Actual source code: biharmonic.c
petsc-3.13.6 2020-09-29
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      2: static char help[] = "Solves biharmonic equation in 1d.\n";

      4: /*
      5:   Solves the equation

      7:     u_t = - kappa  \Delta \Delta u
      8:     Periodic boundary conditions

     10: Evolve the biharmonic heat equation:
     11: ---------------
     12: ./biharmonic -ts_monitor -snes_monitor   -pc_type lu  -draw_pause .1 -snes_converged_reason  -draw_pause -2   -ts_type cn  -da_refine 5 -mymonitor

     14: Evolve with the restriction that -1 <= u <= 1; i.e. as a variational inequality
     15: ---------------
     16: ./biharmonic -ts_monitor -snes_monitor   -pc_type lu  -draw_pause .1 -snes_converged_reason  -draw_pause -2   -ts_type cn   -da_refine 5  -mymonitor

     18:    u_t =  kappa \Delta \Delta u +   6.*u*(u_x)^2 + (3*u^2 - 12) \Delta u
     19:     -1 <= u <= 1
     20:     Periodic boundary conditions

     22: Evolve the Cahn-Hillard equations: double well Initial hump shrinks then grows
     23: ---------------
     24: ./biharmonic -ts_monitor -snes_monitor   -pc_type lu  -draw_pause .1 -snes_converged_reason   -draw_pause -2   -ts_type cn    -da_refine 6   -kappa .00001 -ts_dt 5.96046e-06 -cahn-hillard -ts_monitor_draw_solution --mymonitor

     26: Initial hump neither shrinks nor grows when degenerate (otherwise similar solution)

     28: ./biharmonic -ts_monitor -snes_monitor   -pc_type lu  -draw_pause .1 -snes_converged_reason   -draw_pause -2   -ts_type cn    -da_refine 6   -kappa .00001 -ts_dt 5.96046e-06 -cahn-hillard -degenerate -ts_monitor_draw_solution --mymonitor

     30: ./biharmonic -ts_monitor -snes_monitor   -pc_type lu  -draw_pause .1 -snes_converged_reason   -draw_pause -2   -ts_type cn    -da_refine 6   -kappa .00001 -ts_dt 5.96046e-06 -cahn-hillard -snes_vi_ignore_function_sign -ts_monitor_draw_solution --mymonitor

     32: Evolve the Cahn-Hillard equations: double obstacle
     33: ---------------
     34: ./biharmonic -ts_monitor -snes_monitor  -pc_type lu  -draw_pause .1 -snes_converged_reason   -draw_pause -2   -ts_type cn    -da_refine 5   -kappa .00001 -ts_dt 5.96046e-06 -cahn-hillard -energy 2 -snes_linesearch_monitor    -ts_monitor_draw_solution --mymonitor

     36: Evolve the Cahn-Hillard equations: logarithmic + double well (never shrinks and then grows)
     37: ---------------
     38: ./biharmonic -ts_monitor -snes_monitor  -pc_type lu  --snes_converged_reason  -draw_pause -2   -ts_type cn    -da_refine 5   -kappa .0001 -ts_dt 5.96046e-06 -cahn-hillard -energy 3 -snes_linesearch_monitor -theta .00000001    -ts_monitor_draw_solution --ts_max_time 1. -mymonitor

     40: ./biharmonic -ts_monitor -snes_monitor  -pc_type lu  --snes_converged_reason  -draw_pause -2   -ts_type cn    -da_refine 5   -kappa .0001 -ts_dt 5.96046e-06 -cahn-hillard -energy 3 -snes_linesearch_monitor -theta .00000001    -ts_monitor_draw_solution --ts_max_time 1. -degenerate -mymonitor

     43: Evolve the Cahn-Hillard equations: logarithmic +  double obstacle (never shrinks, never grows)
     44: ---------------
     45: ./biharmonic -ts_monitor -snes_monitor  -pc_type lu  --snes_converged_reason  -draw_pause -2   -ts_type cn    -da_refine 5   -kappa .00001 -ts_dt 5.96046e-06 -cahn-hillard -energy 4 -snes_linesearch_monitor -theta .00000001   -ts_monitor_draw_solution --mymonitor

     50: */
     51:  #include <petscdm.h>
     52:  #include <petscdmda.h>
     53:  #include <petscts.h>
     54:  #include <petscdraw.h>

     56: extern PetscErrorCode FormFunction(TS,PetscReal,Vec,Vec,void*),FormInitialSolution(DM,Vec),MyMonitor(TS,PetscInt,PetscReal,Vec,void*),MyDestroy(void**),FormJacobian(TS,PetscReal,Vec,Mat,Mat,void*);
     57: typedef struct {PetscBool cahnhillard;PetscBool degenerate;PetscReal kappa;PetscInt energy;PetscReal tol;PetscReal theta,theta_c;PetscInt truncation;PetscBool netforce; PetscDrawViewPorts *ports;} UserCtx;

     59: int main(int argc,char **argv)
     60: {
     61:   TS             ts;                 /* nonlinear solver */
     62:   Vec            x,r;                  /* solution, residual vectors */
     63:   Mat            J;                    /* Jacobian matrix */
     64:   PetscInt       steps,Mx;
     66:   DM             da;
     67:   PetscReal      dt;
     68:   PetscBool      mymonitor;
     69:   UserCtx        ctx;

     71:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     72:      Initialize program
     73:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     74:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
     75:   ctx.kappa       = 1.0;
     76:   PetscOptionsGetReal(NULL,NULL,"-kappa",&ctx.kappa,NULL);
     77:   ctx.degenerate  = PETSC_FALSE;
     78:   PetscOptionsGetBool(NULL,NULL,"-degenerate",&ctx.degenerate,NULL);
     79:   ctx.cahnhillard = PETSC_FALSE;
     80:   PetscOptionsGetBool(NULL,NULL,"-cahn-hillard",&ctx.cahnhillard,NULL);
     81:   ctx.netforce    = PETSC_FALSE;
     82:   PetscOptionsGetBool(NULL,NULL,"-netforce",&ctx.netforce,NULL);
     83:   ctx.energy      = 1;
     84:   PetscOptionsGetInt(NULL,NULL,"-energy",&ctx.energy,NULL);
     85:   ctx.tol         = 1.0e-8;
     86:   PetscOptionsGetReal(NULL,NULL,"-tol",&ctx.tol,NULL);
     87:   ctx.theta       = .001;
     88:   ctx.theta_c     = 1.0;
     89:   PetscOptionsGetReal(NULL,NULL,"-theta",&ctx.theta,NULL);
     90:   PetscOptionsGetReal(NULL,NULL,"-theta_c",&ctx.theta_c,NULL);
     91:   ctx.truncation  = 1;
     92:   PetscOptionsGetInt(NULL,NULL,"-truncation",&ctx.truncation,NULL);
     93:   PetscOptionsHasName(NULL,NULL,"-mymonitor",&mymonitor);

     95:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     96:      Create distributed array (DMDA) to manage parallel grid and vectors
     97:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     98:   DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 10,1,2,NULL,&da);
     99:   DMSetFromOptions(da);
    100:   DMSetUp(da);
    101:   DMDASetFieldName(da,0,"Biharmonic heat equation: u");
    102:   DMDAGetInfo(da,0,&Mx,0,0,0,0,0,0,0,0,0,0,0);
    103:   dt   = 1.0/(10.*ctx.kappa*Mx*Mx*Mx*Mx);

    105:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    106:      Extract global vectors from DMDA; then duplicate for remaining
    107:      vectors that are the same types
    108:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    109:   DMCreateGlobalVector(da,&x);
    110:   VecDuplicate(x,&r);

    112:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    113:      Create timestepping solver context
    114:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    115:   TSCreate(PETSC_COMM_WORLD,&ts);
    116:   TSSetDM(ts,da);
    117:   TSSetProblemType(ts,TS_NONLINEAR);
    118:   TSSetRHSFunction(ts,NULL,FormFunction,&ctx);
    119:   DMSetMatType(da,MATAIJ);
    120:   DMCreateMatrix(da,&J);
    121:   TSSetRHSJacobian(ts,J,J,FormJacobian,&ctx);
    122:   TSSetMaxTime(ts,.02);
    123:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_INTERPOLATE);

    125:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    126:      Create matrix data structure; set Jacobian evaluation routine

    128:      Set Jacobian matrix data structure and default Jacobian evaluation
    129:      routine. User can override with:
    130:      -snes_mf : matrix-free Newton-Krylov method with no preconditioning
    131:                 (unless user explicitly sets preconditioner)
    132:      -snes_mf_operator : form preconditioning matrix as set by the user,
    133:                          but use matrix-free approx for Jacobian-vector
    134:                          products within Newton-Krylov method

    136:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    137:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    138:      Customize nonlinear solver
    139:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    140:   TSSetType(ts,TSCN);

    142:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    143:      Set initial conditions
    144:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    145:   FormInitialSolution(da,x);
    146:   TSSetTimeStep(ts,dt);
    147:   TSSetSolution(ts,x);

    149:   if (mymonitor) {
    150:     ctx.ports = NULL;
    151:     TSMonitorSet(ts,MyMonitor,&ctx,MyDestroy);
    152:   }

    154:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    155:      Set runtime options
    156:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    157:   TSSetFromOptions(ts);

    159:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    160:      Solve nonlinear system
    161:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    162:   TSSolve(ts,x);
    163:   TSGetStepNumber(ts,&steps);
    164:   VecView(x,PETSC_VIEWER_BINARY_WORLD);

    166:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    167:      Free work space.  All PETSc objects should be destroyed when they
    168:      are no longer needed.
    169:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    170:   MatDestroy(&J);
    171:   VecDestroy(&x);
    172:   VecDestroy(&r);
    173:   TSDestroy(&ts);
    174:   DMDestroy(&da);

    176:   PetscFinalize();
    177:   return ierr;
    178: }
    179: /* ------------------------------------------------------------------- */
    180: /*
    181:    FormFunction - Evaluates nonlinear function, F(x).

    183:    Input Parameters:
    184: .  ts - the TS context
    185: .  X - input vector
    186: .  ptr - optional user-defined context, as set by SNESSetFunction()

    188:    Output Parameter:
    189: .  F - function vector
    190:  */
    191: PetscErrorCode FormFunction(TS ts,PetscReal ftime,Vec X,Vec F,void *ptr)
    192: {
    193:   DM             da;
    195:   PetscInt       i,Mx,xs,xm;
    196:   PetscReal      hx,sx;
    197:   PetscScalar    *x,*f,c,r,l;
    198:   Vec            localX;
    199:   UserCtx        *ctx = (UserCtx*)ptr;
    200:   PetscReal      tol  = ctx->tol, theta=ctx->theta,theta_c=ctx->theta_c,a,b; /* a and b are used in the cubic truncation of the log function */

    203:   TSGetDM(ts,&da);
    204:   DMGetLocalVector(da,&localX);
    205:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);

    207:   hx = 1.0/(PetscReal)Mx; sx = 1.0/(hx*hx);

    209:   /*
    210:      Scatter ghost points to local vector,using the 2-step process
    211:         DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
    212:      By placing code between these two statements, computations can be
    213:      done while messages are in transition.
    214:   */
    215:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
    216:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);

    218:   /*
    219:      Get pointers to vector data
    220:   */
    221:   DMDAVecGetArrayRead(da,localX,&x);
    222:   DMDAVecGetArray(da,F,&f);

    224:   /*
    225:      Get local grid boundaries
    226:   */
    227:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);

    229:   /*
    230:      Compute function over the locally owned part of the grid
    231:   */
    232:   for (i=xs; i<xs+xm; i++) {
    233:     if (ctx->degenerate) {
    234:       c = (1. - x[i]*x[i])*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    235:       r = (1. - x[i+1]*x[i+1])*(x[i] + x[i+2] - 2.0*x[i+1])*sx;
    236:       l = (1. - x[i-1]*x[i-1])*(x[i-2] + x[i] - 2.0*x[i-1])*sx;
    237:     } else {
    238:       c = (x[i-1] + x[i+1] - 2.0*x[i])*sx;
    239:       r = (x[i] + x[i+2] - 2.0*x[i+1])*sx;
    240:       l = (x[i-2] + x[i] - 2.0*x[i-1])*sx;
    241:     }
    242:     f[i] = -ctx->kappa*(l + r - 2.0*c)*sx;
    243:     if (ctx->cahnhillard) {
    244:       switch (ctx->energy) {
    245:       case 1: /*  double well */
    246:         f[i] += 6.*.25*x[i]*(x[i+1] - x[i-1])*(x[i+1] - x[i-1])*sx + (3.*x[i]*x[i] - 1.)*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    247:         break;
    248:       case 2: /* double obstacle */
    249:         f[i] += -(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    250:         break;
    251:       case 3: /* logarithmic + double well */
    252:         f[i] +=  6.*.25*x[i]*(x[i+1] - x[i-1])*(x[i+1] - x[i-1])*sx + (3.*x[i]*x[i] - 1.)*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    253:         if (ctx->truncation==2) { /* log function with approximated with a quadratic polynomial outside -1.0+2*tol, 1.0-2*tol */
    254:           if (PetscRealPart(x[i]) < -1.0 + 2.0*tol)     f[i] += (.25*theta/(tol-tol*tol))*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    255:           else if (PetscRealPart(x[i]) > 1.0 - 2.0*tol) f[i] += (.25*theta/(tol-tol*tol))*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    256:           else                                          f[i] += 2.0*theta*x[i]/((1.0-x[i]*x[i])*(1.0-x[i]*x[i]))*.25*(x[i+1] - x[i-1])*(x[i+1] - x[i-1])*sx + (theta/(1.0-x[i]*x[i]))*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    257:         } else { /* log function is approximated with a cubic polynomial outside -1.0+2*tol, 1.0-2*tol */
    258:           a = 2.0*theta*(1.0-2.0*tol)/(16.0*tol*tol*(1.0-tol)*(1.0-tol));
    259:           b = theta/(4.0*tol*(1.0-tol)) - a*(1.0-2.0*tol);
    260:           if (PetscRealPart(x[i]) < -1.0 + 2.0*tol)     f[i] += -1.0*a*.25*(x[i+1] - x[i-1])*(x[i+1] - x[i-1])*sx + (-1.0*a*x[i] + b)*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    261:           else if (PetscRealPart(x[i]) > 1.0 - 2.0*tol) f[i] +=  1.0*a*.25*(x[i+1] - x[i-1])*(x[i+1] - x[i-1])*sx + (     a*x[i] + b)*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    262:           else                                          f[i] += 2.0*theta*x[i]/((1.0-x[i]*x[i])*(1.0-x[i]*x[i]))*.25*(x[i+1] - x[i-1])*(x[i+1] - x[i-1])*sx + (theta/(1.0-x[i]*x[i]))*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    263:         }
    264:         break;
    265:       case 4: /* logarithmic + double obstacle */
    266:         f[i] += -theta_c*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    267:         if (ctx->truncation==2) { /* quadratic */
    268:           if (PetscRealPart(x[i]) < -1.0 + 2.0*tol)     f[i] += (.25*theta/(tol-tol*tol))*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    269:           else if (PetscRealPart(x[i]) > 1.0 - 2.0*tol) f[i] += (.25*theta/(tol-tol*tol))*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    270:           else                                          f[i] += 2.0*theta*x[i]/((1.0-x[i]*x[i])*(1.0-x[i]*x[i]))*.25*(x[i+1] - x[i-1])*(x[i+1] - x[i-1])*sx + (theta/(1.0-x[i]*x[i]))*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    271:         } else { /* cubic */
    272:           a = 2.0*theta*(1.0-2.0*tol)/(16.0*tol*tol*(1.0-tol)*(1.0-tol));
    273:           b = theta/(4.0*tol*(1.0-tol)) - a*(1.0-2.0*tol);
    274:           if (PetscRealPart(x[i]) < -1.0 + 2.0*tol)     f[i] += -1.0*a*.25*(x[i+1] - x[i-1])*(x[i+1] - x[i-1])*sx + (-1.0*a*x[i] + b)*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    275:           else if (PetscRealPart(x[i]) > 1.0 - 2.0*tol) f[i] +=  1.0*a*.25*(x[i+1] - x[i-1])*(x[i+1] - x[i-1])*sx + (     a*x[i] + b)*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    276:           else                                          f[i] +=  2.0*theta*x[i]/((1.0-x[i]*x[i])*(1.0-x[i]*x[i]))*.25*(x[i+1] - x[i-1])*(x[i+1] - x[i-1])*sx + (theta/(1.0-x[i]*x[i]))*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    277:         }
    278:         break;
    279:       }
    280:     }

    282:   }

    284:   /*
    285:      Restore vectors
    286:   */
    287:   DMDAVecRestoreArrayRead(da,localX,&x);
    288:   DMDAVecRestoreArray(da,F,&f);
    289:   DMRestoreLocalVector(da,&localX);
    290:   return(0);
    291: }

    293: /* ------------------------------------------------------------------- */
    294: /*
    295:    FormJacobian - Evaluates nonlinear function's Jacobian

    297: */
    298: PetscErrorCode FormJacobian(TS ts,PetscReal ftime,Vec X,Mat A,Mat B,void *ptr)
    299: {
    300:   DM             da;
    302:   PetscInt       i,Mx,xs,xm;
    303:   MatStencil     row,cols[5];
    304:   PetscReal      hx,sx;
    305:   PetscScalar    *x,vals[5];
    306:   Vec            localX;
    307:   UserCtx        *ctx = (UserCtx*)ptr;

    310:   TSGetDM(ts,&da);
    311:   DMGetLocalVector(da,&localX);
    312:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);

    314:   hx = 1.0/(PetscReal)Mx; sx = 1.0/(hx*hx);

    316:   /*
    317:      Scatter ghost points to local vector,using the 2-step process
    318:         DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
    319:      By placing code between these two statements, computations can be
    320:      done while messages are in transition.
    321:   */
    322:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
    323:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);

    325:   /*
    326:      Get pointers to vector data
    327:   */
    328:   DMDAVecGetArrayRead(da,localX,&x);

    330:   /*
    331:      Get local grid boundaries
    332:   */
    333:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);

    335:   /*
    336:      Compute function over the locally owned part of the grid
    337:   */
    338:   for (i=xs; i<xs+xm; i++) {
    339:     row.i = i;
    340:     if (ctx->degenerate) {
    341:       /*PetscScalar c,r,l;
    342:       c = (1. - x[i]*x[i])*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
    343:       r = (1. - x[i+1]*x[i+1])*(x[i] + x[i+2] - 2.0*x[i+1])*sx;
    344:       l = (1. - x[i-1]*x[i-1])*(x[i-2] + x[i] - 2.0*x[i-1])*sx; */
    345:     } else {
    346:       cols[0].i = i - 2; vals[0] = -ctx->kappa*sx*sx;
    347:       cols[1].i = i - 1; vals[1] =  4.0*ctx->kappa*sx*sx;
    348:       cols[2].i = i    ; vals[2] = -6.0*ctx->kappa*sx*sx;
    349:       cols[3].i = i + 1; vals[3] =  4.0*ctx->kappa*sx*sx;
    350:       cols[4].i = i + 2; vals[4] = -ctx->kappa*sx*sx;
    351:     }
    352:     MatSetValuesStencil(B,1,&row,5,cols,vals,INSERT_VALUES);

    354:     if (ctx->cahnhillard) {
    355:       switch (ctx->energy) {
    356:       case 1: /* double well */
    357:         /*  f[i] += 6.*.25*x[i]*(x[i+1] - x[i-1])*(x[i+1] - x[i-1])*sx + (3.*x[i]*x[i] - 1.)*(x[i-1] + x[i+1] - 2.0*x[i])*sx; */
    358:         break;
    359:       case 2: /* double obstacle */
    360:         /*        f[i] += -(x[i-1] + x[i+1] - 2.0*x[i])*sx; */
    361:         break;
    362:       case 3: /* logarithmic + double well */
    363:         break;
    364:       case 4: /* logarithmic + double obstacle */
    365:         break;
    366:       }
    367:     }

    369:   }

    371:   /*
    372:      Restore vectors
    373:   */
    374:   DMDAVecRestoreArrayRead(da,localX,&x);
    375:   DMRestoreLocalVector(da,&localX);
    376:   MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
    377:   MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
    378:   if (A != B) {
    379:     MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    380:     MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
    381:   }
    382:   return(0);
    383: }
    384: /* ------------------------------------------------------------------- */
    385: PetscErrorCode FormInitialSolution(DM da,Vec U)
    386: {
    387:   PetscErrorCode    ierr;
    388:   PetscInt          i,xs,xm,Mx,N,scale;
    389:   PetscScalar       *u;
    390:   PetscReal         r,hx,x;
    391:   const PetscScalar *f;
    392:   Vec               finesolution;
    393:   PetscViewer       viewer;

    396:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);

    398:   hx = 1.0/(PetscReal)Mx;

    400:   /*
    401:      Get pointers to vector data
    402:   */
    403:   DMDAVecGetArray(da,U,&u);

    405:   /*
    406:      Get local grid boundaries
    407:   */
    408:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);

    410:   /*
    411:       Seee heat.c for how to generate InitialSolution.heat
    412:   */
    413:   PetscViewerBinaryOpen(PETSC_COMM_WORLD,"InitialSolution.heat",FILE_MODE_READ,&viewer);
    414:   VecCreate(PETSC_COMM_WORLD,&finesolution);
    415:   VecLoad(finesolution,viewer);
    416:   PetscViewerDestroy(&viewer);
    417:   VecGetSize(finesolution,&N);
    418:   scale = N/Mx;
    419:   VecGetArrayRead(finesolution,&f);

    421:   /*
    422:      Compute function over the locally owned part of the grid
    423:   */
    424:   for (i=xs; i<xs+xm; i++) {
    425:     x = i*hx;
    426:     r = PetscSqrtReal((x-.5)*(x-.5));
    427:     if (r < .125) u[i] = 1.0;
    428:     else u[i] = -.5;

    430:     /* With the initial condition above the method is first order in space */
    431:     /* this is a smooth initial condition so the method becomes second order in space */
    432:     /*u[i] = PetscSinScalar(2*PETSC_PI*x); */
    433:     u[i] = f[scale*i];
    434:   }
    435:   VecRestoreArrayRead(finesolution,&f);
    436:   VecDestroy(&finesolution);

    438:   /*
    439:      Restore vectors
    440:   */
    441:   DMDAVecRestoreArray(da,U,&u);
    442:   return(0);
    443: }

    445: /*
    446:     This routine is not parallel
    447: */
    448: PetscErrorCode  MyMonitor(TS ts,PetscInt step,PetscReal time,Vec U,void *ptr)
    449: {
    450:   UserCtx        *ctx = (UserCtx*)ptr;
    451:   PetscDrawLG    lg;
    453:   PetscScalar    *u,l,r,c;
    454:   PetscInt       Mx,i,xs,xm,cnt;
    455:   PetscReal      x,y,hx,pause,sx,len,max,xx[4],yy[4],xx_netforce,yy_netforce,yup,ydown,y2,len2;
    456:   PetscDraw      draw;
    457:   Vec            localU;
    458:   DM             da;
    459:   int            colors[] = {PETSC_DRAW_YELLOW,PETSC_DRAW_RED,PETSC_DRAW_BLUE,PETSC_DRAW_PLUM,PETSC_DRAW_BLACK};
    460:   /*
    461:   const char *const  legend[3][3] = {{"-kappa (\\grad u,\\grad u)","(1 - u^2)^2"},{"-kappa (\\grad u,\\grad u)","(1 - u^2)"},{"-kappa (\\grad u,\\grad u)","logarithmic"}};
    462:    */
    463:   PetscDrawAxis      axis;
    464:   PetscDrawViewPorts *ports;
    465:   PetscReal          tol = ctx->tol, theta=ctx->theta,theta_c=ctx->theta_c,a,b; /* a and b are used in the cubic truncation of the log function */
    466:   PetscReal          vbounds[] = {-1.1,1.1};

    470:   PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),1,vbounds);
    471:   PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),800,600);
    472:   TSGetDM(ts,&da);
    473:   DMGetLocalVector(da,&localU);
    474:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,
    475:                      PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);
    476:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);
    477:   hx   = 1.0/(PetscReal)Mx; sx = 1.0/(hx*hx);
    478:   DMGlobalToLocalBegin(da,U,INSERT_VALUES,localU);
    479:   DMGlobalToLocalEnd(da,U,INSERT_VALUES,localU);
    480:   DMDAVecGetArrayRead(da,localU,&u);

    482:   PetscViewerDrawGetDrawLG(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),1,&lg);
    483:   PetscDrawLGGetDraw(lg,&draw);
    484:   PetscDrawCheckResizedWindow(draw);
    485:   if (!ctx->ports) {
    486:     PetscDrawViewPortsCreateRect(draw,1,3,&ctx->ports);
    487:   }
    488:   ports = ctx->ports;
    489:   PetscDrawLGGetAxis(lg,&axis);
    490:   PetscDrawLGReset(lg);

    492:   xx[0] = 0.0; xx[1] = 1.0; cnt = 2;
    493:   PetscOptionsGetRealArray(NULL,NULL,"-zoom",xx,&cnt,NULL);
    494:   xs    = xx[0]/hx; xm = (xx[1] - xx[0])/hx;

    496:   /*
    497:       Plot the  energies
    498:   */
    499:   PetscDrawLGSetDimension(lg,1 + (ctx->cahnhillard ? 1 : 0) + (ctx->energy == 3));
    500:   PetscDrawLGSetColors(lg,colors+1);
    501:   PetscDrawViewPortsSet(ports,2);
    502:   x    = hx*xs;
    503:   for (i=xs; i<xs+xm; i++) {
    504:     xx[0] = xx[1]  = xx[2] = x;
    505:     if (ctx->degenerate) yy[0] = PetscRealPart(.25*(1. - u[i]*u[i])*ctx->kappa*(u[i-1] - u[i+1])*(u[i-1] - u[i+1])*sx);
    506:     else                 yy[0] = PetscRealPart(.25*ctx->kappa*(u[i-1] - u[i+1])*(u[i-1] - u[i+1])*sx);

    508:     if (ctx->cahnhillard) {
    509:       switch (ctx->energy) {
    510:       case 1: /* double well */
    511:         yy[1] = .25*PetscRealPart((1. - u[i]*u[i])*(1. - u[i]*u[i]));
    512:         break;
    513:       case 2: /* double obstacle */
    514:         yy[1] = .5*PetscRealPart(1. - u[i]*u[i]);
    515:         break;
    516:       case 3: /* logarithm + double well */
    517:         yy[1] = .25*PetscRealPart((1. - u[i]*u[i])*(1. - u[i]*u[i]));
    518:         if (PetscRealPart(u[i]) < -1.0 + 2.0*tol)     yy[2] = .5*theta*(2.0*tol*PetscLogReal(tol) + PetscRealPart(1.0-u[i])*PetscLogReal(PetscRealPart(1.-u[i])/2.0));
    519:         else if (PetscRealPart(u[i]) > 1.0 - 2.0*tol) yy[2] = .5*theta*(PetscRealPart(1.0+u[i])*PetscLogReal(PetscRealPart(1.0+u[i])/2.0) + 2.0*tol*PetscLogReal(tol));
    520:         else                                          yy[2] = .5*theta*(PetscRealPart(1.0+u[i])*PetscLogReal(PetscRealPart(1.0+u[i])/2.0) + PetscRealPart(1.0-u[i])*PetscLogReal(PetscRealPart(1.0-u[i])/2.0));
    521:         break;
    522:       case 4: /* logarithm + double obstacle */
    523:         yy[1] = .5*theta_c*PetscRealPart(1.0-u[i]*u[i]);
    524:         if (PetscRealPart(u[i]) < -1.0 + 2.0*tol)     yy[2] = .5*theta*(2.0*tol*PetscLogReal(tol) + PetscRealPart(1.0-u[i])*PetscLogReal(PetscRealPart(1.-u[i])/2.0));
    525:         else if (PetscRealPart(u[i]) > 1.0 - 2.0*tol) yy[2] = .5*theta*(PetscRealPart(1.0+u[i])*PetscLogReal(PetscRealPart(1.0+u[i])/2.0) + 2.0*tol*PetscLogReal(tol));
    526:         else                                          yy[2] = .5*theta*(PetscRealPart(1.0+u[i])*PetscLogReal(PetscRealPart(1.0+u[i])/2.0) + PetscRealPart(1.0-u[i])*PetscLogReal(PetscRealPart(1.0-u[i])/2.0));
    527:         break;
    528:       default:
    529:         SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"It will always be one of the values");
    530:         break;
    531:       }
    532:     }
    533:     PetscDrawLGAddPoint(lg,xx,yy);
    534:     x   += hx;
    535:   }
    536:   PetscDrawGetPause(draw,&pause);
    537:   PetscDrawSetPause(draw,0.0);
    538:   PetscDrawAxisSetLabels(axis,"Energy","","");
    539:   /*  PetscDrawLGSetLegend(lg,legend[ctx->energy-1]); */
    540:   PetscDrawLGDraw(lg);

    542:   /*
    543:       Plot the  forces
    544:   */
    545:   PetscDrawLGSetDimension(lg,0 + (ctx->cahnhillard ? 2 : 0) + (ctx->energy == 3));
    546:   PetscDrawLGSetColors(lg,colors+1);
    547:   PetscDrawViewPortsSet(ports,1);
    548:   PetscDrawLGReset(lg);
    549:   x    = xs*hx;
    550:   max  = 0.;
    551:   for (i=xs; i<xs+xm; i++) {
    552:     xx[0] = xx[1] = xx[2] = xx[3] = x;
    553:     xx_netforce = x;
    554:     if (ctx->degenerate) {
    555:       c = (1. - u[i]*u[i])*(u[i-1] + u[i+1] - 2.0*u[i])*sx;
    556:       r = (1. - u[i+1]*u[i+1])*(u[i] + u[i+2] - 2.0*u[i+1])*sx;
    557:       l = (1. - u[i-1]*u[i-1])*(u[i-2] + u[i] - 2.0*u[i-1])*sx;
    558:     } else {
    559:       c = (u[i-1] + u[i+1] - 2.0*u[i])*sx;
    560:       r = (u[i] + u[i+2] - 2.0*u[i+1])*sx;
    561:       l = (u[i-2] + u[i] - 2.0*u[i-1])*sx;
    562:     }
    563:     yy[0]       = PetscRealPart(-ctx->kappa*(l + r - 2.0*c)*sx);
    564:     yy_netforce = yy[0];
    565:     max         = PetscMax(max,PetscAbs(yy[0]));
    566:     if (ctx->cahnhillard) {
    567:       switch (ctx->energy) {
    568:       case 1: /* double well */
    569:         yy[1] = PetscRealPart(6.*.25*u[i]*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (3.*u[i]*u[i] - 1.)*(u[i-1] + u[i+1] - 2.0*u[i])*sx);
    570:         break;
    571:       case 2: /* double obstacle */
    572:         yy[1] = -PetscRealPart(u[i-1] + u[i+1] - 2.0*u[i])*sx;
    573:         break;
    574:       case 3: /* logarithmic + double well */
    575:         yy[1] =  PetscRealPart(6.*.25*u[i]*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (3.*u[i]*u[i] - 1.)*(u[i-1] + u[i+1] - 2.0*u[i])*sx);
    576:         if (ctx->truncation==2) { /* quadratic */
    577:           if (PetscRealPart(u[i]) < -1.0 + 2.0*tol)     yy[2] = (.25*theta/(tol-tol*tol))*PetscRealPart(u[i-1] + u[i+1] - 2.0*u[i])*sx;
    578:           else if (PetscRealPart(u[i]) > 1.0 - 2.0*tol) yy[2] = (.25*theta/(tol-tol*tol))*PetscRealPart(u[i-1] + u[i+1] - 2.0*u[i])*sx;
    579:           else                                          yy[2] = PetscRealPart(2.0*theta*u[i]/((1.0-u[i]*u[i])*(1.0-u[i]*u[i]))*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (theta/(1.0-u[i]*u[i]))*(u[i-1] + u[i+1] - 2.0*u[i])*sx);
    580:         } else { /* cubic */
    581:           a = 2.0*theta*(1.0-2.0*tol)/(16.0*tol*tol*(1.0-tol)*(1.0-tol));
    582:           b = theta/(4.0*tol*(1.0-tol)) - a*(1.0-2.0*tol);
    583:           if (PetscRealPart(u[i]) < -1.0 + 2.0*tol)     yy[2] = PetscRealPart(-1.0*a*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (-1.0*a*u[i] + b)*(u[i-1] + u[i+1] - 2.0*u[i])*sx);
    584:           else if (PetscRealPart(u[i]) > 1.0 - 2.0*tol) yy[2] =  PetscRealPart(1.0*a*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (     a*u[i] + b)*(u[i-1] + u[i+1] - 2.0*u[i])*sx);
    585:           else                                          yy[2] = PetscRealPart(2.0*theta*u[i]/((1.0-u[i]*u[i])*(1.0-u[i]*u[i]))*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (theta/(1.0-u[i]*u[i]))*(u[i-1] + u[i+1] - 2.0*u[i])*sx);
    586:         }
    587:         break;
    588:       case 4: /* logarithmic + double obstacle */
    589:         yy[1] = theta_c*PetscRealPart(-(u[i-1] + u[i+1] - 2.0*u[i]))*sx;
    590:         if (ctx->truncation==2) {
    591:           if (PetscRealPart(u[i]) < -1.0 + 2.0*tol)     yy[2] = (.25*theta/(tol-tol*tol))*PetscRealPart(u[i-1] + u[i+1] - 2.0*u[i])*sx;
    592:           else if (PetscRealPart(u[i]) > 1.0 - 2.0*tol) yy[2] = (.25*theta/(tol-tol*tol))*PetscRealPart(u[i-1] + u[i+1] - 2.0*u[i])*sx;
    593:           else                                          yy[2] = PetscRealPart(2.0*theta*u[i]/((1.0-u[i]*u[i])*(1.0-u[i]*u[i]))*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (theta/(1.0-u[i]*u[i]))*(u[i-1] + u[i+1] - 2.0*u[i])*sx);
    594:         }
    595:         else {
    596:           a = 2.0*theta*(1.0-2.0*tol)/(16.0*tol*tol*(1.0-tol)*(1.0-tol));
    597:           b = theta/(4.0*tol*(1.0-tol)) - a*(1.0-2.0*tol);
    598:           if (PetscRealPart(u[i]) < -1.0 + 2.0*tol)     yy[2] = PetscRealPart(-1.0*a*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (-1.0*a*u[i] + b)*(u[i-1] + u[i+1] - 2.0*u[i])*sx);
    599:           else if (PetscRealPart(u[i]) > 1.0 - 2.0*tol) yy[2] =  PetscRealPart(1.0*a*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (     a*u[i] + b)*(u[i-1] + u[i+1] - 2.0*u[i])*sx);
    600:           else                                          yy[2] =  PetscRealPart(2.0*theta*u[i]/((1.0-u[i]*u[i])*(1.0-u[i]*u[i]))*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (theta/(1.0-u[i]*u[i]))*(u[i-1] + u[i+1] - 2.0*u[i])*sx);
    601:         }
    602:         break;
    603:       default:
    604:         SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"It will always be one of the values");
    605:       break;
    606:       }
    607:       if (ctx->energy < 3) {
    608:         max         = PetscMax(max,PetscAbs(yy[1]));
    609:         yy[2]       = yy[0]+yy[1];
    610:         yy_netforce = yy[2];
    611:       } else {
    612:         max         = PetscMax(max,PetscAbs(yy[1]+yy[2]));
    613:         yy[3]       = yy[0]+yy[1]+yy[2];
    614:         yy_netforce = yy[3];
    615:       }
    616:     }
    617:     if (ctx->netforce) {
    618:       PetscDrawLGAddPoint(lg,&xx_netforce,&yy_netforce);
    619:     } else {
    620:       PetscDrawLGAddPoint(lg,xx,yy);
    621:     }
    622:     x += hx;
    623:     /*if (max > 7200150000.0) */
    624:     /* printf("max very big when i = %d\n",i); */
    625:   }
    626:   PetscDrawAxisSetLabels(axis,"Right hand side","","");
    627:   PetscDrawLGSetLegend(lg,NULL);
    628:   PetscDrawLGDraw(lg);

    630:   /*
    631:         Plot the solution
    632:   */
    633:   PetscDrawLGSetDimension(lg,1);
    634:   PetscDrawViewPortsSet(ports,0);
    635:   PetscDrawLGReset(lg);
    636:   x    = hx*xs;
    637:   PetscDrawLGSetLimits(lg,x,x+(xm-1)*hx,-1.1,1.1);
    638:   PetscDrawLGSetColors(lg,colors);
    639:   for (i=xs; i<xs+xm; i++) {
    640:     xx[0] = x;
    641:     yy[0] = PetscRealPart(u[i]);
    642:     PetscDrawLGAddPoint(lg,xx,yy);
    643:     x    += hx;
    644:   }
    645:   PetscDrawAxisSetLabels(axis,"Solution","","");
    646:   PetscDrawLGDraw(lg);

    648:   /*
    649:       Print the  forces as arrows on the solution
    650:   */
    651:   x   = hx*xs;
    652:   cnt = xm/60;
    653:   cnt = (!cnt) ? 1 : cnt;

    655:   for (i=xs; i<xs+xm; i += cnt) {
    656:     y    = yup = ydown = PetscRealPart(u[i]);
    657:     c    = (u[i-1] + u[i+1] - 2.0*u[i])*sx;
    658:     r    = (u[i] + u[i+2] - 2.0*u[i+1])*sx;
    659:     l    = (u[i-2] + u[i] - 2.0*u[i-1])*sx;
    660:     len  = -.5*PetscRealPart(ctx->kappa*(l + r - 2.0*c)*sx)/max;
    661:     PetscDrawArrow(draw,x,y,x,y+len,PETSC_DRAW_RED);
    662:     if (ctx->cahnhillard) {
    663:       if (len < 0.) ydown += len;
    664:       else yup += len;

    666:       switch (ctx->energy) {
    667:       case 1: /* double well */
    668:         len = .5*PetscRealPart(6.*.25*u[i]*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (3.*u[i]*u[i] - 1.)*(u[i-1] + u[i+1] - 2.0*u[i])*sx)/max;
    669:         break;
    670:       case 2: /* double obstacle */
    671:         len = -.5*PetscRealPart(u[i-1] + u[i+1] - 2.0*u[i])*sx/max;
    672:         break;
    673:       case 3: /* logarithmic + double well */
    674:         len = .5*PetscRealPart(6.*.25*u[i]*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (3.*u[i]*u[i] - 1.)*(u[i-1] + u[i+1] - 2.0*u[i])*sx)/max;
    675:         if (len < 0.) ydown += len;
    676:         else yup += len;

    678:         if (ctx->truncation==2) { /* quadratic */
    679:           if (PetscRealPart(u[i]) < -1.0 + 2.0*tol)     len2 = .5*(.25*theta/(tol-tol*tol))*PetscRealPart(u[i-1] + u[i+1] - 2.0*u[i])*sx/max;
    680:           else if (PetscRealPart(u[i]) > 1.0 - 2.0*tol) len2 = .5*(.25*theta/(tol-tol*tol))*PetscRealPart(u[i-1] + u[i+1] - 2.0*u[i])*sx/max;
    681:           else                                          len2 = PetscRealPart(.5*(2.0*theta*u[i]/((1.0-u[i]*u[i])*(1.0-u[i]*u[i]))*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (theta/(1.0-u[i]*u[i]))*(u[i-1] + u[i+1] - 2.0*u[i])*sx)/max);
    682:         } else { /* cubic */
    683:           a = 2.0*theta*(1.0-2.0*tol)/(16.0*tol*tol*(1.0-tol)*(1.0-tol));
    684:           b = theta/(4.0*tol*(1.0-tol)) - a*(1.0-2.0*tol);
    685:           if (PetscRealPart(u[i]) < -1.0 + 2.0*tol)     len2 = PetscRealPart(.5*(-1.0*a*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (-1.0*a*u[i] + b)*(u[i-1] + u[i+1] - 2.0*u[i])*sx)/max);
    686:           else if (PetscRealPart(u[i]) > 1.0 - 2.0*tol) len2 = PetscRealPart(.5*(a*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (     a*u[i] + b)*(u[i-1] + u[i+1] - 2.0*u[i])*sx)/max);
    687:           else                                          len2 = PetscRealPart(.5*(2.0*theta*u[i]/((1.0-u[i]*u[i])*(1.0-u[i]*u[i]))*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (theta/(1.0-u[i]*u[i]))*(u[i-1] + u[i+1] - 2.0*u[i])*sx)/max);
    688:         }
    689:         y2   = len < 0 ? ydown : yup;
    690:         PetscDrawArrow(draw,x,y2,x,y2+len2,PETSC_DRAW_PLUM);
    691:         break;
    692:       case 4: /* logarithmic + double obstacle */
    693:         len = -.5*theta_c*PetscRealPart(-(u[i-1] + u[i+1] - 2.0*u[i])*sx/max);
    694:         if (len < 0.) ydown += len;
    695:         else yup += len;

    697:         if (ctx->truncation==2) { /* quadratic */
    698:           if (PetscRealPart(u[i]) < -1.0 + 2.0*tol)     len2 = .5*(.25*theta/(tol-tol*tol))*PetscRealPart(u[i-1] + u[i+1] - 2.0*u[i])*sx/max;
    699:           else if (PetscRealPart(u[i]) > 1.0 - 2.0*tol) len2 = .5*(.25*theta/(tol-tol*tol))*PetscRealPart(u[i-1] + u[i+1] - 2.0*u[i])*sx/max;
    700:           else                                          len2 = PetscRealPart(.5*(2.0*theta*u[i]/((1.0-u[i]*u[i])*(1.0-u[i]*u[i]))*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (theta/(1.0-u[i]*u[i]))*(u[i-1] + u[i+1] - 2.0*u[i])*sx)/max);
    701:         } else { /* cubic */
    702:           a = 2.0*theta*(1.0-2.0*tol)/(16.0*tol*tol*(1.0-tol)*(1.0-tol));
    703:           b = theta/(4.0*tol*(1.0-tol)) - a*(1.0-2.0*tol);
    704:           if (PetscRealPart(u[i]) < -1.0 + 2.0*tol)     len2 = .5*PetscRealPart(-1.0*a*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (-1.0*a*u[i] + b)*(u[i-1] + u[i+1] - 2.0*u[i])*sx)/max;
    705:           else if (PetscRealPart(u[i]) > 1.0 - 2.0*tol) len2 =  .5*PetscRealPart(a*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (     a*u[i] + b)*(u[i-1] + u[i+1] - 2.0*u[i])*sx)/max;
    706:           else                                          len2 =  .5*PetscRealPart(2.0*theta*u[i]/((1.0-u[i]*u[i])*(1.0-u[i]*u[i]))*.25*(u[i+1] - u[i-1])*(u[i+1] - u[i-1])*sx + (theta/(1.0-u[i]*u[i]))*(u[i-1] + u[i+1] - 2.0*u[i])*sx)/max;
    707:         }
    708:         y2   = len < 0 ? ydown : yup;
    709:         PetscDrawArrow(draw,x,y2,x,y2+len2,PETSC_DRAW_PLUM);
    710:         break;
    711:       }
    712:       PetscDrawArrow(draw,x,y,x,y+len,PETSC_DRAW_BLUE);
    713:     }
    714:     x += cnt*hx;
    715:   }
    716:   DMDAVecRestoreArrayRead(da,localU,&x);
    717:   DMRestoreLocalVector(da,&localU);
    718:   PetscDrawStringSetSize(draw,.2,.2);
    719:   PetscDrawFlush(draw);
    720:   PetscDrawSetPause(draw,pause);
    721:   PetscDrawPause(draw);
    722:   return(0);
    723: }

    725: PetscErrorCode  MyDestroy(void **ptr)
    726: {
    727:   UserCtx        *ctx = *(UserCtx**)ptr;

    731:   PetscDrawViewPortsDestroy(ctx->ports);
    732:   return(0);
    733: }

    737: /*TEST

    739:    test:
    740:      TODO: currently requires initial condition file generated by heat

    742: TEST*/