -1
4 boundary condition(s) read.
Domain 4: Reading boundary conditions from boundary.inp.
  Boundary                      slip-wall:  Along dimension  0 and face +1
  Boundary                      slip-wall:  Along dimension  0 and face -1
  Boundary                      slip-wall:  Along dimension  1 and face +1
  Boundary                      slip-wall:  Along dimension  1 and face -1
4 boundary condition(s) read.
Domain 5: Reading boundary conditions from boundary.inp.
  Boundary                      slip-wall:  Along dimension  0 and face +1
  Boundary                      slip-wall:  Along dimension  0 and face -1
  Boundary                      slip-wall:  Along dimension  1 and face +1
  Boundary                      slip-wall:  Along dimension  1 and face -1
4 boundary condition(s) read.
Domain 6: Reading boundary conditions from boundary.inp.
  Boundary                      slip-wall:  Along dimension  0 and face +1
  Boundary                      slip-wall:  Along dimension  0 and face -1
  Boundary                      slip-wall:  Along dimension  1 and face +1
  Boundary                      slip-wall:  Along dimension  1 and face -1
4 boundary condition(s) read.
Domain 7: Reading boundary conditions from boundary.inp.
  Boundary                      slip-wall:  Along dimension  0 and face +1
  Boundary                      slip-wall:  Along dimension  0 and face -1
  Boundary                      slip-wall:  Along dimension  1 and face +1
  Boundary                      slip-wall:  Along dimension  1 and face -1
4 boundary condition(s) read.
Domain 8: Reading boundary conditions from boundary.inp.
  Boundary                      slip-wall:  Along dimension  0 and face +1
  Boundary                      slip-wall:  Along dimension  0 and face -1
  Boundary                      slip-wall:  Along dimension  1 and face +1
  Boundary                      slip-wall:  Along dimension  1 and face -1
4 boundary condition(s) read.
Domain 9: Reading boundary conditions from boundary.inp.
  Boundary                      slip-wall:  Along dimension  0 and face +1
  Boundary                      slip-wall:  Along dimension  0 and face -1
  Boundary                      slip-wall:  Along dimension  1 and face +1
  Boundary                      slip-wall:  Along dimension  1 and face -1
4 boundary condition(s) read.
Domain 10: Reading boundary conditions from boundary.inp.
  Boundary                      slip-wall:  Along dimension  0 and face +1
  Boundary                      slip-wall:  Along dimension  0 and face -1
  Boundary                      slip-wall:  Along dimension  1 and face +1
  Boundary                      slip-wall:  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:  1.1234251586382261E+06
     1:  0.0000000000000000E+00
     2:  0.0000000000000000E+00
     3:  2.3801105349946042E+11
Interpolating initial solution to sparse grids domain 1.
  Volume integral of the initial solution on sparse grids domain 1:
     0:  1.1234271613915111E+06
     1:  0.0000000000000000E+00
     2:  0.0000000000000000E+00
     3:  2.3801217265073633E+11
Interpolating initial solution to sparse grids domain 2.
  Volume integral of the initial solution on sparse grids domain 2:
     0:  1.1234277731541058E+06
     1:  0.0000000000000000E+00
     2:  0.0000000000000000E+00
     3:  2.3801245243865408E+11
Interpolating initial solution to sparse grids domain 3.
  Volume integral of the initial solution on sparse grids domain 3:
     0:  1.1234449109078781E+06
     1:  0.0000000000000000E+00
     2:  0.0000000000000000E+00
     3:  2.3801756102382306E+11
Interpolating initial solution to sparse grids domain 4.
  Volume integral of the initial solution on sparse grids domain 4:
     0:  1.1233656900511470E+06
     1:  0.0000000000000000E+00
     2:  0.0000000000000000E+00
     3:  2.3799406404289981E+11
Interpolating initial solution to sparse grids domain 5.
  Volume integral of the initial solution on sparse grids domain 5:
     0:  1.1234285119873849E+06
     1:  0.0000000000000000E+00
     2:  0.0000000000000000E+00
     3:  2.3801254424407306E+11
Interpolating initial solution to sparse grids domain 6.
  Volume integral of the initial solution on sparse grids domain 6:
     0:  1.1234251586382259E+06
     1:  0.0000000000000000E+00
     2:  0.0000000000000000E+00
     3:  2.3801105349946048E+11
Interpolating initial solution to sparse grids domain 7.
  Volume integral of the initial solution on sparse grids domain 7:
     0:  1.1234271612869459E+06
     1:  0.0000000000000000E+00
     2:  0.0000000000000000E+00
     3:  2.3801217265073633E+11
Interpolating initial solution to sparse grids domain 8.
  Volume integral of the initial solution on sparse grids domain 8:
     0:  1.1234277774468665E+06
     1:  0.0000000000000000E+00
     2:  0.0000000000000000E+00
     3:  2.3801245243865405E+11
Interpolating initial solution to sparse grids domain 9.
  Volume integral of the initial solution on sparse grids domain 9:
     0:  1.1234449553994713E+06
     1:  0.0000000000000000E+00
     2:  0.0000000000000000E+00
     3:  2.3801756102382306E+11
Interpolating initial solution to sparse grids domain 10.
  Volume integral of the initial solution on sparse grids domain 10:
     0:  1.1233661938574852E+06
     1:  0.0000000000000000E+00
     2:  0.0000000000000000E+00
     3:  2.3799406404289972E+11
Setting up time integration.
Solving in time (from 0 to 30000 iterations)
Writing solution file op_fg_00000.bin.
--
Iteration:     500,  Time: 5.000e+00
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 6.5813E+00
--

--
Iteration:    1000,  Time: 1.000e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 5.5077E+00
--

Writing solution file op_fg_00001.bin.
--
Iteration:    1500,  Time: 1.500e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 4.9858E+00
--

--
Iteration:    2000,  Time: 2.000e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 4.5788E+00
--

Writing solution file op_fg_00002.bin.
--
Iteration:    2500,  Time: 2.500e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 4.2167E+00
--

--
Iteration:    3000,  Time: 3.000e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 3.9285E+00
--

Writing solution file op_fg_00003.bin.
--
Iteration:    3500,  Time: 3.500e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 9.9974E-01
--

--
Iteration:    4000,  Time: 4.000e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 3.6060E+00
--

Writing solution file op_fg_00004.bin.
--
Iteration:    4500,  Time: 4.500e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 3.5391E+00
--

--
Iteration:    5000,  Time: 5.000e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 3.4880E+00
--

Writing solution file op_fg_00005.bin.
--
Iteration:    5500,  Time: 5.500e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 3.3799E+00
--

--
Iteration:    6000,  Time: 6.000e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 3.2350E+00
--

Writing solution file op_fg_00006.bin.
--
Iteration:    6500,  Time: 6.500e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 3.1020E+00
--

--
Iteration:    7000,  Time: 7.000e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 7.8745E-01
--

Writing solution file op_fg_00007.bin.
--
Iteration:    7500,  Time: 7.500e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.9683E+00
--

--
Iteration:    8000,  Time: 8.000e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.9582E+00
--

Writing solution file op_fg_00008.bin.
--
Iteration:    8500,  Time: 8.500e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.9630E+00
--

--
Iteration:    9000,  Time: 9.000e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.8798E+00
--

Writing solution file op_fg_00009.bin.
--
Iteration:    9500,  Time: 9.500e+01
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.7983E+00
--

--
Iteration:   10000,  Time: 1.000e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.7053E+00
--

Writing solution file op_fg_00010.bin.
--
Iteration:   10500,  Time: 1.050e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 7.4930E-01
--

--
Iteration:   11000,  Time: 1.100e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.6256E+00
--

Writing solution file op_fg_00011.bin.
--
Iteration:   11500,  Time: 1.150e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.6302E+00
--

--
Iteration:   12000,  Time: 1.200e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.6453E+00
--

Writing solution file op_fg_00012.bin.
--
Iteration:   12500,  Time: 1.250e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.5970E+00
--

--
Iteration:   13000,  Time: 1.300e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.5272E+00
--

Writing solution file op_fg_00013.bin.
--
Iteration:   13500,  Time: 1.350e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.4529E+00
--

--
Iteration:   14000,  Time: 1.400e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 7.8316E-01
--

Writing solution file op_fg_00014.bin.
--
Iteration:   14500,  Time: 1.450e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.3973E+00
--

--
Iteration:   15000,  Time: 1.500e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.4074E+00
--

Writing solution file op_fg_00015.bin.
--
Iteration:   15500,  Time: 1.550e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.4236E+00
--

--
Iteration:   16000,  Time: 1.600e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.4106E+00
--

Writing solution file op_fg_00016.bin.
--
Iteration:   16500,  Time: 1.650e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.3345E+00
--

--
Iteration:   17000,  Time: 1.700e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.2712E+00
--

Writing solution file op_fg_00017.bin.
--
Iteration:   17500,  Time: 1.750e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 8.4438E-01
--

--
Iteration:   18000,  Time: 1.800e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.2288E+00
--

Writing solution file op_fg_00018.bin.
--
Iteration:   18500,  Time: 1.850e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.2418E+00
--

--
Iteration:   19000,  Time: 1.900e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.2576E+00
--

Writing solution file op_fg_00019.bin.
--
Iteration:   19500,  Time: 1.950e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.2643E+00
--

--
Iteration:   20000,  Time: 2.000e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.1870E+00
--

Writing solution file op_fg_00020.bin.
--
Iteration:   20500,  Time: 2.050e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.1311E+00
--

--
Iteration:   21000,  Time: 2.100e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 9.1303E-01
--

Writing solution file op_fg_00021.bin.
--
Iteration:   21500,  Time: 2.150e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.0969E+00
--

--
Iteration:   22000,  Time: 2.200e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.1118E+00
--

Writing solution file op_fg_00022.bin.
--
Iteration:   22500,  Time: 2.250e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.1269E+00
--

--
Iteration:   23000,  Time: 2.300e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.1371E+00
--

Writing solution file op_fg_00023.bin.
--
Iteration:   23500,  Time: 2.350e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.0685E+00
--

--
Iteration:   24000,  Time: 2.400e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.0180E+00
--

Writing solution file op_fg_00024.bin.
--
Iteration:   24500,  Time: 2.450e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 9.8161E-01
--

--
Iteration:   25000,  Time: 2.500e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 1.9894E+00
--

Writing solution file op_fg_00025.bin.
--
Iteration:   25500,  Time: 2.550e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.0059E+00
--

--
Iteration:   26000,  Time: 2.600e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.0204E+00
--

Writing solution file op_fg_00026.bin.
--
Iteration:   26500,  Time: 2.650e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 2.0221E+00
--

--
Iteration:   27000,  Time: 2.700e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 1.9700E+00
--

Writing solution file op_fg_00027.bin.
--
Iteration:   27500,  Time: 2.750e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 1.9239E+00
--

--
Iteration:   28000,  Time: 2.800e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 1.0484E+00
--

Writing solution file op_fg_00028.bin.
--
Iteration:   28500,  Time: 2.850e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 1.8991E+00
--

--
Iteration:   29000,  Time: 2.900e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 1.9173E+00
--

Writing solution file op_fg_00029.bin.
--
Iteration:   29500,  Time: 2.950e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 1.9311E+00
--

--
Iteration:   30000,  Time: 3.000e+02
  Max CFL: 4.427E-01,  Max Diff. No.: -1.000E+00,    Norm: 1.9177E+00
--

Writing solution file op_fg_00030.bin.
Completed time integration (Final time: 300.000000).

Solver runtime (in seconds): 6.0051986999999997E+01
Total  runtime (in seconds): 6.0287636999999997E+01
Deallocating arrays.
Finished.