xref: /petsc/src/ts/tutorials/phasefield/biharmonic.c (revision c4762a1b19cd2af06abeed90e8f9d34fb975dd94)

static char help[] = "Solves biharmonic equation in 1d.\n";

/*
  Solves the equation

    u_t = - kappa  \Delta \Delta u
    Periodic boundary conditions

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

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

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

Evolve the Cahn-Hillard equations: double well Initial hump shrinks then grows
---------------
./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

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

./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

./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

Evolve the Cahn-Hillard equations: double obstacle
---------------
./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

Evolve the Cahn-Hillard equations: logarithmic + double well (never shrinks and then grows)
---------------
./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

./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


Evolve the Cahn-Hillard equations: logarithmic +  double obstacle (never shrinks, never grows)
---------------
./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




*/
#include <petscdm.h>
#include <petscdmda.h>
#include <petscts.h>
#include <petscdraw.h>

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*);
typedef struct {PetscBool cahnhillard;PetscBool degenerate;PetscReal kappa;PetscInt energy;PetscReal tol;PetscReal theta,theta_c;PetscInt truncation;PetscBool netforce; PetscDrawViewPorts *ports;} UserCtx;

int main(int argc,char **argv)
{
  TS             ts;                 /* nonlinear solver */
  Vec            x,r;                  /* solution, residual vectors */
  Mat            J;                    /* Jacobian matrix */
  PetscInt       steps,Mx;
  PetscErrorCode ierr;
  DM             da;
  PetscReal      dt;
  PetscBool      mymonitor;
  UserCtx        ctx;

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Initialize program
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
  ctx.kappa       = 1.0;
  ierr            = PetscOptionsGetReal(NULL,NULL,"-kappa",&ctx.kappa,NULL);CHKERRQ(ierr);
  ctx.degenerate  = PETSC_FALSE;
  ierr            = PetscOptionsGetBool(NULL,NULL,"-degenerate",&ctx.degenerate,NULL);CHKERRQ(ierr);
  ctx.cahnhillard = PETSC_FALSE;
  ierr            = PetscOptionsGetBool(NULL,NULL,"-cahn-hillard",&ctx.cahnhillard,NULL);CHKERRQ(ierr);
  ctx.netforce    = PETSC_FALSE;
  ierr            = PetscOptionsGetBool(NULL,NULL,"-netforce",&ctx.netforce,NULL);CHKERRQ(ierr);
  ctx.energy      = 1;
  ierr            = PetscOptionsGetInt(NULL,NULL,"-energy",&ctx.energy,NULL);CHKERRQ(ierr);
  ctx.tol         = 1.0e-8;
  ierr            = PetscOptionsGetReal(NULL,NULL,"-tol",&ctx.tol,NULL);CHKERRQ(ierr);
  ctx.theta       = .001;
  ctx.theta_c     = 1.0;
  ierr            = PetscOptionsGetReal(NULL,NULL,"-theta",&ctx.theta,NULL);CHKERRQ(ierr);
  ierr            = PetscOptionsGetReal(NULL,NULL,"-theta_c",&ctx.theta_c,NULL);CHKERRQ(ierr);
  ctx.truncation  = 1;
  ierr            = PetscOptionsGetInt(NULL,NULL,"-truncation",&ctx.truncation,NULL);CHKERRQ(ierr);
  ierr            = PetscOptionsHasName(NULL,NULL,"-mymonitor",&mymonitor);CHKERRQ(ierr);

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

  /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Extract global vectors from DMDA; then duplicate for remaining
     vectors that are the same types
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = DMCreateGlobalVector(da,&x);CHKERRQ(ierr);
  ierr = VecDuplicate(x,&r);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create timestepping solver context
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr);
  ierr = TSSetDM(ts,da);CHKERRQ(ierr);
  ierr = TSSetProblemType(ts,TS_NONLINEAR);CHKERRQ(ierr);
  ierr = TSSetRHSFunction(ts,NULL,FormFunction,&ctx);CHKERRQ(ierr);
  ierr = DMSetMatType(da,MATAIJ);CHKERRQ(ierr);
  ierr = DMCreateMatrix(da,&J);CHKERRQ(ierr);
  ierr = TSSetRHSJacobian(ts,J,J,FormJacobian,&ctx);CHKERRQ(ierr);
  ierr = TSSetMaxTime(ts,.02);CHKERRQ(ierr);
  ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_INTERPOLATE);CHKERRQ(ierr);

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

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

     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Customize nonlinear solver
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = TSSetType(ts,TSCN);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set initial conditions
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = FormInitialSolution(da,x);CHKERRQ(ierr);
  ierr = TSSetTimeStep(ts,dt);CHKERRQ(ierr);
  ierr = TSSetSolution(ts,x);CHKERRQ(ierr);

  if (mymonitor) {
    ctx.ports = NULL;
    ierr      = TSMonitorSet(ts,MyMonitor,&ctx,MyDestroy);CHKERRQ(ierr);
  }

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set runtime options
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = TSSetFromOptions(ts);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Solve nonlinear system
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = TSSolve(ts,x);CHKERRQ(ierr);
  ierr = TSGetStepNumber(ts,&steps);CHKERRQ(ierr);
  ierr = VecView(x,PETSC_VIEWER_BINARY_WORLD);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Free work space.  All PETSc objects should be destroyed when they
     are no longer needed.
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = MatDestroy(&J);CHKERRQ(ierr);
  ierr = VecDestroy(&x);CHKERRQ(ierr);
  ierr = VecDestroy(&r);CHKERRQ(ierr);
  ierr = TSDestroy(&ts);CHKERRQ(ierr);
  ierr = DMDestroy(&da);CHKERRQ(ierr);

  ierr = PetscFinalize();
  return ierr;
}
/* ------------------------------------------------------------------- */
/*
   FormFunction - Evaluates nonlinear function, F(x).

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

   Output Parameter:
.  F - function vector
 */
PetscErrorCode FormFunction(TS ts,PetscReal ftime,Vec X,Vec F,void *ptr)
{
  DM             da;
  PetscErrorCode ierr;
  PetscInt       i,Mx,xs,xm;
  PetscReal      hx,sx;
  PetscScalar    *x,*f,c,r,l;
  Vec            localX;
  UserCtx        *ctx = (UserCtx*)ptr;
  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 */

  PetscFunctionBegin;
  ierr = TSGetDM(ts,&da);CHKERRQ(ierr);
  ierr = DMGetLocalVector(da,&localX);CHKERRQ(ierr);
  ierr = 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);CHKERRQ(ierr);

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

  /*
     Scatter ghost points to local vector,using the 2-step process
        DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
     By placing code between these two statements, computations can be
     done while messages are in transition.
  */
  ierr = DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);CHKERRQ(ierr);
  ierr = DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);CHKERRQ(ierr);

  /*
     Get pointers to vector data
  */
  ierr = DMDAVecGetArrayRead(da,localX,&x);CHKERRQ(ierr);
  ierr = DMDAVecGetArray(da,F,&f);CHKERRQ(ierr);

  /*
     Get local grid boundaries
  */
  ierr = DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);CHKERRQ(ierr);

  /*
     Compute function over the locally owned part of the grid
  */
  for (i=xs; i<xs+xm; i++) {
    if (ctx->degenerate) {
      c = (1. - x[i]*x[i])*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
      r = (1. - x[i+1]*x[i+1])*(x[i] + x[i+2] - 2.0*x[i+1])*sx;
      l = (1. - x[i-1]*x[i-1])*(x[i-2] + x[i] - 2.0*x[i-1])*sx;
    } else {
      c = (x[i-1] + x[i+1] - 2.0*x[i])*sx;
      r = (x[i] + x[i+2] - 2.0*x[i+1])*sx;
      l = (x[i-2] + x[i] - 2.0*x[i-1])*sx;
    }
    f[i] = -ctx->kappa*(l + r - 2.0*c)*sx;
    if (ctx->cahnhillard) {
      switch (ctx->energy) {
      case 1: /*  double well */
        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;
        break;
      case 2: /* double obstacle */
        f[i] += -(x[i-1] + x[i+1] - 2.0*x[i])*sx;
        break;
      case 3: /* logarithmic + double well */
        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;
        if (ctx->truncation==2) { /* log function with approximated with a quadratic polynomial outside -1.0+2*tol, 1.0-2*tol */
          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;
          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;
          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;
        } else { /* log function is approximated with a cubic polynomial outside -1.0+2*tol, 1.0-2*tol */
          a = 2.0*theta*(1.0-2.0*tol)/(16.0*tol*tol*(1.0-tol)*(1.0-tol));
          b = theta/(4.0*tol*(1.0-tol)) - a*(1.0-2.0*tol);
          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;
          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;
          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;
        }
        break;
      case 4: /* logarithmic + double obstacle */
        f[i] += -theta_c*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
        if (ctx->truncation==2) { /* quadratic */
          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;
          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;
          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;
        } else { /* cubic */
          a = 2.0*theta*(1.0-2.0*tol)/(16.0*tol*tol*(1.0-tol)*(1.0-tol));
          b = theta/(4.0*tol*(1.0-tol)) - a*(1.0-2.0*tol);
          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;
          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;
          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;
        }
        break;
      }
    }

  }

  /*
     Restore vectors
  */
  ierr = DMDAVecRestoreArrayRead(da,localX,&x);CHKERRQ(ierr);
  ierr = DMDAVecRestoreArray(da,F,&f);CHKERRQ(ierr);
  ierr = DMRestoreLocalVector(da,&localX);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* ------------------------------------------------------------------- */
/*
   FormJacobian - Evaluates nonlinear function's Jacobian

*/
PetscErrorCode FormJacobian(TS ts,PetscReal ftime,Vec X,Mat A,Mat B,void *ptr)
{
  DM             da;
  PetscErrorCode ierr;
  PetscInt       i,Mx,xs,xm;
  MatStencil     row,cols[5];
  PetscReal      hx,sx;
  PetscScalar    *x,vals[5];
  Vec            localX;
  UserCtx        *ctx = (UserCtx*)ptr;

  PetscFunctionBegin;
  ierr = TSGetDM(ts,&da);CHKERRQ(ierr);
  ierr = DMGetLocalVector(da,&localX);CHKERRQ(ierr);
  ierr = 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);CHKERRQ(ierr);

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

  /*
     Scatter ghost points to local vector,using the 2-step process
        DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
     By placing code between these two statements, computations can be
     done while messages are in transition.
  */
  ierr = DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);CHKERRQ(ierr);
  ierr = DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);CHKERRQ(ierr);

  /*
     Get pointers to vector data
  */
  ierr = DMDAVecGetArrayRead(da,localX,&x);CHKERRQ(ierr);

  /*
     Get local grid boundaries
  */
  ierr = DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);CHKERRQ(ierr);

  /*
     Compute function over the locally owned part of the grid
  */
  for (i=xs; i<xs+xm; i++) {
    row.i = i;
    if (ctx->degenerate) {
      /*PetscScalar c,r,l;
      c = (1. - x[i]*x[i])*(x[i-1] + x[i+1] - 2.0*x[i])*sx;
      r = (1. - x[i+1]*x[i+1])*(x[i] + x[i+2] - 2.0*x[i+1])*sx;
      l = (1. - x[i-1]*x[i-1])*(x[i-2] + x[i] - 2.0*x[i-1])*sx; */
    } else {
      cols[0].i = i - 2; vals[0] = -ctx->kappa*sx*sx;
      cols[1].i = i - 1; vals[1] =  4.0*ctx->kappa*sx*sx;
      cols[2].i = i    ; vals[2] = -6.0*ctx->kappa*sx*sx;
      cols[3].i = i + 1; vals[3] =  4.0*ctx->kappa*sx*sx;
      cols[4].i = i + 2; vals[4] = -ctx->kappa*sx*sx;
    }
    ierr = MatSetValuesStencil(B,1,&row,5,cols,vals,INSERT_VALUES);CHKERRQ(ierr);

    if (ctx->cahnhillard) {
      switch (ctx->energy) {
      case 1: /* double well */
        /*  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; */
        break;
      case 2: /* double obstacle */
        /*        f[i] += -(x[i-1] + x[i+1] - 2.0*x[i])*sx; */
        break;
      case 3: /* logarithmic + double well */
        break;
      case 4: /* logarithmic + double obstacle */
        break;
      }
    }

  }

  /*
     Restore vectors
  */
  ierr = DMDAVecRestoreArrayRead(da,localX,&x);CHKERRQ(ierr);
  ierr = DMRestoreLocalVector(da,&localX);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  if (A != B) {
    ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
    ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}
/* ------------------------------------------------------------------- */
PetscErrorCode FormInitialSolution(DM da,Vec U)
{
  PetscErrorCode    ierr;
  PetscInt          i,xs,xm,Mx,N,scale;
  PetscScalar       *u;
  PetscReal         r,hx,x;
  const PetscScalar *f;
  Vec               finesolution;
  PetscViewer       viewer;

  PetscFunctionBegin;
  ierr = 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);CHKERRQ(ierr);

  hx = 1.0/(PetscReal)Mx;

  /*
     Get pointers to vector data
  */
  ierr = DMDAVecGetArray(da,U,&u);CHKERRQ(ierr);

  /*
     Get local grid boundaries
  */
  ierr = DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);CHKERRQ(ierr);

  /*
      Seee heat.c for how to generate InitialSolution.heat
  */
  ierr  = PetscViewerBinaryOpen(PETSC_COMM_WORLD,"InitialSolution.heat",FILE_MODE_READ,&viewer);CHKERRQ(ierr);
  ierr  = VecCreate(PETSC_COMM_WORLD,&finesolution);CHKERRQ(ierr);
  ierr  = VecLoad(finesolution,viewer);CHKERRQ(ierr);
  ierr  = PetscViewerDestroy(&viewer);CHKERRQ(ierr);
  ierr  = VecGetSize(finesolution,&N);CHKERRQ(ierr);
  scale = N/Mx;
  ierr  = VecGetArrayRead(finesolution,&f);CHKERRQ(ierr);

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

    /* With the initial condition above the method is first order in space */
    /* this is a smooth initial condition so the method becomes second order in space */
    /*u[i] = PetscSinScalar(2*PETSC_PI*x); */
    u[i] = f[scale*i];
  }
  ierr = VecRestoreArrayRead(finesolution,&f);CHKERRQ(ierr);
  ierr = VecDestroy(&finesolution);CHKERRQ(ierr);

  /*
     Restore vectors
  */
  ierr = DMDAVecRestoreArray(da,U,&u);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*
    This routine is not parallel
*/
PetscErrorCode  MyMonitor(TS ts,PetscInt step,PetscReal time,Vec U,void *ptr)
{
  UserCtx        *ctx = (UserCtx*)ptr;
  PetscDrawLG    lg;
  PetscErrorCode ierr;
  PetscScalar    *u,l,r,c;
  PetscInt       Mx,i,xs,xm,cnt;
  PetscReal      x,y,hx,pause,sx,len,max,xx[4],yy[4],xx_netforce,yy_netforce,yup,ydown,y2,len2;
  PetscDraw      draw;
  Vec            localU;
  DM             da;
  int            colors[] = {PETSC_DRAW_YELLOW,PETSC_DRAW_RED,PETSC_DRAW_BLUE,PETSC_DRAW_PLUM,PETSC_DRAW_BLACK};
  /*
  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"}};
   */
  PetscDrawAxis      axis;
  PetscDrawViewPorts *ports;
  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 */
  PetscReal          vbounds[] = {-1.1,1.1};


  PetscFunctionBegin;
  ierr = PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),1,vbounds);CHKERRQ(ierr);
  ierr = PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),800,600);CHKERRQ(ierr);
  ierr = TSGetDM(ts,&da);CHKERRQ(ierr);
  ierr = DMGetLocalVector(da,&localU);CHKERRQ(ierr);
  ierr = 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);CHKERRQ(ierr);
  ierr = DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);CHKERRQ(ierr);
  hx   = 1.0/(PetscReal)Mx; sx = 1.0/(hx*hx);
  ierr = DMGlobalToLocalBegin(da,U,INSERT_VALUES,localU);CHKERRQ(ierr);
  ierr = DMGlobalToLocalEnd(da,U,INSERT_VALUES,localU);CHKERRQ(ierr);
  ierr = DMDAVecGetArrayRead(da,localU,&u);CHKERRQ(ierr);

  ierr = PetscViewerDrawGetDrawLG(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),1,&lg);CHKERRQ(ierr);
  ierr = PetscDrawLGGetDraw(lg,&draw);CHKERRQ(ierr);
  ierr = PetscDrawCheckResizedWindow(draw);CHKERRQ(ierr);
  if (!ctx->ports) {
    ierr = PetscDrawViewPortsCreateRect(draw,1,3,&ctx->ports);CHKERRQ(ierr);
  }
  ports = ctx->ports;
  ierr  = PetscDrawLGGetAxis(lg,&axis);CHKERRQ(ierr);
  ierr  = PetscDrawLGReset(lg);CHKERRQ(ierr);

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

  /*
      Plot the  energies
  */
  ierr = PetscDrawLGSetDimension(lg,1 + (ctx->cahnhillard ? 1 : 0) + (ctx->energy == 3));CHKERRQ(ierr);
  ierr = PetscDrawLGSetColors(lg,colors+1);CHKERRQ(ierr);
  ierr = PetscDrawViewPortsSet(ports,2);CHKERRQ(ierr);
  x    = hx*xs;
  for (i=xs; i<xs+xm; i++) {
    xx[0] = xx[1]  = xx[2] = x;
    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);
    else                 yy[0] = PetscRealPart(.25*ctx->kappa*(u[i-1] - u[i+1])*(u[i-1] - u[i+1])*sx);

    if (ctx->cahnhillard) {
      switch (ctx->energy) {
      case 1: /* double well */
        yy[1] = .25*PetscRealPart((1. - u[i]*u[i])*(1. - u[i]*u[i]));
        break;
      case 2: /* double obstacle */
        yy[1] = .5*PetscRealPart(1. - u[i]*u[i]);
        break;
      case 3: /* logarithm + double well */
        yy[1] = .25*PetscRealPart((1. - u[i]*u[i])*(1. - u[i]*u[i]));
        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));
        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));
        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));
        break;
      case 4: /* logarithm + double obstacle */
        yy[1] = .5*theta_c*PetscRealPart(1.0-u[i]*u[i]);
        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));
        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));
        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));
        break;
      default:
        SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"It will always be one of the values");
        break;
      }
    }
    ierr = PetscDrawLGAddPoint(lg,xx,yy);CHKERRQ(ierr);
    x   += hx;
  }
  ierr = PetscDrawGetPause(draw,&pause);CHKERRQ(ierr);
  ierr = PetscDrawSetPause(draw,0.0);CHKERRQ(ierr);
  ierr = PetscDrawAxisSetLabels(axis,"Energy","","");CHKERRQ(ierr);
  /*  ierr = PetscDrawLGSetLegend(lg,legend[ctx->energy-1]);CHKERRQ(ierr); */
  ierr = PetscDrawLGDraw(lg);CHKERRQ(ierr);

  /*
      Plot the  forces
  */
  ierr = PetscDrawLGSetDimension(lg,0 + (ctx->cahnhillard ? 2 : 0) + (ctx->energy == 3));CHKERRQ(ierr);
  ierr = PetscDrawLGSetColors(lg,colors+1);CHKERRQ(ierr);
  ierr = PetscDrawViewPortsSet(ports,1);CHKERRQ(ierr);
  ierr = PetscDrawLGReset(lg);CHKERRQ(ierr);
  x    = xs*hx;
  max  = 0.;
  for (i=xs; i<xs+xm; i++) {
    xx[0] = xx[1] = xx[2] = xx[3] = x;
    xx_netforce = x;
    if (ctx->degenerate) {
      c = (1. - u[i]*u[i])*(u[i-1] + u[i+1] - 2.0*u[i])*sx;
      r = (1. - u[i+1]*u[i+1])*(u[i] + u[i+2] - 2.0*u[i+1])*sx;
      l = (1. - u[i-1]*u[i-1])*(u[i-2] + u[i] - 2.0*u[i-1])*sx;
    } else {
      c = (u[i-1] + u[i+1] - 2.0*u[i])*sx;
      r = (u[i] + u[i+2] - 2.0*u[i+1])*sx;
      l = (u[i-2] + u[i] - 2.0*u[i-1])*sx;
    }
    yy[0]       = PetscRealPart(-ctx->kappa*(l + r - 2.0*c)*sx);
    yy_netforce = yy[0];
    max         = PetscMax(max,PetscAbs(yy[0]));
    if (ctx->cahnhillard) {
      switch (ctx->energy) {
      case 1: /* double well */
        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);
        break;
      case 2: /* double obstacle */
        yy[1] = -PetscRealPart(u[i-1] + u[i+1] - 2.0*u[i])*sx;
        break;
      case 3: /* logarithmic + double well */
        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);
        if (ctx->truncation==2) { /* quadratic */
          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;
          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;
          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);
        } else { /* cubic */
          a = 2.0*theta*(1.0-2.0*tol)/(16.0*tol*tol*(1.0-tol)*(1.0-tol));
          b = theta/(4.0*tol*(1.0-tol)) - a*(1.0-2.0*tol);
          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);
          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);
          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);
        }
        break;
      case 4: /* logarithmic + double obstacle */
        yy[1] = theta_c*PetscRealPart(-(u[i-1] + u[i+1] - 2.0*u[i]))*sx;
        if (ctx->truncation==2) {
          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;
          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;
          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);
        }
        else {
          a = 2.0*theta*(1.0-2.0*tol)/(16.0*tol*tol*(1.0-tol)*(1.0-tol));
          b = theta/(4.0*tol*(1.0-tol)) - a*(1.0-2.0*tol);
          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);
          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);
          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);
        }
        break;
      default:
        SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"It will always be one of the values");
      break;
      }
      if (ctx->energy < 3) {
        max         = PetscMax(max,PetscAbs(yy[1]));
        yy[2]       = yy[0]+yy[1];
        yy_netforce = yy[2];
      } else {
        max         = PetscMax(max,PetscAbs(yy[1]+yy[2]));
        yy[3]       = yy[0]+yy[1]+yy[2];
        yy_netforce = yy[3];
      }
    }
    if (ctx->netforce) {
      ierr = PetscDrawLGAddPoint(lg,&xx_netforce,&yy_netforce);CHKERRQ(ierr);
    } else {
      ierr = PetscDrawLGAddPoint(lg,xx,yy);CHKERRQ(ierr);
    }
    x += hx;
    /*if (max > 7200150000.0) */
    /* printf("max very big when i = %d\n",i); */
  }
  ierr = PetscDrawAxisSetLabels(axis,"Right hand side","","");CHKERRQ(ierr);
  ierr = PetscDrawLGSetLegend(lg,NULL);CHKERRQ(ierr);
  ierr = PetscDrawLGDraw(lg);CHKERRQ(ierr);

  /*
        Plot the solution
  */
  ierr = PetscDrawLGSetDimension(lg,1);CHKERRQ(ierr);
  ierr = PetscDrawViewPortsSet(ports,0);CHKERRQ(ierr);
  ierr = PetscDrawLGReset(lg);CHKERRQ(ierr);
  x    = hx*xs;
  ierr = PetscDrawLGSetLimits(lg,x,x+(xm-1)*hx,-1.1,1.1);CHKERRQ(ierr);
  ierr = PetscDrawLGSetColors(lg,colors);CHKERRQ(ierr);
  for (i=xs; i<xs+xm; i++) {
    xx[0] = x;
    yy[0] = PetscRealPart(u[i]);
    ierr  = PetscDrawLGAddPoint(lg,xx,yy);CHKERRQ(ierr);
    x    += hx;
  }
  ierr = PetscDrawAxisSetLabels(axis,"Solution","","");CHKERRQ(ierr);
  ierr = PetscDrawLGDraw(lg);CHKERRQ(ierr);

  /*
      Print the  forces as arrows on the solution
  */
  x   = hx*xs;
  cnt = xm/60;
  cnt = (!cnt) ? 1 : cnt;

  for (i=xs; i<xs+xm; i += cnt) {
    y    = yup = ydown = PetscRealPart(u[i]);
    c    = (u[i-1] + u[i+1] - 2.0*u[i])*sx;
    r    = (u[i] + u[i+2] - 2.0*u[i+1])*sx;
    l    = (u[i-2] + u[i] - 2.0*u[i-1])*sx;
    len  = -.5*PetscRealPart(ctx->kappa*(l + r - 2.0*c)*sx)/max;
    ierr = PetscDrawArrow(draw,x,y,x,y+len,PETSC_DRAW_RED);CHKERRQ(ierr);
    if (ctx->cahnhillard) {
      if (len < 0.) ydown += len;
      else yup += len;

      switch (ctx->energy) {
      case 1: /* double well */
        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;
        break;
      case 2: /* double obstacle */
        len = -.5*PetscRealPart(u[i-1] + u[i+1] - 2.0*u[i])*sx/max;
        break;
      case 3: /* logarithmic + double well */
        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;
        if (len < 0.) ydown += len;
        else yup += len;

        if (ctx->truncation==2) { /* quadratic */
          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;
          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;
          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);
        } else { /* cubic */
          a = 2.0*theta*(1.0-2.0*tol)/(16.0*tol*tol*(1.0-tol)*(1.0-tol));
          b = theta/(4.0*tol*(1.0-tol)) - a*(1.0-2.0*tol);
          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);
          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);
          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);
        }
        y2   = len < 0 ? ydown : yup;
        ierr = PetscDrawArrow(draw,x,y2,x,y2+len2,PETSC_DRAW_PLUM);CHKERRQ(ierr);
        break;
      case 4: /* logarithmic + double obstacle */
        len = -.5*theta_c*PetscRealPart(-(u[i-1] + u[i+1] - 2.0*u[i])*sx/max);
        if (len < 0.) ydown += len;
        else yup += len;

        if (ctx->truncation==2) { /* quadratic */
          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;
          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;
          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);
        } else { /* cubic */
          a = 2.0*theta*(1.0-2.0*tol)/(16.0*tol*tol*(1.0-tol)*(1.0-tol));
          b = theta/(4.0*tol*(1.0-tol)) - a*(1.0-2.0*tol);
          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;
          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;
          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;
        }
        y2   = len < 0 ? ydown : yup;
        ierr = PetscDrawArrow(draw,x,y2,x,y2+len2,PETSC_DRAW_PLUM);CHKERRQ(ierr);
        break;
      }
      ierr = PetscDrawArrow(draw,x,y,x,y+len,PETSC_DRAW_BLUE);CHKERRQ(ierr);
    }
    x += cnt*hx;
  }
  ierr = DMDAVecRestoreArrayRead(da,localU,&x);CHKERRQ(ierr);
  ierr = DMRestoreLocalVector(da,&localU);CHKERRQ(ierr);
  ierr = PetscDrawStringSetSize(draw,.2,.2);CHKERRQ(ierr);
  ierr = PetscDrawFlush(draw);CHKERRQ(ierr);
  ierr = PetscDrawSetPause(draw,pause);CHKERRQ(ierr);
  ierr = PetscDrawPause(draw);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode  MyDestroy(void **ptr)
{
  UserCtx        *ctx = *(UserCtx**)ptr;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscDrawViewPortsDestroy(ctx->ports);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}



/*TEST

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

TEST*/