2D Euler Equations - Riemann Problem Case 4 (Time Windowed DMD)
See 2D Euler Equations - Riemann Problem Case 4 to familiarize yourself with this case.
Location: hypar/Examples/2D/NavierStokes2D/Riemann2DCase4_libROM_DMD_Train (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.)
Reduced Order Modeling: This example trains a DMD object and then predicts the solution using the DMD at the same times that the actual HyPar solution is written at.
Reference:
- P. Lax and X.-D. Liu, "Solution of two-dimensional Riemann problems of gas dynamics by positive schemes," SIAM J Sci Comp 19 (1998), 319–340.
Domain: \(-0.5 \le x,y \le 0.5\), "extrapolate" (_EXTRAPOLATE_) boundary conditions.
Initial solution: see Case 4 in the reference.
Numerical method:
- Spatial discretization (hyperbolic): Characteristic-based 5th order WENO (Interp1PrimFifthOrderWENOChar())
- Time integration: SSPRK3 (TimeRK(), _RK_SSP3_)
Reduced Order Modeling:
- Mode: train (libROMInterface::m_mode, _ROM_MODE_TRAIN_)
- Component Mode: monolithic (libROMInterface::m_comp_mode, _ROM_COMP_MODE_MONOLITHIC_)
- Type: Dynamic Mode Decomposition (DMD) with time windowing (libROMInterface::m_rom_type)
- Latent subspace dimension: 16 (DMDROMObject::m_rdim)
- Number of samples per time window: 50 (DMDROMObject::m_num_window_samples)
Input files required:
librom.inp
begin
rdim 16
sampling_frequency 1
mode train
dmd_num_win_samples 50
end
solver.inp
begin
ndims 2
nvars 4
size 201 201
iproc 4 4
ghost 3
n_iter 1000
time_scheme rk
time_scheme_type ssprk3
hyp_space_scheme weno5
hyp_interp_type characteristic
conservation_check yes
dt 0.00025
screen_op_iter 20
file_op_iter 100
op_file_format binary
op_overwrite no
model navierstokes2d
end
boundary.inp
4
extrapolate 0 1 0 0 -0.5 0.5
extrapolate 0 -1 0 0 -0.5 0.5
extrapolate 1 1 -0.5 0.5 0 0
extrapolate 1 -1 -0.5 0.5 0 0
physics.inp
begin
gamma 1.4
upwinding rf-char
end
weno.inp (optional)
begin
mapped 1
borges 0
yc 0
no_limiting 0
epsilon 0.000001
p 2.0
rc 0.3
xi 0.001
end
To generate initial.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(){
double GAMMA = 1.4;
int NI=101,NJ=101,ndims=2;
char ip_file_type[50];
strcpy(ip_file_type,"ascii");
FILE *in, *out;
printf("Reading file \"solver.inp\"...\n");
in = fopen("solver.inp","r");
if (!in) printf("Error: Input file \"solver.inp\" not found. Default values will be used.\n");
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);
if (ndims != 2) {
printf("ndims is not 2 in solver.inp. this code is to generate 2D initial conditions\n");
return(0);
}
printf("Grid:\t\t\t%d X %d\n",NI,NJ);
int i,j;
double dx = 1.0 / ((double)(NI-1));
double dy = 1.0 / ((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));
for (i = 0; i < NI; i++){
for (j = 0; j < NJ; j++){
x[i] = -0.5 + i*dx;
y[j] = -0.5 + j*dy;
int p = NJ*i + j;
double rho, u, v, P;
if ((x[i] < 0) && (y[j] < 0)) {
rho = 1.1;
P = 1.1;
u = 0.8939;
v = 0.8939;
} else if ((x[i] >= 0) && (y[j] < 0)) {
rho = 0.5065;
P = 0.35;
u = 0.0;
v = 0.8939;
} else if ((x[i] < 0) && (y[j] >= 0)) {
rho = 0.5065;
P = 0.35;
u = 0.8939;
v = 0.0;
} else if ((x[i] >= 0) && (y[j] >= 0)) {
rho = 1.1;
P = 1.1;
u = 0.0;
v = 0.0;
}
u0[p] = rho;
u1[p] = rho*u;
u2[p] = rho*v;
u3[p] = P/(GAMMA-1.0) + 0.5*rho*(u*u+v*v);
}
}
if (!strcmp(ip_file_type,"ascii")) {
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("Error: Writing binary initial solution file not implemented. ");
printf("Please choose ip_file_type in solver.inp as \"ascii\".\n");
}
free(x);
free(y);
free(u0);
free(u1);
free(u2);
free(u3);
return(0);
}
Output:
Note that iproc is set to
4 4
in solver.inp (i.e., 4 processors along x, and 4 processors along y). Thus, this example should be run with 16 MPI ranks (or change iproc).
After running the code, there should be the following output files:
- 11 output files op_00000.bin, op_00001.bin, ... op_00010.bin; these are the HyPar solutions.
- 11 output files op_rom_00000.bin, op_rom_00001.bin, ... op_rom_00010.bin; these are the predicted solutions from the DMD object(s).
The first of each of these file sets is the solution at \(t=0\) and the final one is the solution at \(t=0.25\). Since HyPar::op_overwrite is set to no in solver.inp, a separate file is written for solutions at each output time. All the files are binary (HyPar::op_file_format is set to binary in solver.inp).
The provided Python script (plotSolution.py) can be used to generate plots from the binary files that compare the HyPar and DMD solutions. Alternatively, HyPar::op_file_format can be set to tecplot2d, and Tecplot/VisIt or something similar can be used to plot the resulting text files.
The following plot shows the final solution (density) - FOM (full-order model) refers to the HyPar solution, ROM (reduced-order model) refers to the DMD solution, and Diff is the difference between the two.
Wall clock times:
- PDE solution: 111.1 seconds
- DMD training time: 17.6 seconds
- DMD prediction/query time: 1.4 seconds
The L1, L2, and Linf norms of the diff between the HyPar and ROM solution at the final time are calculated and reported on screen (see below) as well as pde_rom_diff.dat:
201 201 4 4 2.5000000000000001E-04 5.6479508453806618E-05 3.6234649516234460E-04 8.2275499561386429E-03 1.4052938399999999E+02 1.4097253200000000E+02
The numbers are: number of grid points in each dimension (HyPar::dim_global), number of processors in each dimension (MPIVariables::iproc), time step size (HyPar::dt), L1, L2, and L-infinity norms of the diff (HyPar::rom_diff_norms), solver wall time (seconds) (i.e., not accounting for initialization, and cleaning up), and total wall time.
By default, the code will write the trained DMD object(s) to files in a subdirectory (DMDROMObject::m_dirname - default value is "DMD"). If the subdirectory does not exist, the code may not report an error (or give some HDF5 file-writing error); the DMD objects will not be written! If the subdirectory exists, several files will exist after the simulation is complete - they are in a format that is readable by libROM.
Expected screen output:
HyPar - Parallel (MPI) version with 16 processes
Compiled with PETSc time integration.
Allocated simulation object(s).
Reading solver inputs from file "solver.inp".
No. of dimensions : 2
No. of variables : 4
Domain size : 201 201
Processes along each dimension : 4 4
Exact solution domain size : 201 201
No. of ghosts pts : 3
No. of iter. : 1000
Restart iteration : 0
Time integration scheme : rk (ssprk3)
Spatial discretization scheme (hyperbolic) : weno5
Split hyperbolic flux term? : no
Interpolation type for hyperbolic term : characteristic
Spatial discretization type (parabolic ) : nonconservative-1stage
Spatial discretization scheme (parabolic ) : 2
Time Step : 2.500000E-04
Check for conservation : yes
Screen output iterations : 20
File output iterations : 100
Initial solution file type : ascii
Initial solution read mode : serial
Solution file write mode : serial
Solution file format : binary
Overwrite solution file : no
Physical model : navierstokes2d
Partitioning domain and allocating data arrays.
Reading array from ASCII file initial.inp (Serial mode).
Volume integral of the initial solution:
0: 8.1130999999999365E-01
1: 3.6014440000000125E-01
2: 3.6014440000000125E-01
3: 2.1526268050000308E+00
Reading boundary conditions from boundary.inp.
Boundary extrapolate: Along dimension 0 and face +1
Boundary extrapolate: Along dimension 0 and face -1
Boundary extrapolate: Along dimension 1 and face +1
Boundary extrapolate: Along dimension 1 and face -1
4 boundary condition(s) read.
Initializing solvers.
Reading WENO parameters from weno.inp.
Initializing physics. Model = "navierstokes2d"
Reading physical model inputs from file "physics.inp".
Setting up time integration.
Setting up libROM interface.
libROM inputs and parameters:
reduced model dimensionality: 16
sampling frequency: 1
mode: train
type: DMD
save to file: true
local vector size: 10000
libROM DMD inputs:
number of samples per window: 50
directory name for DMD onjects: DMD
Solving in time (from 0 to 1000 iterations)
Writing solution file op_00000.bin.
DMDROMObject::takeSample() - creating new DMD object, t=0.000000 (total: 1).
iter= 20 t=5.000E-03 CFL=1.041E-01 norm=1.2145E-02 wctime: 1.1E-01 (s) cons_err=4.5558E-15
iter= 40 t=1.000E-02 CFL=1.048E-01 norm=1.2094E-02 wctime: 1.1E-01 (s) cons_err=6.3667E-15
DMDROMObject::takeSample() - creating new DMD object, t=0.012500 (total: 2).
iter= 60 t=1.500E-02 CFL=1.046E-01 norm=1.1487E-02 wctime: 1.1E-01 (s) cons_err=9.0036E-15
iter= 80 t=2.000E-02 CFL=1.044E-01 norm=1.1625E-02 wctime: 1.1E-01 (s) cons_err=1.0664E-14
iter= 100 t=2.500E-02 CFL=1.064E-01 norm=1.2017E-02 wctime: 1.1E-01 (s) cons_err=1.0241E-14
Writing solution file op_00001.bin.
DMDROMObject::takeSample() - creating new DMD object, t=0.025000 (total: 3).
iter= 120 t=3.000E-02 CFL=1.082E-01 norm=1.1761E-02 wctime: 1.1E-01 (s) cons_err=1.2432E-14
iter= 140 t=3.500E-02 CFL=1.094E-01 norm=1.2143E-02 wctime: 1.1E-01 (s) cons_err=1.2424E-14
DMDROMObject::takeSample() - creating new DMD object, t=0.037500 (total: 4).
iter= 160 t=4.000E-02 CFL=1.110E-01 norm=1.2351E-02 wctime: 1.1E-01 (s) cons_err=1.4344E-14
iter= 180 t=4.500E-02 CFL=1.119E-01 norm=1.2074E-02 wctime: 1.1E-01 (s) cons_err=1.2662E-14
iter= 200 t=5.000E-02 CFL=1.136E-01 norm=1.2484E-02 wctime: 1.1E-01 (s) cons_err=1.2776E-14
Writing solution file op_00002.bin.
DMDROMObject::takeSample() - creating new DMD object, t=0.050000 (total: 5).
iter= 220 t=5.500E-02 CFL=1.146E-01 norm=1.2759E-02 wctime: 1.1E-01 (s) cons_err=1.4045E-14
iter= 240 t=6.000E-02 CFL=1.153E-01 norm=1.2462E-02 wctime: 1.1E-01 (s) cons_err=1.3024E-14
DMDROMObject::takeSample() - creating new DMD object, t=0.062500 (total: 6).
iter= 260 t=6.500E-02 CFL=1.169E-01 norm=1.2843E-02 wctime: 1.1E-01 (s) cons_err=1.0347E-14
iter= 280 t=7.000E-02 CFL=1.185E-01 norm=1.3032E-02 wctime: 1.1E-01 (s) cons_err=1.1335E-14
iter= 300 t=7.500E-02 CFL=1.192E-01 norm=1.2918E-02 wctime: 1.1E-01 (s) cons_err=1.6423E-14
Writing solution file op_00003.bin.
DMDROMObject::takeSample() - creating new DMD object, t=0.075000 (total: 7).
iter= 320 t=8.000E-02 CFL=1.197E-01 norm=1.3307E-02 wctime: 1.1E-01 (s) cons_err=1.1796E-14
iter= 340 t=8.500E-02 CFL=1.216E-01 norm=1.3303E-02 wctime: 1.1E-01 (s) cons_err=1.4160E-14
DMDROMObject::takeSample() - creating new DMD object, t=0.087500 (total: 8).
iter= 360 t=9.000E-02 CFL=1.229E-01 norm=1.3197E-02 wctime: 1.2E-01 (s) cons_err=1.3388E-14
iter= 380 t=9.500E-02 CFL=1.231E-01 norm=1.3656E-02 wctime: 1.1E-01 (s) cons_err=1.3445E-14
iter= 400 t=1.000E-01 CFL=1.229E-01 norm=1.3688E-02 wctime: 1.1E-01 (s) cons_err=1.7038E-14
Writing solution file op_00004.bin.
DMDROMObject::takeSample() - creating new DMD object, t=0.100000 (total: 9).
iter= 420 t=1.050E-01 CFL=1.245E-01 norm=1.3567E-02 wctime: 1.1E-01 (s) cons_err=1.6050E-14
iter= 440 t=1.100E-01 CFL=1.255E-01 norm=1.3981E-02 wctime: 1.1E-01 (s) cons_err=1.1911E-14
DMDROMObject::takeSample() - creating new DMD object, t=0.112500 (total: 10).
iter= 460 t=1.150E-01 CFL=1.256E-01 norm=1.3918E-02 wctime: 1.1E-01 (s) cons_err=1.4287E-14
iter= 480 t=1.200E-01 CFL=1.269E-01 norm=1.3982E-02 wctime: 1.1E-01 (s) cons_err=1.3570E-14
iter= 500 t=1.250E-01 CFL=1.272E-01 norm=1.4267E-02 wctime: 1.1E-01 (s) cons_err=1.5399E-14
Writing solution file op_00005.bin.
DMDROMObject::takeSample() - creating new DMD object, t=0.125000 (total: 11).
iter= 520 t=1.300E-01 CFL=1.276E-01 norm=1.4185E-02 wctime: 1.1E-01 (s) cons_err=1.3997E-14
iter= 540 t=1.350E-01 CFL=1.289E-01 norm=1.4258E-02 wctime: 1.1E-01 (s) cons_err=1.4545E-14
DMDROMObject::takeSample() - creating new DMD object, t=0.137500 (total: 12).
iter= 560 t=1.400E-01 CFL=1.298E-01 norm=1.4746E-02 wctime: 1.1E-01 (s) cons_err=1.3437E-14
iter= 580 t=1.450E-01 CFL=1.299E-01 norm=1.4432E-02 wctime: 1.1E-01 (s) cons_err=1.6812E-14
iter= 600 t=1.500E-01 CFL=1.297E-01 norm=1.4578E-02 wctime: 1.1E-01 (s) cons_err=1.3464E-14
Writing solution file op_00006.bin.
DMDROMObject::takeSample() - creating new DMD object, t=0.150000 (total: 13).
iter= 620 t=1.550E-01 CFL=1.308E-01 norm=1.4862E-02 wctime: 1.1E-01 (s) cons_err=1.4884E-14
iter= 640 t=1.600E-01 CFL=1.314E-01 norm=1.4870E-02 wctime: 1.1E-01 (s) cons_err=1.8206E-14
DMDROMObject::takeSample() - creating new DMD object, t=0.162500 (total: 14).
iter= 660 t=1.650E-01 CFL=1.313E-01 norm=1.4850E-02 wctime: 1.1E-01 (s) cons_err=1.9133E-14
iter= 680 t=1.700E-01 CFL=1.315E-01 norm=1.5221E-02 wctime: 1.1E-01 (s) cons_err=1.9239E-14
iter= 700 t=1.750E-01 CFL=1.314E-01 norm=1.4952E-02 wctime: 1.1E-01 (s) cons_err=1.9522E-14
Writing solution file op_00007.bin.
DMDROMObject::takeSample() - creating new DMD object, t=0.175000 (total: 15).
iter= 720 t=1.800E-01 CFL=1.321E-01 norm=1.5306E-02 wctime: 1.1E-01 (s) cons_err=2.2289E-14
iter= 740 t=1.850E-01 CFL=1.328E-01 norm=1.5367E-02 wctime: 1.1E-01 (s) cons_err=1.9030E-14
DMDROMObject::takeSample() - creating new DMD object, t=0.187500 (total: 16).
iter= 760 t=1.900E-01 CFL=1.330E-01 norm=1.5257E-02 wctime: 1.1E-01 (s) cons_err=2.1492E-14
iter= 780 t=1.950E-01 CFL=1.328E-01 norm=1.5386E-02 wctime: 1.1E-01 (s) cons_err=2.1584E-14
iter= 800 t=2.000E-01 CFL=1.332E-01 norm=1.5763E-02 wctime: 1.1E-01 (s) cons_err=2.5972E-14
Writing solution file op_00008.bin.
DMDROMObject::takeSample() - creating new DMD object, t=0.200000 (total: 17).
iter= 820 t=2.050E-01 CFL=1.334E-01 norm=1.5559E-02 wctime: 1.1E-01 (s) cons_err=1.7745E-14
iter= 840 t=2.100E-01 CFL=1.334E-01 norm=1.5729E-02 wctime: 1.1E-01 (s) cons_err=2.1150E-14
DMDROMObject::takeSample() - creating new DMD object, t=0.212500 (total: 18).
iter= 860 t=2.150E-01 CFL=1.331E-01 norm=1.5928E-02 wctime: 1.1E-01 (s) cons_err=1.6984E-14
iter= 880 t=2.200E-01 CFL=1.332E-01 norm=1.5801E-02 wctime: 1.1E-01 (s) cons_err=2.1588E-14
iter= 900 t=2.250E-01 CFL=1.335E-01 norm=1.6104E-02 wctime: 1.1E-01 (s) cons_err=1.9666E-14
Writing solution file op_00009.bin.
DMDROMObject::takeSample() - creating new DMD object, t=0.225000 (total: 19).
iter= 920 t=2.300E-01 CFL=1.338E-01 norm=1.6060E-02 wctime: 1.1E-01 (s) cons_err=1.8516E-14
iter= 940 t=2.350E-01 CFL=1.337E-01 norm=1.6161E-02 wctime: 1.1E-01 (s) cons_err=1.5606E-14
DMDROMObject::takeSample() - creating new DMD object, t=0.237500 (total: 20).
iter= 960 t=2.400E-01 CFL=1.336E-01 norm=1.6273E-02 wctime: 1.1E-01 (s) cons_err=1.6180E-14
iter= 980 t=2.450E-01 CFL=1.338E-01 norm=1.6568E-02 wctime: 1.1E-01 (s) cons_err=1.7095E-14
iter= 1000 t=2.500E-01 CFL=1.338E-01 norm=1.6215E-02 wctime: 1.1E-01 (s) cons_err=1.8917E-14
Completed time integration (Final time: 0.250000), total wctime: 111.100796 (seconds).
Writing solution file op_00010.bin.
libROM: Training ROM.
DMDROMObject::train() - training DMD object 0 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 1 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 2 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 3 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 4 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 5 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 6 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 7 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 8 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 9 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 10 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 11 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 12 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 13 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 14 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 15 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 16 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 17 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 18 with 51 samples.
Using 16 basis vectors out of 50.
DMDROMObject::train() - training DMD object 19 with 50 samples.
Using 16 basis vectors out of 49.
libROM: total training wallclock time: 17.589843 (seconds).
libROM: Predicting solution at time 0.0000e+00 using ROM.
libROM: wallclock time: 0.131456 (seconds).
Writing solution file op_rom_00000.bin.
libROM: Predicting solution at time 2.5000e-02 using ROM.
libROM: wallclock time: 0.134897 (seconds).
Writing solution file op_rom_00001.bin.
libROM: Predicting solution at time 5.0000e-02 using ROM.
libROM: wallclock time: 0.131684 (seconds).
Writing solution file op_rom_00002.bin.
libROM: Predicting solution at time 7.5000e-02 using ROM.
libROM: wallclock time: 0.130372 (seconds).
Writing solution file op_rom_00003.bin.
libROM: Predicting solution at time 1.0000e-01 using ROM.
libROM: wallclock time: 0.130186 (seconds).
Writing solution file op_rom_00004.bin.
libROM: Predicting solution at time 1.2500e-01 using ROM.
libROM: wallclock time: 0.130239 (seconds).
Writing solution file op_rom_00005.bin.
libROM: Predicting solution at time 1.5000e-01 using ROM.
libROM: wallclock time: 0.130202 (seconds).
Writing solution file op_rom_00006.bin.
libROM: Predicting solution at time 1.7500e-01 using ROM.
libROM: wallclock time: 0.130257 (seconds).
Writing solution file op_rom_00007.bin.
libROM: Predicting solution at time 2.0000e-01 using ROM.
libROM: wallclock time: 0.130218 (seconds).
Writing solution file op_rom_00008.bin.
libROM: Predicting solution at time 2.2500e-01 using ROM.
libROM: wallclock time: 0.135114 (seconds).
Writing solution file op_rom_00009.bin.
libROM: Predicting solution at time 2.5000e-01 using ROM.
libROM: wallclock time: 0.130282 (seconds).
libROM: Calculating diff between PDE and ROM solutions.
Writing solution file op_rom_00010.bin.
libROM: total prediction/query wallclock time: 1.444907 (seconds).
libROMInterface::saveROM() - saving ROM objects.
Saving DMD object with filename root DMD/dmdobj_0000.
Saving DMD object with filename root DMD/dmdobj_0001.
Saving DMD object with filename root DMD/dmdobj_0002.
Saving DMD object with filename root DMD/dmdobj_0003.
Saving DMD object with filename root DMD/dmdobj_0004.
Saving DMD object with filename root DMD/dmdobj_0005.
Saving DMD object with filename root DMD/dmdobj_0006.
Saving DMD object with filename root DMD/dmdobj_0007.
Saving DMD object with filename root DMD/dmdobj_0008.
Saving DMD object with filename root DMD/dmdobj_0009.
Saving DMD object with filename root DMD/dmdobj_0010.
Saving DMD object with filename root DMD/dmdobj_0011.
Saving DMD object with filename root DMD/dmdobj_0012.
Saving DMD object with filename root DMD/dmdobj_0013.
Saving DMD object with filename root DMD/dmdobj_0014.
Saving DMD object with filename root DMD/dmdobj_0015.
Saving DMD object with filename root DMD/dmdobj_0016.
Saving DMD object with filename root DMD/dmdobj_0017.
Saving DMD object with filename root DMD/dmdobj_0018.
Saving DMD object with filename root DMD/dmdobj_0019.
Conservation Errors:
6.1062266354383610E-15
1.1657341758564144E-15
2.1094237467877974E-15
1.7741891886877642E-14
Norms of the diff between ROM and PDE solutions for domain 0:
L1 Norm : 5.6479508453806618E-05
L2 Norm : 3.6234649516234460E-04
Linfinity Norm : 8.2275499561386429E-03
Solver runtime (in seconds): 1.4601398699999999E+02
Total runtime (in seconds): 1.4684907300000000E+02
Deallocating arrays.
Finished.