static char help[] = "Time-dependent PDE in 2d. Simplified from ex7.c for illustrating how to use TS on a structured domain. \n";
/*
   u_t = uxx + uyy
   0 < x < 1, 0 < y < 1;
   At t=0: u(x,y) = exp(c*r*r*r), if r=PetscSqrtReal((x-.5)*(x-.5) + (y-.5)*(y-.5)) < .125
           u(x,y) = 0.0           if r >= .125

    mpiexec -n 2 ./ex13 -da_grid_x 40 -da_grid_y 40 -ts_max_steps 2 -snes_monitor -ksp_monitor
    mpiexec -n 1 ./ex13 -snes_fd_color -ts_monitor_draw_solution
    mpiexec -n 2 ./ex13 -ts_type sundials -ts_monitor
*/

#include <petscdm.h>
#include <petscdmda.h>
#include <petscts.h>

/*
   User-defined data structures and routines
*/
typedef struct {
  PetscReal c;
} AppCtx;

extern PetscErrorCode RHSFunction(TS,PetscReal,Vec,Vec,void*);
extern PetscErrorCode RHSJacobian(TS,PetscReal,Vec,Mat,Mat,void*);
extern PetscErrorCode FormInitialSolution(DM,Vec,void*);

int main(int argc,char **argv)
{
  TS             ts;                   /* nonlinear solver */
  Vec            u,r;                  /* solution, residual vector */
  Mat            J;                    /* Jacobian matrix */
  PetscInt       steps;                /* iterations for convergence */
  PetscErrorCode ierr;
  DM             da;
  PetscReal      ftime,dt;
  AppCtx         user;              /* user-defined work context */

  ierr = PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create distributed array (DMDA) to manage parallel grid and vectors
  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,8,8,PETSC_DECIDE,PETSC_DECIDE,1,1,NULL,NULL,&da);CHKERRQ(ierr);
  ierr = DMSetFromOptions(da);CHKERRQ(ierr);
  ierr = DMSetUp(da);CHKERRQ(ierr);

  /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Extract global vectors from DMDA;
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = DMCreateGlobalVector(da,&u);CHKERRQ(ierr);
  ierr = VecDuplicate(u,&r);CHKERRQ(ierr);

  /* Initialize user application context */
  user.c = -30.0;

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create timestepping solver context
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr);
  ierr = TSSetDM(ts,da);CHKERRQ(ierr);
  ierr = TSSetType(ts,TSBEULER);CHKERRQ(ierr);
  ierr = TSSetRHSFunction(ts,r,RHSFunction,&user);CHKERRQ(ierr);

  /* Set Jacobian */
  ierr = DMSetMatType(da,MATAIJ);CHKERRQ(ierr);
  ierr = DMCreateMatrix(da,&J);CHKERRQ(ierr);
  ierr = TSSetRHSJacobian(ts,J,J,RHSJacobian,NULL);CHKERRQ(ierr);

  ftime = 1.0;
  ierr = TSSetMaxTime(ts,ftime);CHKERRQ(ierr);
  ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set initial conditions
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = FormInitialSolution(da,u,&user);CHKERRQ(ierr);
  dt   = .01;
  ierr = TSSetTimeStep(ts,dt);CHKERRQ(ierr);

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

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Solve nonlinear system
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = TSSolve(ts,u);CHKERRQ(ierr);
  ierr = TSGetSolveTime(ts,&ftime);CHKERRQ(ierr);
  ierr = TSGetStepNumber(ts,&steps);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Free work space.
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = MatDestroy(&J);CHKERRQ(ierr);
  ierr = VecDestroy(&u);CHKERRQ(ierr);
  ierr = VecDestroy(&r);CHKERRQ(ierr);
  ierr = TSDestroy(&ts);CHKERRQ(ierr);
  ierr = DMDestroy(&da);CHKERRQ(ierr);

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

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

   Output Parameter:
.  F - function vector
 */
PetscErrorCode RHSFunction(TS ts,PetscReal ftime,Vec U,Vec F,void *ptr)
{
  /* PETSC_UNUSED AppCtx *user=(AppCtx*)ptr; */
  DM             da;
  PetscErrorCode ierr;
  PetscInt       i,j,Mx,My,xs,ys,xm,ym;
  PetscReal      two = 2.0,hx,hy,sx,sy;
  PetscScalar    u,uxx,uyy,**uarray,**f;
  Vec            localU;

  PetscFunctionBeginUser;
  ierr = TSGetDM(ts,&da);CHKERRQ(ierr);
  ierr = DMGetLocalVector(da,&localU);CHKERRQ(ierr);
  ierr = 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);CHKERRQ(ierr);

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

  /*
     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,U,INSERT_VALUES,localU);CHKERRQ(ierr);
  ierr = DMGlobalToLocalEnd(da,U,INSERT_VALUES,localU);CHKERRQ(ierr);

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

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

  /* Compute function over the locally owned part of the grid */
  for (j=ys; j<ys+ym; j++) {
    for (i=xs; i<xs+xm; i++) {
      if (i == 0 || j == 0 || i == Mx-1 || j == My-1) {
        f[j][i] = uarray[j][i];
        continue;
      }
      u       = uarray[j][i];
      uxx     = (-two*u + uarray[j][i-1] + uarray[j][i+1])*sx;
      uyy     = (-two*u + uarray[j-1][i] + uarray[j+1][i])*sy;
      f[j][i] = uxx + uyy;
    }
  }

  /* Restore vectors */
  ierr = DMDAVecRestoreArrayRead(da,localU,&uarray);CHKERRQ(ierr);
  ierr = DMDAVecRestoreArray(da,F,&f);CHKERRQ(ierr);
  ierr = DMRestoreLocalVector(da,&localU);CHKERRQ(ierr);
  ierr = PetscLogFlops(11.0*ym*xm);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* --------------------------------------------------------------------- */
/*
   RHSJacobian - User-provided routine to compute the Jacobian of
   the nonlinear right-hand-side function of the ODE.

   Input Parameters:
   ts - the TS context
   t - current time
   U - global input vector
   dummy - optional user-defined context, as set by TSetRHSJacobian()

   Output Parameters:
   J - Jacobian matrix
   Jpre - optionally different preconditioning matrix
   str - flag indicating matrix structure
*/
PetscErrorCode RHSJacobian(TS ts,PetscReal t,Vec U,Mat J,Mat Jpre,void *ctx)
{
  PetscErrorCode ierr;
  DM             da;
  DMDALocalInfo  info;
  PetscInt       i,j;
  PetscReal      hx,hy,sx,sy;

  PetscFunctionBeginUser;
  ierr = TSGetDM(ts,&da);CHKERRQ(ierr);
  ierr = DMDAGetLocalInfo(da,&info);CHKERRQ(ierr);
  hx   = 1.0/(PetscReal)(info.mx-1); sx = 1.0/(hx*hx);
  hy   = 1.0/(PetscReal)(info.my-1); sy = 1.0/(hy*hy);
  for (j=info.ys; j<info.ys+info.ym; j++) {
    for (i=info.xs; i<info.xs+info.xm; i++) {
      PetscInt    nc = 0;
      MatStencil  row,col[5];
      PetscScalar val[5];
      row.i = i; row.j = j;
      if (i == 0 || j == 0 || i == info.mx-1 || j == info.my-1) {
        col[nc].i = i; col[nc].j = j; val[nc++] = 1.0;
      } else {
        col[nc].i = i-1; col[nc].j = j;   val[nc++] = sx;
        col[nc].i = i+1; col[nc].j = j;   val[nc++] = sx;
        col[nc].i = i;   col[nc].j = j-1; val[nc++] = sy;
        col[nc].i = i;   col[nc].j = j+1; val[nc++] = sy;
        col[nc].i = i;   col[nc].j = j;   val[nc++] = -2*sx - 2*sy;
      }
      ierr = MatSetValuesStencil(Jpre,1,&row,nc,col,val,INSERT_VALUES);CHKERRQ(ierr);
    }
  }
  ierr = MatAssemblyBegin(Jpre,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(Jpre,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  if (J != Jpre) {
    ierr = MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
    ierr = MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

/* ------------------------------------------------------------------- */
PetscErrorCode FormInitialSolution(DM da,Vec U,void* ptr)
{
  AppCtx         *user=(AppCtx*)ptr;
  PetscReal      c=user->c;
  PetscErrorCode ierr;
  PetscInt       i,j,xs,ys,xm,ym,Mx,My;
  PetscScalar    **u;
  PetscReal      hx,hy,x,y,r;

  PetscFunctionBeginUser;
  ierr = 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);CHKERRQ(ierr);

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

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

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

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

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

/*TEST

    test:
      args: -ts_max_steps 5 -ts_monitor

    test:
      suffix: 2
      args: -ts_max_steps 5 -ts_monitor

    test:
      suffix: 3
      args: -ts_max_steps 5 -snes_fd_color -ts_monitor

TEST*/