Date: 2022-08-02T03:15:03+00:00

Actual source code: ex15.c

      2: static char help[] = "Time-dependent PDE in 2d. Modified from ex13.c for illustrating how to solve DAEs. \n";
      3: /*
      4:    u_t = uxx + uyy
      5:    0 < x < 1, 0 < y < 1;
      6:    At t=0: u(x,y) = exp(c*r*r*r), if r=PetscSqrtReal((x-.5)*(x-.5) + (y-.5)*(y-.5)) < .125
      7:            u(x,y) = 0.0           if r >= .125

      9:    Boundary conditions:
     10:    Drichlet BC:
     11:    At x=0, x=1, y=0, y=1: u = 0.0

     13:    Neumann BC:
     14:    At x=0, x=1: du(x,y,t)/dx = 0
     15:    At y=0, y=1: du(x,y,t)/dy = 0

     17:    mpiexec -n 2 ./ex15 -da_grid_x 40 -da_grid_y 40 -ts_max_steps 2 -snes_monitor -ksp_monitor
     18:          ./ex15 -da_grid_x 40 -da_grid_y 40  -draw_pause .1 -boundary 1 -ts_monitor_draw_solution
     19:          ./ex15 -da_grid_x 40 -da_grid_y 40  -draw_pause .1 -boundary 1 -Jtype 2 -nstencilpts 9

     21: */

     23: #include <petscdm.h>
     24: #include <petscdmda.h>
     25: #include <petscts.h>

     27: /*
     28:    User-defined data structures and routines
     29: */

     31: /* AppCtx: used by FormIFunction() and FormIJacobian() */
     32: typedef struct {
     33:   DM        da;
     34:   PetscInt  nstencilpts;         /* number of stencil points: 5 or 9 */
     35:   PetscReal c;
     36:   PetscInt  boundary;            /* Type of boundary condition */
     37:   PetscBool viewJacobian;
     38: } AppCtx;

     40: extern PetscErrorCode FormIFunction(TS,PetscReal,Vec,Vec,Vec,void*);
     41: extern PetscErrorCode FormIJacobian(TS,PetscReal,Vec,Vec,PetscReal,Mat,Mat,void*);
     42: extern PetscErrorCode FormInitialSolution(Vec,void*);

     44: int main(int argc,char **argv)
     45: {
     46:   TS             ts;                   /* nonlinear solver */
     47:   Vec            u,r;                  /* solution, residual vectors */
     48:   Mat            J,Jmf = NULL;   /* Jacobian matrices */
     49:   DM             da;
     50:   PetscReal      dt;
     51:   AppCtx         user;              /* user-defined work context */
     52:   SNES           snes;
     53:   PetscInt       Jtype; /* Jacobian type
     54:                             0: user provide Jacobian;
     55:                             1: slow finite difference;
     56:                             2: fd with coloring; */

     58:   PetscInitialize(&argc,&argv,(char*)0,help);
     59:   /* Initialize user application context */
     60:   user.da           = NULL;
     61:   user.nstencilpts  = 5;
     62:   user.c            = -30.0;
     63:   user.boundary     = 0;  /* 0: Drichlet BC; 1: Neumann BC */
     64:   user.viewJacobian = PETSC_FALSE;

     66:   PetscOptionsGetInt(NULL,NULL,"-nstencilpts",&user.nstencilpts,NULL);
     67:   PetscOptionsGetInt(NULL,NULL,"-boundary",&user.boundary,NULL);
     68:   PetscOptionsHasName(NULL,NULL,"-viewJacobian",&user.viewJacobian);

     70:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     71:      Create distributed array (DMDA) to manage parallel grid and vectors
     72:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     73:   if (user.nstencilpts == 5) {
     74:     DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,11,11,PETSC_DECIDE,PETSC_DECIDE,1,1,NULL,NULL,&da);
     75:   } else if (user.nstencilpts == 9) {
     76:     DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE,DMDA_STENCIL_BOX,11,11,PETSC_DECIDE,PETSC_DECIDE,1,1,NULL,NULL,&da);
     77:   } else SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_SUP,"nstencilpts %d is not supported",user.nstencilpts);
     78:   DMSetFromOptions(da);
     79:   DMSetUp(da);
     80:   user.da = da;

     82:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     83:      Extract global vectors from DMDA;
     84:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     85:   DMCreateGlobalVector(da,&u);
     86:   VecDuplicate(u,&r);

     88:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     89:      Create timestepping solver context
     90:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     91:   TSCreate(PETSC_COMM_WORLD,&ts);
     92:   TSSetProblemType(ts,TS_NONLINEAR);
     93:   TSSetType(ts,TSBEULER);
     94:   TSSetDM(ts,da);
     95:   TSSetIFunction(ts,r,FormIFunction,&user);
     96:   TSSetMaxTime(ts,1.0);
     97:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);

     99:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    100:      Set initial conditions
    101:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    102:   FormInitialSolution(u,&user);
    103:   TSSetSolution(ts,u);
    104:   dt   = .01;
    105:   TSSetTimeStep(ts,dt);

    107:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    108:    Set Jacobian evaluation routine
    109:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    110:   DMSetMatType(da,MATAIJ);
    111:   DMCreateMatrix(da,&J);
    112:   Jtype = 0;
    113:   PetscOptionsGetInt(NULL,NULL, "-Jtype",&Jtype,NULL);
    114:   if (Jtype == 0) { /* use user provided Jacobian evaluation routine */
    116:     TSSetIJacobian(ts,J,J,FormIJacobian,&user);
    117:   } else { /* use finite difference Jacobian J as preconditioner and '-snes_mf_operator' for Mat*vec */
    118:     TSGetSNES(ts,&snes);
    119:     MatCreateSNESMF(snes,&Jmf);
    120:     if (Jtype == 1) { /* slow finite difference J; */
    121:       SNESSetJacobian(snes,Jmf,J,SNESComputeJacobianDefault,NULL);
    122:     } else if (Jtype == 2) { /* Use coloring to compute  finite difference J efficiently */
    123:       SNESSetJacobian(snes,Jmf,J,SNESComputeJacobianDefaultColor,0);
    124:     } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Jtype is not supported");
    125:   }

    127:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    128:    Sets various TS parameters from user options
    129:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    130:   TSSetFromOptions(ts);

    132:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    133:      Solve nonlinear system
    134:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    135:   TSSolve(ts,u);

    137:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    138:      Free work space.
    139:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    140:   MatDestroy(&J);
    141:   MatDestroy(&Jmf);
    142:   VecDestroy(&u);
    143:   VecDestroy(&r);
    144:   TSDestroy(&ts);
    145:   DMDestroy(&da);

    147:   PetscFinalize();
    148:   return 0;
    149: }

    151: /* --------------------------------------------------------------------- */
    152: /*
    153:   FormIFunction = Udot - RHSFunction
    154: */
    155: PetscErrorCode FormIFunction(TS ts,PetscReal t,Vec U,Vec Udot,Vec F,void *ctx)
    156: {
    157:   AppCtx         *user=(AppCtx*)ctx;
    158:   DM             da   = (DM)user->da;
    159:   PetscInt       i,j,Mx,My,xs,ys,xm,ym;
    160:   PetscReal      hx,hy,sx,sy;
    161:   PetscScalar    u,uxx,uyy,**uarray,**f,**udot;
    162:   Vec            localU;

    165:   DMGetLocalVector(da,&localU);
    166:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,&My,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);

    168:   hx = 1.0/(PetscReal)(Mx-1); sx = 1.0/(hx*hx);
    169:   hy = 1.0/(PetscReal)(My-1); sy = 1.0/(hy*hy);

    172:   /*
    173:      Scatter ghost points to local vector,using the 2-step process
    174:         DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
    175:      By placing code between these two statements, computations can be
    176:      done while messages are in transition.
    177:   */
    178:   DMGlobalToLocalBegin(da,U,INSERT_VALUES,localU);
    179:   DMGlobalToLocalEnd(da,U,INSERT_VALUES,localU);

    181:   /* Get pointers to vector data */
    182:   DMDAVecGetArrayRead(da,localU,&uarray);
    183:   DMDAVecGetArray(da,F,&f);
    184:   DMDAVecGetArray(da,Udot,&udot);

    186:   /* Get local grid boundaries */
    187:   DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);

    189:   /* Compute function over the locally owned part of the grid */
    190:   for (j=ys; j<ys+ym; j++) {
    191:     for (i=xs; i<xs+xm; i++) {
    192:       /* Boundary conditions */
    193:       if (i == 0 || j == 0 || i == Mx-1 || j == My-1) {
    194:         if (user->boundary == 0) { /* Drichlet BC */
    195:           f[j][i] = uarray[j][i]; /* F = U */
    196:         } else {                  /* Neumann BC */
    197:           if (i == 0 && j == 0) {              /* SW corner */
    198:             f[j][i] = uarray[j][i] - uarray[j+1][i+1];
    199:           } else if (i == Mx-1 && j == 0) {    /* SE corner */
    200:             f[j][i] = uarray[j][i] - uarray[j+1][i-1];
    201:           } else if (i == 0 && j == My-1) {    /* NW corner */
    202:             f[j][i] = uarray[j][i] - uarray[j-1][i+1];
    203:           } else if (i == Mx-1 && j == My-1) { /* NE corner */
    204:             f[j][i] = uarray[j][i] - uarray[j-1][i-1];
    205:           } else if (i == 0) {                  /* Left */
    206:             f[j][i] = uarray[j][i] - uarray[j][i+1];
    207:           } else if (i == Mx-1) {               /* Right */
    208:             f[j][i] = uarray[j][i] - uarray[j][i-1];
    209:           } else if (j == 0) {                 /* Bottom */
    210:             f[j][i] = uarray[j][i] - uarray[j+1][i];
    211:           } else if (j == My-1) {               /* Top */
    212:             f[j][i] = uarray[j][i] - uarray[j-1][i];
    213:           }
    214:         }
    215:       } else { /* Interior */
    216:         u = uarray[j][i];
    217:         /* 5-point stencil */
    218:         uxx = (-2.0*u + uarray[j][i-1] + uarray[j][i+1]);
    219:         uyy = (-2.0*u + uarray[j-1][i] + uarray[j+1][i]);
    220:         if (user->nstencilpts == 9) {
    221:           /* 9-point stencil: assume hx=hy */
    222:           uxx = 2.0*uxx/3.0 + (0.5*(uarray[j-1][i-1]+uarray[j-1][i+1]+uarray[j+1][i-1]+uarray[j+1][i+1]) - 2.0*u)/6.0;
    223:           uyy = 2.0*uyy/3.0 + (0.5*(uarray[j-1][i-1]+uarray[j-1][i+1]+uarray[j+1][i-1]+uarray[j+1][i+1]) - 2.0*u)/6.0;
    224:         }
    225:         f[j][i] = udot[j][i] - (uxx*sx + uyy*sy);
    226:       }
    227:     }
    228:   }

    230:   /* Restore vectors */
    231:   DMDAVecRestoreArrayRead(da,localU,&uarray);
    232:   DMDAVecRestoreArray(da,F,&f);
    233:   DMDAVecRestoreArray(da,Udot,&udot);
    234:   DMRestoreLocalVector(da,&localU);
    235:   PetscLogFlops(11.0*ym*xm);
    236:   return 0;
    237: }

    239: /* --------------------------------------------------------------------- */
    240: /*
    241:   FormIJacobian() - Compute IJacobian = dF/dU + a dF/dUdot
    242:   This routine is not used with option '-use_coloring'
    243: */
    244: PetscErrorCode FormIJacobian(TS ts,PetscReal t,Vec U,Vec Udot,PetscReal a,Mat J,Mat Jpre,void *ctx)
    245: {
    246:   PetscInt       i,j,Mx,My,xs,ys,xm,ym,nc;
    247:   AppCtx         *user = (AppCtx*)ctx;
    248:   DM             da    = (DM)user->da;
    249:   MatStencil     col[5],row;
    250:   PetscScalar    vals[5],hx,hy,sx,sy;

    253:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,&My,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);
    254:   DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);

    256:   hx = 1.0/(PetscReal)(Mx-1); sx = 1.0/(hx*hx);
    257:   hy = 1.0/(PetscReal)(My-1); sy = 1.0/(hy*hy);

    259:   for (j=ys; j<ys+ym; j++) {
    260:     for (i=xs; i<xs+xm; i++) {
    261:       nc    = 0;
    262:       row.j = j; row.i = i;
    263:       if (user->boundary == 0 && (i == 0 || i == Mx-1 || j == 0 || j == My-1)) {
    264:         col[nc].j = j; col[nc].i = i; vals[nc++] = 1.0;

    266:       } else if (user->boundary > 0 && i == 0) {  /* Left Neumann */
    267:         col[nc].j = j; col[nc].i = i;   vals[nc++] = 1.0;
    268:         col[nc].j = j; col[nc].i = i+1; vals[nc++] = -1.0;
    269:       } else if (user->boundary > 0 && i == Mx-1) { /* Right Neumann */
    270:         col[nc].j = j; col[nc].i = i;   vals[nc++] = 1.0;
    271:         col[nc].j = j; col[nc].i = i-1; vals[nc++] = -1.0;
    272:       } else if (user->boundary > 0 && j == 0) {  /* Bottom Neumann */
    273:         col[nc].j = j;   col[nc].i = i; vals[nc++] = 1.0;
    274:         col[nc].j = j+1; col[nc].i = i; vals[nc++] = -1.0;
    275:       } else if (user->boundary > 0 && j == My-1) { /* Top Neumann */
    276:         col[nc].j = j;   col[nc].i = i;  vals[nc++] = 1.0;
    277:         col[nc].j = j-1; col[nc].i = i;  vals[nc++] = -1.0;
    278:       } else {   /* Interior */
    279:         col[nc].j = j-1; col[nc].i = i;   vals[nc++] = -sy;
    280:         col[nc].j = j;   col[nc].i = i-1; vals[nc++] = -sx;
    281:         col[nc].j = j;   col[nc].i = i;   vals[nc++] = 2.0*(sx + sy) + a;
    282:         col[nc].j = j;   col[nc].i = i+1; vals[nc++] = -sx;
    283:         col[nc].j = j+1; col[nc].i = i;   vals[nc++] = -sy;
    284:       }
    285:       MatSetValuesStencil(Jpre,1,&row,nc,col,vals,INSERT_VALUES);
    286:     }
    287:   }
    288:   MatAssemblyBegin(Jpre,MAT_FINAL_ASSEMBLY);
    289:   MatAssemblyEnd(Jpre,MAT_FINAL_ASSEMBLY);
    290:   if (J != Jpre) {
    291:     MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);
    292:     MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);
    293:   }

    295:   if (user->viewJacobian) {
    296:     PetscPrintf(PetscObjectComm((PetscObject)Jpre),"Jpre:\n");
    297:     MatView(Jpre,PETSC_VIEWER_STDOUT_WORLD);
    298:   }
    299:   return 0;
    300: }

    302: /* ------------------------------------------------------------------- */
    303: PetscErrorCode FormInitialSolution(Vec U,void *ptr)
    304: {
    305:   AppCtx         *user=(AppCtx*)ptr;
    306:   DM             da   =user->da;
    307:   PetscReal      c    =user->c;
    308:   PetscInt       i,j,xs,ys,xm,ym,Mx,My;
    309:   PetscScalar    **u;
    310:   PetscReal      hx,hy,x,y,r;

    313:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,&My,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);

    315:   hx = 1.0/(PetscReal)(Mx-1);
    316:   hy = 1.0/(PetscReal)(My-1);

    318:   /* Get pointers to vector data */
    319:   DMDAVecGetArray(da,U,&u);

    321:   /* Get local grid boundaries */
    322:   DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);

    324:   /* Compute function over the locally owned part of the grid */
    325:   for (j=ys; j<ys+ym; j++) {
    326:     y = j*hy;
    327:     for (i=xs; i<xs+xm; i++) {
    328:       x = i*hx;
    329:       r = PetscSqrtReal((x-.5)*(x-.5) + (y-.5)*(y-.5));
    330:       if (r < .125) u[j][i] = PetscExpReal(c*r*r*r);
    331:       else u[j][i] = 0.0;
    332:     }
    333:   }

    335:   /* Restore vectors */
    336:   DMDAVecRestoreArray(da,U,&u);
    337:   return 0;
    338: }

    340: /*TEST

    342:     test:
    343:       args: -da_grid_x 20 -da_grid_y 20 -boundary 0 -ts_max_steps 10 -ts_monitor

    345:     test:
    346:       suffix: 2
    347:       args: -da_grid_x 20 -da_grid_y 20 -boundary 0 -ts_max_steps 10 -Jtype 2 -ts_monitor

    349:     test:
    350:       suffix: 3
    351:       requires: !single
    352:       args: -da_grid_x 20 -da_grid_y 20 -boundary 1 -ts_max_steps 10 -ts_monitor

    354:     test:
    355:       suffix: 4
    356:       requires: !single
    357:       nsize: 2
    358:       args: -da_grid_x 20 -da_grid_y 20 -boundary 1 -ts_max_steps 10 -ts_monitor

    360:     test:
    361:       suffix: 5
    362:       nsize: 1
    363:       args: -da_grid_x 20 -da_grid_y 20 -boundary 0 -ts_max_steps 10 -Jtype 1 -ts_monitor

    365: TEST*/