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

/*
  Solves the equation biharmonic equation in split form

    w = -kappa \Delta u
    u_t =  \Delta w
    -1  <= u <= 1
    Periodic boundary conditions

Evolve the biharmonic heat equation with bounds:  (same as biharmonic)
---------------
./biharmonic2 -ts_monitor -snes_monitor -ts_monitor_draw_solution  -pc_type lu  -draw_pause .1 -snes_converged_reason  -ts_type beuler  -da_refine 5 -draw_fields 1 -ts_time_step 9.53674e-9

    w = -kappa \Delta u  + u^3  - u
    u_t =  \Delta w
    -1  <= u <= 1
    Periodic boundary conditions

Evolve the Cahn-Hillard equations: (this fails after a few timesteps 12/17/2017)
---------------
./biharmonic2 -ts_monitor -snes_monitor -ts_monitor_draw_solution  -pc_type lu  -draw_pause .1 -snes_converged_reason   -ts_type beuler    -da_refine 6  -draw_fields 1  -kappa .00001 -ts_time_step 5.96046e-06 -cahn-hillard

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

/*
   User-defined routines
*/
extern PetscErrorCode FormFunction(TS, PetscReal, Vec, Vec, Vec, void *), FormInitialSolution(DM, Vec, PetscReal);
typedef struct {
  PetscBool cahnhillard;
  PetscReal kappa;
  PetscInt  energy;
  PetscReal tol;
  PetscReal theta;
  PetscReal theta_c;
} UserCtx;

int main(int argc, char **argv)
{
  TS            ts;   /* nonlinear solver */
  Vec           x, r; /* solution, residual vectors */
  Mat           J;    /* Jacobian matrix */
  PetscInt      steps, Mx;
  DM            da;
  MatFDColoring matfdcoloring;
  ISColoring    iscoloring;
  PetscReal     dt;
  PetscReal     vbounds[] = {-100000, 100000, -1.1, 1.1};
  SNES          snes;
  UserCtx       ctx;

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Initialize program
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, NULL, help));
  ctx.kappa = 1.0;
  PetscCall(PetscOptionsGetReal(NULL, NULL, "-kappa", &ctx.kappa, NULL));
  ctx.cahnhillard = PETSC_FALSE;

  PetscCall(PetscOptionsGetBool(NULL, NULL, "-cahn-hillard", &ctx.cahnhillard, NULL));
  PetscCall(PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 2, vbounds));
  PetscCall(PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 600, 600));
  ctx.energy = 1;
  /*PetscCall(PetscOptionsGetInt(NULL,NULL,"-energy",&ctx.energy,NULL));*/
  PetscCall(PetscOptionsGetInt(NULL, NULL, "-energy", &ctx.energy, NULL));
  ctx.tol = 1.0e-8;
  PetscCall(PetscOptionsGetReal(NULL, NULL, "-tol", &ctx.tol, NULL));
  ctx.theta   = .001;
  ctx.theta_c = 1.0;
  PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta", &ctx.theta, NULL));
  PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta_c", &ctx.theta_c, NULL));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create distributed array (DMDA) to manage parallel grid and vectors
  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 10, 2, 2, NULL, &da));
  PetscCall(DMSetFromOptions(da));
  PetscCall(DMSetUp(da));
  PetscCall(DMDASetFieldName(da, 0, "Biharmonic heat equation: w = -kappa*u_xx"));
  PetscCall(DMDASetFieldName(da, 1, "Biharmonic heat equation: u"));
  PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
  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
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(DMCreateGlobalVector(da, &x));
  PetscCall(VecDuplicate(x, &r));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create timestepping solver context
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
  PetscCall(TSSetDM(ts, da));
  PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
  PetscCall(TSSetIFunction(ts, NULL, FormFunction, &ctx));
  PetscCall(TSSetMaxTime(ts, .02));
  PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_INTERPOLATE));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     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 matrix used to construct the preconditioner as set by the user,
                         but use matrix-free approx for Jacobian-vector
                         products within Newton-Krylov method

     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSGetSNES(ts, &snes));
  PetscCall(DMCreateColoring(da, IS_COLORING_GLOBAL, &iscoloring));
  PetscCall(DMSetMatType(da, MATAIJ));
  PetscCall(DMCreateMatrix(da, &J));
  PetscCall(MatFDColoringCreate(J, iscoloring, &matfdcoloring));
  PetscCall(MatFDColoringSetFunction(matfdcoloring, (MatFDColoringFn *)SNESTSFormFunction, ts));
  PetscCall(MatFDColoringSetFromOptions(matfdcoloring));
  PetscCall(MatFDColoringSetUp(J, iscoloring, matfdcoloring));
  PetscCall(ISColoringDestroy(&iscoloring));
  PetscCall(SNESSetJacobian(snes, J, J, SNESComputeJacobianDefaultColor, matfdcoloring));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Customize nonlinear solver
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSSetType(ts, TSBEULER));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set initial conditions
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(FormInitialSolution(da, x, ctx.kappa));
  PetscCall(TSSetTimeStep(ts, dt));
  PetscCall(TSSetSolution(ts, x));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set runtime options
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSSetFromOptions(ts));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Solve nonlinear system
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSSolve(ts, x));
  PetscCall(TSGetStepNumber(ts, &steps));

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

  PetscCall(PetscFinalize());
  return 0;
}

typedef struct {
  PetscScalar w, u;
} Field;
/* ------------------------------------------------------------------- */
/*
   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 Xdot, Vec F, void *ptr)
{
  DM        da;
  PetscInt  i, Mx, xs, xm;
  PetscReal hx, sx;
  Field    *x, *xdot, *f;
  Vec       localX, localXdot;
  UserCtx  *ctx = (UserCtx *)ptr;

  PetscFunctionBegin;
  PetscCall(TSGetDM(ts, &da));
  PetscCall(DMGetLocalVector(da, &localX));
  PetscCall(DMGetLocalVector(da, &localXdot));
  PetscCall(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));

  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.
  */
  PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX));
  PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX));
  PetscCall(DMGlobalToLocalBegin(da, Xdot, INSERT_VALUES, localXdot));
  PetscCall(DMGlobalToLocalEnd(da, Xdot, INSERT_VALUES, localXdot));

  /*
     Get pointers to vector data
  */
  PetscCall(DMDAVecGetArrayRead(da, localX, &x));
  PetscCall(DMDAVecGetArrayRead(da, localXdot, &xdot));
  PetscCall(DMDAVecGetArray(da, F, &f));

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

  /*
     Compute function over the locally owned part of the grid
  */
  for (i = xs; i < xs + xm; i++) {
    f[i].w = x[i].w + ctx->kappa * (x[i - 1].u + x[i + 1].u - 2.0 * x[i].u) * sx;
    if (ctx->cahnhillard) {
      switch (ctx->energy) {
      case 1: /* double well */
        f[i].w += -x[i].u * x[i].u * x[i].u + x[i].u;
        break;
      case 2: /* double obstacle */
        f[i].w += x[i].u;
        break;
      case 3: /* logarithmic */
        if (PetscRealPart(x[i].u) < -1.0 + 2.0 * ctx->tol) f[i].w += .5 * ctx->theta * (-PetscLogReal(ctx->tol) + PetscLogScalar((1.0 - x[i].u) / 2.0)) + ctx->theta_c * x[i].u;
        else if (PetscRealPart(x[i].u) > 1.0 - 2.0 * ctx->tol) f[i].w += .5 * ctx->theta * (-PetscLogScalar((1.0 + x[i].u) / 2.0) + PetscLogReal(ctx->tol)) + ctx->theta_c * x[i].u;
        else f[i].w += .5 * ctx->theta * (-PetscLogScalar((1.0 + x[i].u) / 2.0) + PetscLogScalar((1.0 - x[i].u) / 2.0)) + ctx->theta_c * x[i].u;
        break;
      }
    }
    f[i].u = xdot[i].u - (x[i - 1].w + x[i + 1].w - 2.0 * x[i].w) * sx;
  }

  /*
     Restore vectors
  */
  PetscCall(DMDAVecRestoreArrayRead(da, localXdot, &xdot));
  PetscCall(DMDAVecRestoreArrayRead(da, localX, &x));
  PetscCall(DMDAVecRestoreArray(da, F, &f));
  PetscCall(DMRestoreLocalVector(da, &localX));
  PetscCall(DMRestoreLocalVector(da, &localXdot));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* ------------------------------------------------------------------- */
PetscErrorCode FormInitialSolution(DM da, Vec X, PetscReal kappa)
{
  PetscInt  i, xs, xm, Mx, xgs, xgm;
  Field    *x;
  PetscReal hx, xx, r, sx;
  Vec       Xg;

  PetscFunctionBegin;
  PetscCall(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));

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

  /*
     Get pointers to vector data
  */
  PetscCall(DMCreateLocalVector(da, &Xg));
  PetscCall(DMDAVecGetArray(da, Xg, &x));

  /*
     Get local grid boundaries
  */
  PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL));
  PetscCall(DMDAGetGhostCorners(da, &xgs, NULL, NULL, &xgm, NULL, NULL));

  /*
     Compute u function over the locally owned part of the grid including ghost points
  */
  for (i = xgs; i < xgs + xgm; i++) {
    xx = i * hx;
    r  = PetscSqrtReal((xx - .5) * (xx - .5));
    if (r < .125) x[i].u = 1.0;
    else x[i].u = -.50;
    /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */
    x[i].w = 0;
  }
  for (i = xs; i < xs + xm; i++) x[i].w = -kappa * (x[i - 1].u + x[i + 1].u - 2.0 * x[i].u) * sx;

  /*
     Restore vectors
  */
  PetscCall(DMDAVecRestoreArray(da, Xg, &x));

  /* Grab only the global part of the vector */
  PetscCall(VecSet(X, 0));
  PetscCall(DMLocalToGlobalBegin(da, Xg, ADD_VALUES, X));
  PetscCall(DMLocalToGlobalEnd(da, Xg, ADD_VALUES, X));
  PetscCall(VecDestroy(&Xg));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*TEST

   build:
     requires: !complex !single

   test:
     args: -ts_monitor -snes_monitor -pc_type lu -snes_converged_reason -ts_type beuler -da_refine 5 -ts_time_step 9.53674e-9 -ts_max_steps 50
     requires: x

TEST*/
