1 2 static char help[] = "Time-dependent PDE in 2d for calculating joint PDF. \n"; 3 /* 4 p_t = -x_t*p_x -y_t*p_y + f(t)*p_yy 5 xmin < x < xmax, ymin < y < ymax; 6 7 Boundary conditions Neumman using mirror values 8 9 Note that x_t and y_t in the above are given functions of x and y; they are not derivatives of x and y. 10 x_t = (y - ws) y_t = (ws/2H)*(Pm - Pmax*sin(x)) 11 12 */ 13 14 #include <petscdm.h> 15 #include <petscdmda.h> 16 #include <petscts.h> 17 18 /* 19 User-defined data structures and routines 20 */ 21 typedef struct { 22 PetscScalar ws; /* Synchronous speed */ 23 PetscScalar H; /* Inertia constant */ 24 PetscScalar D; /* Damping constant */ 25 PetscScalar Pmax; /* Maximum power output of generator */ 26 PetscScalar PM_min; /* Mean mechanical power input */ 27 PetscScalar lambda; /* correlation time */ 28 PetscScalar q; /* noise strength */ 29 PetscScalar mux; /* Initial average angle */ 30 PetscScalar sigmax; /* Standard deviation of initial angle */ 31 PetscScalar muy; /* Average speed */ 32 PetscScalar sigmay; /* standard deviation of initial speed */ 33 PetscScalar rho; /* Cross-correlation coefficient */ 34 PetscScalar xmin; /* left boundary of angle */ 35 PetscScalar xmax; /* right boundary of angle */ 36 PetscScalar ymin; /* bottom boundary of speed */ 37 PetscScalar ymax; /* top boundary of speed */ 38 PetscScalar dx; /* x step size */ 39 PetscScalar dy; /* y step size */ 40 PetscScalar disper_coe; /* Dispersion coefficient */ 41 DM da; 42 PetscInt st_width; /* Stencil width */ 43 DMBoundaryType bx; /* x boundary type */ 44 DMBoundaryType by; /* y boundary type */ 45 PetscBool nonoiseinitial; 46 } AppCtx; 47 48 PetscErrorCode Parameter_settings(AppCtx*); 49 PetscErrorCode ini_bou(Vec,AppCtx*); 50 PetscErrorCode IFunction(TS,PetscReal,Vec,Vec,Vec,void*); 51 PetscErrorCode IJacobian(TS,PetscReal,Vec,Vec,PetscReal,Mat,Mat,void*); 52 PetscErrorCode PostStep(TS); 53 54 int main(int argc, char **argv) 55 { 56 Vec x; /* Solution vector */ 57 TS ts; /* Time-stepping context */ 58 AppCtx user; /* Application context */ 59 PetscViewer viewer; 60 61 PetscCall(PetscInitialize(&argc,&argv,"petscopt_ex7", help)); 62 63 /* Get physics and time parameters */ 64 PetscCall(Parameter_settings(&user)); 65 /* Create a 2D DA with dof = 1 */ 66 PetscCall(DMDACreate2d(PETSC_COMM_WORLD,user.bx,user.by,DMDA_STENCIL_STAR,4,4,PETSC_DECIDE,PETSC_DECIDE,1,user.st_width,NULL,NULL,&user.da)); 67 PetscCall(DMSetFromOptions(user.da)); 68 PetscCall(DMSetUp(user.da)); 69 /* Set x and y coordinates */ 70 PetscCall(DMDASetUniformCoordinates(user.da,user.xmin,user.xmax,user.ymin,user.ymax,0,0)); 71 PetscCall(DMDASetCoordinateName(user.da,0,"X - the angle")); 72 PetscCall(DMDASetCoordinateName(user.da,1,"Y - the speed")); 73 74 /* Get global vector x from DM */ 75 PetscCall(DMCreateGlobalVector(user.da,&x)); 76 77 PetscCall(ini_bou(x,&user)); 78 PetscCall(PetscViewerBinaryOpen(PETSC_COMM_WORLD,"ini_x",FILE_MODE_WRITE,&viewer)); 79 PetscCall(VecView(x,viewer)); 80 PetscCall(PetscViewerDestroy(&viewer)); 81 82 PetscCall(TSCreate(PETSC_COMM_WORLD,&ts)); 83 PetscCall(TSSetDM(ts,user.da)); 84 PetscCall(TSSetProblemType(ts,TS_NONLINEAR)); 85 PetscCall(TSSetType(ts,TSARKIMEX)); 86 PetscCall(TSSetIFunction(ts,NULL,IFunction,&user)); 87 /* PetscCall(TSSetIJacobian(ts,NULL,NULL,IJacobian,&user)); */ 88 PetscCall(TSSetApplicationContext(ts,&user)); 89 PetscCall(TSSetTimeStep(ts,.005)); 90 PetscCall(TSSetFromOptions(ts)); 91 PetscCall(TSSetPostStep(ts,PostStep)); 92 PetscCall(TSSolve(ts,x)); 93 94 PetscCall(PetscViewerBinaryOpen(PETSC_COMM_WORLD,"fin_x",FILE_MODE_WRITE,&viewer)); 95 PetscCall(VecView(x,viewer)); 96 PetscCall(PetscViewerDestroy(&viewer)); 97 98 PetscCall(VecDestroy(&x)); 99 PetscCall(DMDestroy(&user.da)); 100 PetscCall(TSDestroy(&ts)); 101 PetscCall(PetscFinalize()); 102 return 0; 103 } 104 105 PetscErrorCode PostStep(TS ts) 106 { 107 Vec X,gc; 108 AppCtx *user; 109 PetscScalar sum = 0,asum; 110 PetscReal t,**p; 111 DMDACoor2d **coors; 112 DM cda; 113 PetscInt i,j,xs,ys,xm,ym; 114 115 PetscFunctionBegin; 116 PetscCall(TSGetApplicationContext(ts,&user)); 117 PetscCall(TSGetTime(ts,&t)); 118 PetscCall(TSGetSolution(ts,&X)); 119 120 PetscCall(DMGetCoordinateDM(user->da,&cda)); 121 PetscCall(DMDAGetCorners(cda,&xs,&ys,0,&xm,&ym,0)); 122 PetscCall(DMGetCoordinates(user->da,&gc)); 123 PetscCall(DMDAVecGetArrayRead(cda,gc,&coors)); 124 PetscCall(DMDAVecGetArrayRead(user->da,X,&p)); 125 for (i=xs; i < xs+xm; i++) { 126 for (j=ys; j < ys+ym; j++) { 127 if (coors[j][i].y < 5) sum += p[j][i]; 128 } 129 } 130 PetscCall(DMDAVecRestoreArrayRead(cda,gc,&coors)); 131 PetscCall(DMDAVecRestoreArrayRead(user->da,X,&p)); 132 PetscCallMPI(MPI_Allreduce(&sum,&asum,1,MPIU_SCALAR,MPIU_SUM,PetscObjectComm((PetscObject)ts))); 133 PetscCall(PetscPrintf(PETSC_COMM_WORLD,"sum(p)*dw*dtheta at t = %f = %f\n",(double)t,(double)(asum))); 134 if (sum < 1.0e-2) { 135 PetscCall(TSSetConvergedReason(ts,TS_CONVERGED_USER)); 136 PetscCall(PetscPrintf(PETSC_COMM_WORLD,"Exiting TS as the integral of PDF is almost zero\n")); 137 } 138 PetscFunctionReturn(0); 139 } 140 141 PetscErrorCode ini_bou(Vec X,AppCtx* user) 142 { 143 DM cda; 144 DMDACoor2d **coors; 145 PetscScalar **p; 146 Vec gc; 147 PetscInt i,j; 148 PetscInt xs,ys,xm,ym,M,N; 149 PetscScalar xi,yi; 150 PetscScalar sigmax=user->sigmax,sigmay=user->sigmay; 151 PetscScalar rho =user->rho; 152 PetscScalar muy=user->muy,mux; 153 PetscMPIInt rank; 154 PetscScalar sum; 155 156 PetscFunctionBeginUser; 157 PetscCallMPI(MPI_Comm_rank(PETSC_COMM_WORLD,&rank)); 158 PetscCall(DMDAGetInfo(user->da,NULL,&M,&N,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL)); 159 user->dx = (user->xmax - user->xmin)/(M-1); user->dy = (user->ymax - user->ymin)/(N-1); 160 PetscCall(DMGetCoordinateDM(user->da,&cda)); 161 PetscCall(DMGetCoordinates(user->da,&gc)); 162 PetscCall(DMDAVecGetArray(cda,gc,&coors)); 163 PetscCall(DMDAVecGetArray(user->da,X,&p)); 164 PetscCall(DMDAGetCorners(cda,&xs,&ys,0,&xm,&ym,0)); 165 166 /* mux and muy need to be grid points in the x and y-direction otherwise the solution goes unstable 167 muy is set by choosing the y domain, no. of grid points along y-direction so that muy is a grid point 168 in the y-direction. We only modify mux here 169 */ 170 mux = user->mux = coors[0][M/2+10].x; /* For -pi < x < pi, this should be some angle between 0 and pi/2 */ 171 if (user->nonoiseinitial) { 172 for (i=xs; i < xs+xm; i++) { 173 for (j=ys; j < ys+ym; j++) { 174 xi = coors[j][i].x; yi = coors[j][i].y; 175 if ((xi == mux) && (yi == muy)) { 176 p[j][i] = 1.0; 177 } 178 } 179 } 180 } else { 181 /* Change PM_min accordingly */ 182 user->PM_min = user->Pmax*PetscSinScalar(mux); 183 for (i=xs; i < xs+xm; i++) { 184 for (j=ys; j < ys+ym; j++) { 185 xi = coors[j][i].x; yi = coors[j][i].y; 186 p[j][i] = (0.5/(PETSC_PI*sigmax*sigmay*PetscSqrtScalar(1.0-rho*rho)))*PetscExpScalar(-0.5/(1-rho*rho)*(PetscPowScalar((xi-mux)/sigmax,2) + PetscPowScalar((yi-muy)/sigmay,2) - 2*rho*(xi-mux)*(yi-muy)/(sigmax*sigmay))); 187 } 188 } 189 } 190 PetscCall(DMDAVecRestoreArray(cda,gc,&coors)); 191 PetscCall(DMDAVecRestoreArray(user->da,X,&p)); 192 PetscCall(VecSum(X,&sum)); 193 PetscCall(VecScale(X,1.0/sum)); 194 PetscFunctionReturn(0); 195 } 196 197 /* First advection term */ 198 PetscErrorCode adv1(PetscScalar **p,PetscScalar y,PetscInt i,PetscInt j,PetscInt M,PetscScalar *p1,AppCtx *user) 199 { 200 PetscScalar f,fpos,fneg; 201 PetscFunctionBegin; 202 f = (y - user->ws); 203 fpos = PetscMax(f,0); 204 fneg = PetscMin(f,0); 205 if (user->st_width == 1) { 206 *p1 = fpos*(p[j][i] - p[j][i-1])/user->dx + fneg*(p[j][i+1] - p[j][i])/user->dx; 207 } else if (user->st_width == 2) { 208 *p1 = fpos*(3*p[j][i] - 4*p[j][i-1] + p[j][i-2])/(2*user->dx) + fneg*(-p[j][i+2] + 4*p[j][i+1] - 3*p[j][i])/(2*user->dx); 209 } else if (user->st_width == 3) { 210 *p1 = fpos*(2*p[j][i+1] + 3*p[j][i] - 6*p[j][i-1] + p[j][i-2])/(6*user->dx) + fneg*(-p[j][i+2] + 6*p[j][i+1] - 3*p[j][i] - 2*p[j][i-1])/(6*user->dx); 211 } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"No support for wider stencils"); 212 PetscFunctionReturn(0); 213 } 214 215 /* Second advection term */ 216 PetscErrorCode adv2(PetscScalar **p,PetscScalar x,PetscInt i,PetscInt j,PetscInt N,PetscScalar *p2,AppCtx *user) 217 { 218 PetscScalar f,fpos,fneg; 219 PetscFunctionBegin; 220 f = (user->ws/(2*user->H))*(user->PM_min - user->Pmax*PetscSinScalar(x)); 221 fpos = PetscMax(f,0); 222 fneg = PetscMin(f,0); 223 if (user->st_width == 1) { 224 *p2 = fpos*(p[j][i] - p[j-1][i])/user->dy + fneg*(p[j+1][i] - p[j][i])/user->dy; 225 } else if (user->st_width ==2) { 226 *p2 = fpos*(3*p[j][i] - 4*p[j-1][i] + p[j-2][i])/(2*user->dy) + fneg*(-p[j+2][i] + 4*p[j+1][i] - 3*p[j][i])/(2*user->dy); 227 } else if (user->st_width == 3) { 228 *p2 = fpos*(2*p[j+1][i] + 3*p[j][i] - 6*p[j-1][i] + p[j-2][i])/(6*user->dy) + fneg*(-p[j+2][i] + 6*p[j+1][i] - 3*p[j][i] - 2*p[j-1][i])/(6*user->dy); 229 } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"No support for wider stencils"); 230 PetscFunctionReturn(0); 231 } 232 233 /* Diffusion term */ 234 PetscErrorCode diffuse(PetscScalar **p,PetscInt i,PetscInt j,PetscReal t,PetscScalar *p_diff,AppCtx * user) 235 { 236 PetscFunctionBeginUser; 237 if (user->st_width == 1) { 238 *p_diff = user->disper_coe*((p[j-1][i] - 2*p[j][i] + p[j+1][i])/(user->dy*user->dy)); 239 } else if (user->st_width == 2) { 240 *p_diff = user->disper_coe*((-p[j-2][i] + 16*p[j-1][i] - 30*p[j][i] + 16*p[j+1][i] - p[j+2][i])/(12.0*user->dy*user->dy)); 241 } else if (user->st_width == 3) { 242 *p_diff = user->disper_coe*((2*p[j-3][i] - 27*p[j-2][i] + 270*p[j-1][i] - 490*p[j][i] + 270*p[j+1][i] - 27*p[j+2][i] + 2*p[j+3][i])/(180.0*user->dy*user->dy)); 243 } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"No support for wider stencils"); 244 PetscFunctionReturn(0); 245 } 246 247 PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx) 248 { 249 AppCtx *user=(AppCtx*)ctx; 250 DM cda; 251 DMDACoor2d **coors; 252 PetscScalar **p,**f,**pdot; 253 PetscInt i,j; 254 PetscInt xs,ys,xm,ym,M,N; 255 Vec localX,gc,localXdot; 256 PetscScalar p_adv1 = 0.0,p_adv2 = 0.0,p_diff; /* initialize to prevent incorrect compiler warnings */ 257 258 PetscFunctionBeginUser; 259 PetscCall(DMDAGetInfo(user->da,NULL,&M,&N,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL)); 260 PetscCall(DMGetCoordinateDM(user->da,&cda)); 261 PetscCall(DMDAGetCorners(cda,&xs,&ys,0,&xm,&ym,0)); 262 263 PetscCall(DMGetLocalVector(user->da,&localX)); 264 PetscCall(DMGetLocalVector(user->da,&localXdot)); 265 266 PetscCall(DMGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX)); 267 PetscCall(DMGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX)); 268 PetscCall(DMGlobalToLocalBegin(user->da,Xdot,INSERT_VALUES,localXdot)); 269 PetscCall(DMGlobalToLocalEnd(user->da,Xdot,INSERT_VALUES,localXdot)); 270 271 PetscCall(DMGetCoordinatesLocal(user->da,&gc)); 272 273 PetscCall(DMDAVecGetArrayRead(cda,gc,&coors)); 274 PetscCall(DMDAVecGetArrayRead(user->da,localX,&p)); 275 PetscCall(DMDAVecGetArrayRead(user->da,localXdot,&pdot)); 276 PetscCall(DMDAVecGetArray(user->da,F,&f)); 277 278 user->disper_coe = PetscPowScalar((user->lambda*user->ws)/(2*user->H),2)*user->q*(1.0-PetscExpScalar(-t/user->lambda)); 279 for (i=xs; i < xs+xm; i++) { 280 for (j=ys; j < ys+ym; j++) { 281 PetscCall(adv1(p,coors[j][i].y,i,j,M,&p_adv1,user)); 282 PetscCall(adv2(p,coors[j][i].x,i,j,N,&p_adv2,user)); 283 PetscCall(diffuse(p,i,j,t,&p_diff,user)); 284 f[j][i] = -p_adv1 - p_adv2 + p_diff - pdot[j][i]; 285 } 286 } 287 PetscCall(DMDAVecRestoreArrayRead(user->da,localX,&p)); 288 PetscCall(DMDAVecRestoreArrayRead(user->da,localX,&pdot)); 289 PetscCall(DMRestoreLocalVector(user->da,&localX)); 290 PetscCall(DMRestoreLocalVector(user->da,&localXdot)); 291 PetscCall(DMDAVecRestoreArray(user->da,F,&f)); 292 PetscCall(DMDAVecRestoreArrayRead(cda,gc,&coors)); 293 294 PetscFunctionReturn(0); 295 } 296 297 PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat J,Mat Jpre,void *ctx) 298 { 299 AppCtx *user=(AppCtx*)ctx; 300 DM cda; 301 DMDACoor2d **coors; 302 PetscInt i,j; 303 PetscInt xs,ys,xm,ym,M,N; 304 Vec gc; 305 PetscScalar val[5],xi,yi; 306 MatStencil row,col[5]; 307 PetscScalar c1,c3,c5,c1pos,c1neg,c3pos,c3neg; 308 309 PetscFunctionBeginUser; 310 PetscCall(DMDAGetInfo(user->da,NULL,&M,&N,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL)); 311 PetscCall(DMGetCoordinateDM(user->da,&cda)); 312 PetscCall(DMDAGetCorners(cda,&xs,&ys,0,&xm,&ym,0)); 313 314 PetscCall(DMGetCoordinatesLocal(user->da,&gc)); 315 PetscCall(DMDAVecGetArrayRead(cda,gc,&coors)); 316 for (i=xs; i < xs+xm; i++) { 317 for (j=ys; j < ys+ym; j++) { 318 PetscInt nc = 0; 319 xi = coors[j][i].x; yi = coors[j][i].y; 320 row.i = i; row.j = j; 321 c1 = (yi-user->ws)/user->dx; 322 c1pos = PetscMax(c1,0); 323 c1neg = PetscMin(c1,0); 324 c3 = (user->ws/(2.0*user->H))*(user->PM_min - user->Pmax*PetscSinScalar(xi))/user->dy; 325 c3pos = PetscMax(c3,0); 326 c3neg = PetscMin(c3,0); 327 c5 = (PetscPowScalar((user->lambda*user->ws)/(2*user->H),2)*user->q*(1.0-PetscExpScalar(-t/user->lambda)))/(user->dy*user->dy); 328 col[nc].i = i-1; col[nc].j = j; val[nc++] = c1pos; 329 col[nc].i = i+1; col[nc].j = j; val[nc++] = -c1neg; 330 col[nc].i = i; col[nc].j = j-1; val[nc++] = c3pos + c5; 331 col[nc].i = i; col[nc].j = j+1; val[nc++] = -c3neg + c5; 332 col[nc].i = i; col[nc].j = j; val[nc++] = -c1pos + c1neg -c3pos + c3neg -2*c5 -a; 333 PetscCall(MatSetValuesStencil(Jpre,1,&row,nc,col,val,INSERT_VALUES)); 334 } 335 } 336 PetscCall(DMDAVecRestoreArrayRead(cda,gc,&coors)); 337 338 PetscCall(MatAssemblyBegin(Jpre,MAT_FINAL_ASSEMBLY)); 339 PetscCall(MatAssemblyEnd(Jpre,MAT_FINAL_ASSEMBLY)); 340 if (J != Jpre) { 341 PetscCall(MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY)); 342 PetscCall(MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY)); 343 } 344 PetscFunctionReturn(0); 345 } 346 347 PetscErrorCode Parameter_settings(AppCtx *user) 348 { 349 PetscBool flg; 350 351 PetscFunctionBeginUser; 352 353 /* Set default parameters */ 354 user->ws = 1.0; 355 user->H = 5.0; 356 user->Pmax = 2.1; 357 user->PM_min = 1.0; 358 user->lambda = 0.1; 359 user->q = 1.0; 360 user->mux = PetscAsinScalar(user->PM_min/user->Pmax); 361 user->sigmax = 0.1; 362 user->sigmay = 0.1; 363 user->rho = 0.0; 364 user->xmin = -PETSC_PI; 365 user->xmax = PETSC_PI; 366 user->bx = DM_BOUNDARY_PERIODIC; 367 user->by = DM_BOUNDARY_MIRROR; 368 user->nonoiseinitial = PETSC_FALSE; 369 370 /* 371 ymin of -3 seems to let the unstable solution move up and leave a zero in its wake 372 with an ymin of -1 the wake is never exactly zero 373 */ 374 user->ymin = -3.0; 375 user->ymax = 10.0; 376 user->st_width = 1; 377 378 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-ws",&user->ws,&flg)); 379 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-Inertia",&user->H,&flg)); 380 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-Pmax",&user->Pmax,&flg)); 381 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-PM_min",&user->PM_min,&flg)); 382 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-lambda",&user->lambda,&flg)); 383 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-q",&user->q,&flg)); 384 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-mux",&user->mux,&flg)); 385 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-sigmax",&user->sigmax,&flg)); 386 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-muy",&user->muy,&flg)); 387 if (flg == 0) { 388 user->muy = user->ws; 389 } 390 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-sigmay",&user->sigmay,&flg)); 391 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-rho",&user->rho,&flg)); 392 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-xmin",&user->xmin,&flg)); 393 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-xmax",&user->xmax,&flg)); 394 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-ymin",&user->ymin,&flg)); 395 PetscCall(PetscOptionsGetScalar(NULL,NULL,"-ymax",&user->ymax,&flg)); 396 PetscCall(PetscOptionsGetInt(NULL,NULL,"-stencil_width",&user->st_width,&flg)); 397 PetscCall(PetscOptionsGetEnum(NULL,NULL,"-bx",DMBoundaryTypes,(PetscEnum*)&user->bx,&flg)); 398 PetscCall(PetscOptionsGetEnum(NULL,NULL,"-by",DMBoundaryTypes,(PetscEnum*)&user->by,&flg)); 399 PetscCall(PetscOptionsGetBool(NULL,NULL,"-nonoiseinitial",&user->nonoiseinitial,&flg)); 400 401 PetscFunctionReturn(0); 402 } 403 404 /*TEST 405 406 build: 407 requires: !complex !single 408 409 test: 410 args: -ts_max_steps 2 411 localrunfiles: petscopt_ex7 412 413 test: 414 suffix: 2 415 args: -ts_max_steps 2 -snes_mf_operator 416 output_file: output/ex7_1.out 417 localrunfiles: petscopt_ex7 418 timeoutfactor: 2 419 420 test: 421 suffix: 3 422 args: -ts_max_steps 2 -snes_mf -pc_type none 423 output_file: output/ex7_1.out 424 localrunfiles: petscopt_ex7 425 timeoutfactor: 2 426 427 TEST*/ 428