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 Vec x; /* Solution vector */ 56 TS ts; /* Time-stepping context */ 57 AppCtx user; /* Application context */ 58 PetscViewer viewer; 59 60 PetscFunctionBeginUser; 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 Vec X, gc; 107 AppCtx *user; 108 PetscScalar sum = 0, asum; 109 PetscReal t, **p; 110 DMDACoor2d **coors; 111 DM cda; 112 PetscInt i, j, xs, ys, xm, ym; 113 114 PetscFunctionBegin; 115 PetscCall(TSGetApplicationContext(ts, &user)); 116 PetscCall(TSGetTime(ts, &t)); 117 PetscCall(TSGetSolution(ts, &X)); 118 119 PetscCall(DMGetCoordinateDM(user->da, &cda)); 120 PetscCall(DMDAGetCorners(cda, &xs, &ys, 0, &xm, &ym, 0)); 121 PetscCall(DMGetCoordinates(user->da, &gc)); 122 PetscCall(DMDAVecGetArrayRead(cda, gc, &coors)); 123 PetscCall(DMDAVecGetArrayRead(user->da, X, &p)); 124 for (i = xs; i < xs + xm; i++) { 125 for (j = ys; j < ys + ym; j++) { 126 if (coors[j][i].y < 5) sum += p[j][i]; 127 } 128 } 129 PetscCall(DMDAVecRestoreArrayRead(cda, gc, &coors)); 130 PetscCall(DMDAVecRestoreArrayRead(user->da, X, &p)); 131 PetscCallMPI(MPI_Allreduce(&sum, &asum, 1, MPIU_SCALAR, MPIU_SUM, PetscObjectComm((PetscObject)ts))); 132 PetscCall(PetscPrintf(PETSC_COMM_WORLD, "sum(p)*dw*dtheta at t = %f = %f\n", (double)t, (double)(asum))); 133 if (sum < 1.0e-2) { 134 PetscCall(TSSetConvergedReason(ts, TS_CONVERGED_USER)); 135 PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Exiting TS as the integral of PDF is almost zero\n")); 136 } 137 PetscFunctionReturn(0); 138 } 139 140 PetscErrorCode ini_bou(Vec X, AppCtx *user) { 141 DM cda; 142 DMDACoor2d **coors; 143 PetscScalar **p; 144 Vec gc; 145 PetscInt i, j; 146 PetscInt xs, ys, xm, ym, M, N; 147 PetscScalar xi, yi; 148 PetscScalar sigmax = user->sigmax, sigmay = user->sigmay; 149 PetscScalar rho = user->rho; 150 PetscScalar muy = user->muy, mux; 151 PetscMPIInt rank; 152 PetscScalar sum; 153 154 PetscFunctionBeginUser; 155 PetscCallMPI(MPI_Comm_rank(PETSC_COMM_WORLD, &rank)); 156 PetscCall(DMDAGetInfo(user->da, NULL, &M, &N, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL)); 157 user->dx = (user->xmax - user->xmin) / (M - 1); 158 user->dy = (user->ymax - user->ymin) / (N - 1); 159 PetscCall(DMGetCoordinateDM(user->da, &cda)); 160 PetscCall(DMGetCoordinates(user->da, &gc)); 161 PetscCall(DMDAVecGetArray(cda, gc, &coors)); 162 PetscCall(DMDAVecGetArray(user->da, X, &p)); 163 PetscCall(DMDAGetCorners(cda, &xs, &ys, 0, &xm, &ym, 0)); 164 165 /* mux and muy need to be grid points in the x and y-direction otherwise the solution goes unstable 166 muy is set by choosing the y domain, no. of grid points along y-direction so that muy is a grid point 167 in the y-direction. We only modify mux here 168 */ 169 mux = user->mux = coors[0][M / 2 + 10].x; /* For -pi < x < pi, this should be some angle between 0 and pi/2 */ 170 if (user->nonoiseinitial) { 171 for (i = xs; i < xs + xm; i++) { 172 for (j = ys; j < ys + ym; j++) { 173 xi = coors[j][i].x; 174 yi = coors[j][i].y; 175 if ((xi == mux) && (yi == muy)) { p[j][i] = 1.0; } 176 } 177 } 178 } else { 179 /* Change PM_min accordingly */ 180 user->PM_min = user->Pmax * PetscSinScalar(mux); 181 for (i = xs; i < xs + xm; i++) { 182 for (j = ys; j < ys + ym; j++) { 183 xi = coors[j][i].x; 184 yi = coors[j][i].y; 185 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))); 186 } 187 } 188 } 189 PetscCall(DMDAVecRestoreArray(cda, gc, &coors)); 190 PetscCall(DMDAVecRestoreArray(user->da, X, &p)); 191 PetscCall(VecSum(X, &sum)); 192 PetscCall(VecScale(X, 1.0 / sum)); 193 PetscFunctionReturn(0); 194 } 195 196 /* First advection term */ 197 PetscErrorCode adv1(PetscScalar **p, PetscScalar y, PetscInt i, PetscInt j, PetscInt M, PetscScalar *p1, AppCtx *user) { 198 PetscScalar f, fpos, fneg; 199 PetscFunctionBegin; 200 f = (y - user->ws); 201 fpos = PetscMax(f, 0); 202 fneg = PetscMin(f, 0); 203 if (user->st_width == 1) { 204 *p1 = fpos * (p[j][i] - p[j][i - 1]) / user->dx + fneg * (p[j][i + 1] - p[j][i]) / user->dx; 205 } else if (user->st_width == 2) { 206 *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); 207 } else if (user->st_width == 3) { 208 *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); 209 } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "No support for wider stencils"); 210 PetscFunctionReturn(0); 211 } 212 213 /* Second advection term */ 214 PetscErrorCode adv2(PetscScalar **p, PetscScalar x, PetscInt i, PetscInt j, PetscInt N, PetscScalar *p2, AppCtx *user) { 215 PetscScalar f, fpos, fneg; 216 PetscFunctionBegin; 217 f = (user->ws / (2 * user->H)) * (user->PM_min - user->Pmax * PetscSinScalar(x)); 218 fpos = PetscMax(f, 0); 219 fneg = PetscMin(f, 0); 220 if (user->st_width == 1) { 221 *p2 = fpos * (p[j][i] - p[j - 1][i]) / user->dy + fneg * (p[j + 1][i] - p[j][i]) / user->dy; 222 } else if (user->st_width == 2) { 223 *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); 224 } else if (user->st_width == 3) { 225 *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); 226 } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "No support for wider stencils"); 227 PetscFunctionReturn(0); 228 } 229 230 /* Diffusion term */ 231 PetscErrorCode diffuse(PetscScalar **p, PetscInt i, PetscInt j, PetscReal t, PetscScalar *p_diff, AppCtx *user) { 232 PetscFunctionBeginUser; 233 if (user->st_width == 1) { 234 *p_diff = user->disper_coe * ((p[j - 1][i] - 2 * p[j][i] + p[j + 1][i]) / (user->dy * user->dy)); 235 } else if (user->st_width == 2) { 236 *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)); 237 } else if (user->st_width == 3) { 238 *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)); 239 } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "No support for wider stencils"); 240 PetscFunctionReturn(0); 241 } 242 243 PetscErrorCode IFunction(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, void *ctx) { 244 AppCtx *user = (AppCtx *)ctx; 245 DM cda; 246 DMDACoor2d **coors; 247 PetscScalar **p, **f, **pdot; 248 PetscInt i, j; 249 PetscInt xs, ys, xm, ym, M, N; 250 Vec localX, gc, localXdot; 251 PetscScalar p_adv1 = 0.0, p_adv2 = 0.0, p_diff; /* initialize to prevent incorrect compiler warnings */ 252 253 PetscFunctionBeginUser; 254 PetscCall(DMDAGetInfo(user->da, NULL, &M, &N, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL)); 255 PetscCall(DMGetCoordinateDM(user->da, &cda)); 256 PetscCall(DMDAGetCorners(cda, &xs, &ys, 0, &xm, &ym, 0)); 257 258 PetscCall(DMGetLocalVector(user->da, &localX)); 259 PetscCall(DMGetLocalVector(user->da, &localXdot)); 260 261 PetscCall(DMGlobalToLocalBegin(user->da, X, INSERT_VALUES, localX)); 262 PetscCall(DMGlobalToLocalEnd(user->da, X, INSERT_VALUES, localX)); 263 PetscCall(DMGlobalToLocalBegin(user->da, Xdot, INSERT_VALUES, localXdot)); 264 PetscCall(DMGlobalToLocalEnd(user->da, Xdot, INSERT_VALUES, localXdot)); 265 266 PetscCall(DMGetCoordinatesLocal(user->da, &gc)); 267 268 PetscCall(DMDAVecGetArrayRead(cda, gc, &coors)); 269 PetscCall(DMDAVecGetArrayRead(user->da, localX, &p)); 270 PetscCall(DMDAVecGetArrayRead(user->da, localXdot, &pdot)); 271 PetscCall(DMDAVecGetArray(user->da, F, &f)); 272 273 user->disper_coe = PetscPowScalar((user->lambda * user->ws) / (2 * user->H), 2) * user->q * (1.0 - PetscExpScalar(-t / user->lambda)); 274 for (i = xs; i < xs + xm; i++) { 275 for (j = ys; j < ys + ym; j++) { 276 PetscCall(adv1(p, coors[j][i].y, i, j, M, &p_adv1, user)); 277 PetscCall(adv2(p, coors[j][i].x, i, j, N, &p_adv2, user)); 278 PetscCall(diffuse(p, i, j, t, &p_diff, user)); 279 f[j][i] = -p_adv1 - p_adv2 + p_diff - pdot[j][i]; 280 } 281 } 282 PetscCall(DMDAVecRestoreArrayRead(user->da, localX, &p)); 283 PetscCall(DMDAVecRestoreArrayRead(user->da, localX, &pdot)); 284 PetscCall(DMRestoreLocalVector(user->da, &localX)); 285 PetscCall(DMRestoreLocalVector(user->da, &localXdot)); 286 PetscCall(DMDAVecRestoreArray(user->da, F, &f)); 287 PetscCall(DMDAVecRestoreArrayRead(cda, gc, &coors)); 288 289 PetscFunctionReturn(0); 290 } 291 292 PetscErrorCode IJacobian(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat J, Mat Jpre, void *ctx) { 293 AppCtx *user = (AppCtx *)ctx; 294 DM cda; 295 DMDACoor2d **coors; 296 PetscInt i, j; 297 PetscInt xs, ys, xm, ym, M, N; 298 Vec gc; 299 PetscScalar val[5], xi, yi; 300 MatStencil row, col[5]; 301 PetscScalar c1, c3, c5, c1pos, c1neg, c3pos, c3neg; 302 303 PetscFunctionBeginUser; 304 PetscCall(DMDAGetInfo(user->da, NULL, &M, &N, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL)); 305 PetscCall(DMGetCoordinateDM(user->da, &cda)); 306 PetscCall(DMDAGetCorners(cda, &xs, &ys, 0, &xm, &ym, 0)); 307 308 PetscCall(DMGetCoordinatesLocal(user->da, &gc)); 309 PetscCall(DMDAVecGetArrayRead(cda, gc, &coors)); 310 for (i = xs; i < xs + xm; i++) { 311 for (j = ys; j < ys + ym; j++) { 312 PetscInt nc = 0; 313 xi = coors[j][i].x; 314 yi = coors[j][i].y; 315 row.i = i; 316 row.j = j; 317 c1 = (yi - user->ws) / user->dx; 318 c1pos = PetscMax(c1, 0); 319 c1neg = PetscMin(c1, 0); 320 c3 = (user->ws / (2.0 * user->H)) * (user->PM_min - user->Pmax * PetscSinScalar(xi)) / user->dy; 321 c3pos = PetscMax(c3, 0); 322 c3neg = PetscMin(c3, 0); 323 c5 = (PetscPowScalar((user->lambda * user->ws) / (2 * user->H), 2) * user->q * (1.0 - PetscExpScalar(-t / user->lambda))) / (user->dy * user->dy); 324 col[nc].i = i - 1; 325 col[nc].j = j; 326 val[nc++] = c1pos; 327 col[nc].i = i + 1; 328 col[nc].j = j; 329 val[nc++] = -c1neg; 330 col[nc].i = i; 331 col[nc].j = j - 1; 332 val[nc++] = c3pos + c5; 333 col[nc].i = i; 334 col[nc].j = j + 1; 335 val[nc++] = -c3neg + c5; 336 col[nc].i = i; 337 col[nc].j = j; 338 val[nc++] = -c1pos + c1neg - c3pos + c3neg - 2 * c5 - a; 339 PetscCall(MatSetValuesStencil(Jpre, 1, &row, nc, col, val, INSERT_VALUES)); 340 } 341 } 342 PetscCall(DMDAVecRestoreArrayRead(cda, gc, &coors)); 343 344 PetscCall(MatAssemblyBegin(Jpre, MAT_FINAL_ASSEMBLY)); 345 PetscCall(MatAssemblyEnd(Jpre, MAT_FINAL_ASSEMBLY)); 346 if (J != Jpre) { 347 PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY)); 348 PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY)); 349 } 350 PetscFunctionReturn(0); 351 } 352 353 PetscErrorCode Parameter_settings(AppCtx *user) { 354 PetscBool flg; 355 356 PetscFunctionBeginUser; 357 358 /* Set default parameters */ 359 user->ws = 1.0; 360 user->H = 5.0; 361 user->Pmax = 2.1; 362 user->PM_min = 1.0; 363 user->lambda = 0.1; 364 user->q = 1.0; 365 user->mux = PetscAsinScalar(user->PM_min / user->Pmax); 366 user->sigmax = 0.1; 367 user->sigmay = 0.1; 368 user->rho = 0.0; 369 user->xmin = -PETSC_PI; 370 user->xmax = PETSC_PI; 371 user->bx = DM_BOUNDARY_PERIODIC; 372 user->by = DM_BOUNDARY_MIRROR; 373 user->nonoiseinitial = PETSC_FALSE; 374 375 /* 376 ymin of -3 seems to let the unstable solution move up and leave a zero in its wake 377 with an ymin of -1 the wake is never exactly zero 378 */ 379 user->ymin = -3.0; 380 user->ymax = 10.0; 381 user->st_width = 1; 382 383 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-ws", &user->ws, &flg)); 384 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-Inertia", &user->H, &flg)); 385 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-Pmax", &user->Pmax, &flg)); 386 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-PM_min", &user->PM_min, &flg)); 387 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-lambda", &user->lambda, &flg)); 388 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-q", &user->q, &flg)); 389 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-mux", &user->mux, &flg)); 390 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-sigmax", &user->sigmax, &flg)); 391 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-muy", &user->muy, &flg)); 392 if (flg == 0) { user->muy = user->ws; } 393 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-sigmay", &user->sigmay, &flg)); 394 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-rho", &user->rho, &flg)); 395 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-xmin", &user->xmin, &flg)); 396 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-xmax", &user->xmax, &flg)); 397 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-ymin", &user->ymin, &flg)); 398 PetscCall(PetscOptionsGetScalar(NULL, NULL, "-ymax", &user->ymax, &flg)); 399 PetscCall(PetscOptionsGetInt(NULL, NULL, "-stencil_width", &user->st_width, &flg)); 400 PetscCall(PetscOptionsGetEnum(NULL, NULL, "-bx", DMBoundaryTypes, (PetscEnum *)&user->bx, &flg)); 401 PetscCall(PetscOptionsGetEnum(NULL, NULL, "-by", DMBoundaryTypes, (PetscEnum *)&user->by, &flg)); 402 PetscCall(PetscOptionsGetBool(NULL, NULL, "-nonoiseinitial", &user->nonoiseinitial, &flg)); 403 404 PetscFunctionReturn(0); 405 } 406 407 /*TEST 408 409 build: 410 requires: !complex !single 411 412 test: 413 args: -ts_max_steps 2 414 localrunfiles: petscopt_ex7 415 416 test: 417 suffix: 2 418 args: -ts_max_steps 2 -snes_mf_operator 419 output_file: output/ex7_1.out 420 localrunfiles: petscopt_ex7 421 timeoutfactor: 2 422 423 test: 424 suffix: 3 425 args: -ts_max_steps 2 -snes_mf -pc_type none 426 output_file: output/ex7_1.out 427 localrunfiles: petscopt_ex7 428 timeoutfactor: 2 429 430 TEST*/ 431