static char help[] = "Solves a simple time-dependent linear PDE (the heat equation).\n\
Input parameters include:\n\
  -m <points>, where <points> = number of grid points\n\
  -time_dependent_rhs : Treat the problem as having a time-dependent right-hand side\n\
  -debug              : Activate debugging printouts\n\
  -nox                : Deactivate x-window graphics\n\n";

/* ------------------------------------------------------------------------

   This program solves the one-dimensional heat equation (also called the
   diffusion equation),
       u_t = u_xx,
   on the domain 0 <= x <= 1, with the boundary conditions
       u(t,0) = 0, u(t,1) = 0,
   and the initial condition
       u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
   This is a linear, second-order, parabolic equation.

   We discretize the right-hand side using finite differences with
   uniform grid spacing h:
       u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
   We then demonstrate time evolution using the various TS methods by
   running the program via
       mpiexec -n <procs> ex3 -ts_type <timestepping solver>

   We compare the approximate solution with the exact solution, given by
       u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
                      3*exp(-4*pi*pi*t) * sin(2*pi*x)

   Notes:
   This code demonstrates the TS solver interface to two variants of
   linear problems, u_t = f(u,t), namely
     - time-dependent f:   f(u,t) is a function of t
     - time-independent f: f(u,t) is simply f(u)

    The uniprocessor version of this code is ts/tutorials/ex3.c

  ------------------------------------------------------------------------- */

/*
   Include "petscdmda.h" so that we can use distributed arrays (DMDAs) to manage
   the parallel grid.  Include "petscts.h" so that we can use TS solvers.
   Note that this file automatically includes:
     petscsys.h       - base PETSc routines   petscvec.h  - vectors
     petscmat.h  - matrices
     petscis.h     - index sets            petscksp.h  - Krylov subspace methods
     petscviewer.h - viewers               petscpc.h   - preconditioners
     petscksp.h   - linear solvers        petscsnes.h - nonlinear solvers
*/

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

/*
   User-defined application context - contains data needed by the
   application-provided call-back routines.
*/
typedef struct {
  MPI_Comm    comm;             /* communicator */
  DM          da;               /* distributed array data structure */
  Vec         localwork;        /* local ghosted work vector */
  Vec         u_local;          /* local ghosted approximate solution vector */
  Vec         solution;         /* global exact solution vector */
  PetscInt    m;                /* total number of grid points */
  PetscReal   h;                /* mesh width h = 1/(m-1) */
  PetscBool   debug;            /* flag (1 indicates activation of debugging printouts) */
  PetscViewer viewer1, viewer2; /* viewers for the solution and error */
  PetscReal   norm_2, norm_max; /* error norms */
} AppCtx;

/*
   User-defined routines
*/
extern PetscErrorCode InitialConditions(Vec, AppCtx *);
extern PetscErrorCode RHSMatrixHeat(TS, PetscReal, Vec, Mat, Mat, void *);
extern PetscErrorCode RHSFunctionHeat(TS, PetscReal, Vec, Vec, void *);
extern PetscErrorCode Monitor(TS, PetscInt, PetscReal, Vec, void *);
extern PetscErrorCode ExactSolution(PetscReal, Vec, AppCtx *);

int main(int argc, char **argv)
{
  AppCtx        appctx;               /* user-defined application context */
  TS            ts;                   /* timestepping context */
  Mat           A;                    /* matrix data structure */
  Vec           u;                    /* approximate solution vector */
  PetscReal     time_total_max = 1.0; /* default max total time */
  PetscInt      time_steps_max = 100; /* default max timesteps */
  PetscDraw     draw;                 /* drawing context */
  PetscInt      steps, m;
  PetscMPIInt   size;
  PetscReal     dt, ftime;
  PetscBool     flg;
  TSProblemType tsproblem = TS_LINEAR;

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Initialize program and set problem parameters
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, NULL, help));
  appctx.comm = PETSC_COMM_WORLD;

  m = 60;
  PetscCall(PetscOptionsGetInt(NULL, NULL, "-m", &m, NULL));
  PetscCall(PetscOptionsHasName(NULL, NULL, "-debug", &appctx.debug));
  appctx.m        = m;
  appctx.h        = 1.0 / (m - 1.0);
  appctx.norm_2   = 0.0;
  appctx.norm_max = 0.0;

  PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Solving a linear TS problem, number of processors = %d\n", size));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create vector data structures
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  /*
     Create distributed array (DMDA) to manage parallel grid and vectors
     and to set up the ghost point communication pattern.  There are M
     total grid values spread equally among all the processors.
  */

  PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, m, 1, 1, NULL, &appctx.da));
  PetscCall(DMSetFromOptions(appctx.da));
  PetscCall(DMSetUp(appctx.da));

  /*
     Extract global and local vectors from DMDA; we use these to store the
     approximate solution.  Then duplicate these for remaining vectors that
     have the same types.
  */
  PetscCall(DMCreateGlobalVector(appctx.da, &u));
  PetscCall(DMCreateLocalVector(appctx.da, &appctx.u_local));

  /*
     Create local work vector for use in evaluating right-hand-side function;
     create global work vector for storing exact solution.
  */
  PetscCall(VecDuplicate(appctx.u_local, &appctx.localwork));
  PetscCall(VecDuplicate(u, &appctx.solution));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set up displays to show graphs of the solution and error
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  PetscCall(PetscViewerDrawOpen(PETSC_COMM_WORLD, 0, "", 80, 380, 400, 160, &appctx.viewer1));
  PetscCall(PetscViewerDrawGetDraw(appctx.viewer1, 0, &draw));
  PetscCall(PetscDrawSetDoubleBuffer(draw));
  PetscCall(PetscViewerDrawOpen(PETSC_COMM_WORLD, 0, "", 80, 0, 400, 160, &appctx.viewer2));
  PetscCall(PetscViewerDrawGetDraw(appctx.viewer2, 0, &draw));
  PetscCall(PetscDrawSetDoubleBuffer(draw));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create timestepping solver context
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));

  flg = PETSC_FALSE;
  PetscCall(PetscOptionsGetBool(NULL, NULL, "-nonlinear", &flg, NULL));
  PetscCall(TSSetProblemType(ts, flg ? TS_NONLINEAR : TS_LINEAR));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set optional user-defined monitoring routine
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSMonitorSet(ts, Monitor, &appctx, NULL));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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

  PetscCall(MatCreate(PETSC_COMM_WORLD, &A));
  PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, m, m));
  PetscCall(MatSetFromOptions(A));
  PetscCall(MatSetUp(A));

  flg = PETSC_FALSE;
  PetscCall(PetscOptionsGetBool(NULL, NULL, "-time_dependent_rhs", &flg, NULL));
  if (flg) {
    /*
       For linear problems with a time-dependent f(u,t) in the equation
       u_t = f(u,t), the user provides the discretized right-hand side
       as a time-dependent matrix.
    */
    PetscCall(TSSetRHSFunction(ts, NULL, TSComputeRHSFunctionLinear, &appctx));
    PetscCall(TSSetRHSJacobian(ts, A, A, RHSMatrixHeat, &appctx));
  } else {
    /*
       For linear problems with a time-independent f(u) in the equation
       u_t = f(u), the user provides the discretized right-hand side
       as a matrix only once, and then sets a null matrix evaluation
       routine.
    */
    PetscCall(RHSMatrixHeat(ts, 0.0, u, A, A, &appctx));
    PetscCall(TSSetRHSFunction(ts, NULL, TSComputeRHSFunctionLinear, &appctx));
    PetscCall(TSSetRHSJacobian(ts, A, A, TSComputeRHSJacobianConstant, &appctx));
  }

  if (tsproblem == TS_NONLINEAR) {
    SNES snes;
    PetscCall(TSSetRHSFunction(ts, NULL, RHSFunctionHeat, &appctx));
    PetscCall(TSGetSNES(ts, &snes));
    PetscCall(SNESSetJacobian(snes, NULL, NULL, SNESComputeJacobianDefault, NULL));
  }

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set solution vector and initial timestep
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  dt = appctx.h * appctx.h / 2.0;
  PetscCall(TSSetTimeStep(ts, dt));
  PetscCall(TSSetSolution(ts, u));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Customize timestepping solver:
       - Set the solution method to be the Backward Euler method.
       - Set timestepping duration info
     Then set runtime options, which can override these defaults.
     For example,
          -ts_max_steps <maxsteps> -ts_max_time <maxtime>
     to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  PetscCall(TSSetMaxSteps(ts, time_steps_max));
  PetscCall(TSSetMaxTime(ts, time_total_max));
  PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
  PetscCall(TSSetFromOptions(ts));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Solve the problem
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  /*
     Evaluate initial conditions
  */
  PetscCall(InitialConditions(u, &appctx));

  /*
     Run the timestepping solver
  */
  PetscCall(TSSolve(ts, u));
  PetscCall(TSGetSolveTime(ts, &ftime));
  PetscCall(TSGetStepNumber(ts, &steps));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     View timestepping solver info
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Total timesteps %" PetscInt_FMT ", Final time %g\n", steps, (double)ftime));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Avg. error (2 norm) = %g Avg. error (max norm) = %g\n", (double)(appctx.norm_2 / steps), (double)(appctx.norm_max / steps)));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Free work space.  All PETSc objects should be destroyed when they
     are no longer needed.
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  PetscCall(TSDestroy(&ts));
  PetscCall(MatDestroy(&A));
  PetscCall(VecDestroy(&u));
  PetscCall(PetscViewerDestroy(&appctx.viewer1));
  PetscCall(PetscViewerDestroy(&appctx.viewer2));
  PetscCall(VecDestroy(&appctx.localwork));
  PetscCall(VecDestroy(&appctx.solution));
  PetscCall(VecDestroy(&appctx.u_local));
  PetscCall(DMDestroy(&appctx.da));

  /*
     Always call PetscFinalize() before exiting a program.  This routine
       - finalizes the PETSc libraries as well as MPI
       - provides summary and diagnostic information if certain runtime
         options are chosen (e.g., -log_view).
  */
  PetscCall(PetscFinalize());
  return 0;
}
/* --------------------------------------------------------------------- */
/*
   InitialConditions - Computes the solution at the initial time.

   Input Parameter:
   u - uninitialized solution vector (global)
   appctx - user-defined application context

   Output Parameter:
   u - vector with solution at initial time (global)
*/
PetscErrorCode InitialConditions(Vec u, AppCtx *appctx)
{
  PetscScalar *u_localptr, h = appctx->h;
  PetscInt     i, mybase, myend;

  PetscFunctionBeginUser;
  /*
     Determine starting point of each processor's range of
     grid values.
  */
  PetscCall(VecGetOwnershipRange(u, &mybase, &myend));

  /*
    Get a pointer to vector data.
    - For default PETSc vectors, VecGetArray() returns a pointer to
      the data array.  Otherwise, the routine is implementation dependent.
    - You MUST call VecRestoreArray() when you no longer need access to
      the array.
    - Note that the Fortran interface to VecGetArray() differs from the
      C version.  See the users manual for details.
  */
  PetscCall(VecGetArray(u, &u_localptr));

  /*
     We initialize the solution array by simply writing the solution
     directly into the array locations.  Alternatively, we could use
     VecSetValues() or VecSetValuesLocal().
  */
  for (i = mybase; i < myend; i++) u_localptr[i - mybase] = PetscSinScalar(PETSC_PI * i * 6. * h) + 3. * PetscSinScalar(PETSC_PI * i * 2. * h);

  /*
     Restore vector
  */
  PetscCall(VecRestoreArray(u, &u_localptr));

  /*
     Print debugging information if desired
  */
  if (appctx->debug) {
    PetscCall(PetscPrintf(appctx->comm, "initial guess vector\n"));
    PetscCall(VecView(u, PETSC_VIEWER_STDOUT_WORLD));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}
/* --------------------------------------------------------------------- */
/*
   ExactSolution - Computes the exact solution at a given time.

   Input Parameters:
   t - current time
   solution - vector in which exact solution will be computed
   appctx - user-defined application context

   Output Parameter:
   solution - vector with the newly computed exact solution
*/
PetscErrorCode ExactSolution(PetscReal t, Vec solution, AppCtx *appctx)
{
  PetscScalar *s_localptr, h = appctx->h, ex1, ex2, sc1, sc2;
  PetscInt     i, mybase, myend;

  PetscFunctionBeginUser;
  /*
     Determine starting and ending points of each processor's
     range of grid values
  */
  PetscCall(VecGetOwnershipRange(solution, &mybase, &myend));

  /*
     Get a pointer to vector data.
  */
  PetscCall(VecGetArray(solution, &s_localptr));

  /*
     Simply write the solution directly into the array locations.
     Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
  */
  ex1 = PetscExpReal(-36. * PETSC_PI * PETSC_PI * t);
  ex2 = PetscExpReal(-4. * PETSC_PI * PETSC_PI * t);
  sc1 = PETSC_PI * 6. * h;
  sc2 = PETSC_PI * 2. * h;
  for (i = mybase; i < myend; i++) s_localptr[i - mybase] = PetscSinScalar(sc1 * (PetscReal)i) * ex1 + 3. * PetscSinScalar(sc2 * (PetscReal)i) * ex2;

  /*
     Restore vector
  */
  PetscCall(VecRestoreArray(solution, &s_localptr));
  PetscFunctionReturn(PETSC_SUCCESS);
}
/* --------------------------------------------------------------------- */
/*
   Monitor - User-provided routine to monitor the solution computed at
   each timestep.  This example plots the solution and computes the
   error in two different norms.

   Input Parameters:
   ts     - the timestep context
   step   - the count of the current step (with 0 meaning the
             initial condition)
   time   - the current time
   u      - the solution at this timestep
   ctx    - the user-provided context for this monitoring routine.
            In this case we use the application context which contains
            information about the problem size, workspace and the exact
            solution.
*/
PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal time, Vec u, void *ctx)
{
  AppCtx   *appctx = (AppCtx *)ctx; /* user-defined application context */
  PetscReal norm_2, norm_max;

  PetscFunctionBeginUser;
  /*
     View a graph of the current iterate
  */
  PetscCall(VecView(u, appctx->viewer2));

  /*
     Compute the exact solution
  */
  PetscCall(ExactSolution(time, appctx->solution, appctx));

  /*
     Print debugging information if desired
  */
  if (appctx->debug) {
    PetscCall(PetscPrintf(appctx->comm, "Computed solution vector\n"));
    PetscCall(VecView(u, PETSC_VIEWER_STDOUT_WORLD));
    PetscCall(PetscPrintf(appctx->comm, "Exact solution vector\n"));
    PetscCall(VecView(appctx->solution, PETSC_VIEWER_STDOUT_WORLD));
  }

  /*
     Compute the 2-norm and max-norm of the error
  */
  PetscCall(VecAXPY(appctx->solution, -1.0, u));
  PetscCall(VecNorm(appctx->solution, NORM_2, &norm_2));
  norm_2 = PetscSqrtReal(appctx->h) * norm_2;
  PetscCall(VecNorm(appctx->solution, NORM_MAX, &norm_max));
  if (norm_2 < 1e-14) norm_2 = 0;
  if (norm_max < 1e-14) norm_max = 0;

  /*
     PetscPrintf() causes only the first processor in this
     communicator to print the timestep information.
  */
  PetscCall(PetscPrintf(appctx->comm, "Timestep %" PetscInt_FMT ": time = %g 2-norm error = %g max norm error = %g\n", step, (double)time, (double)norm_2, (double)norm_max));
  appctx->norm_2 += norm_2;
  appctx->norm_max += norm_max;

  /*
     View a graph of the error
  */
  PetscCall(VecView(appctx->solution, appctx->viewer1));

  /*
     Print debugging information if desired
  */
  if (appctx->debug) {
    PetscCall(PetscPrintf(appctx->comm, "Error vector\n"));
    PetscCall(VecView(appctx->solution, PETSC_VIEWER_STDOUT_WORLD));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* --------------------------------------------------------------------- */
/*
   RHSMatrixHeat - User-provided routine to compute the right-hand-side
   matrix for the heat equation.

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

   Output Parameters:
   AA - Jacobian matrix
   BB - optionally different matrix used to construct the preconditioner

  Notes:
  RHSMatrixHeat computes entries for the locally owned part of the system.
   - Currently, all PETSc parallel matrix formats are partitioned by
     contiguous chunks of rows across the processors.
   - Each processor needs to insert only elements that it owns
     locally (but any non-local elements will be sent to the
     appropriate processor during matrix assembly).
   - Always specify global row and columns of matrix entries when
     using MatSetValues(); we could alternatively use MatSetValuesLocal().
   - Here, we set all entries for a particular row at once.
   - Note that MatSetValues() uses 0-based row and column numbers
     in Fortran as well as in C.
*/
PetscErrorCode RHSMatrixHeat(TS ts, PetscReal t, Vec X, Mat AA, Mat BB, void *ctx)
{
  Mat         A      = AA;            /* Jacobian matrix */
  AppCtx     *appctx = (AppCtx *)ctx; /* user-defined application context */
  PetscInt    i, mstart, mend, idx[3];
  PetscScalar v[3], stwo = -2. / (appctx->h * appctx->h), sone = -.5 * stwo;

  PetscFunctionBeginUser;
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Compute entries for the locally owned part of the matrix
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  PetscCall(MatGetOwnershipRange(A, &mstart, &mend));

  /*
     Set matrix rows corresponding to boundary data
  */

  if (mstart == 0) { /* first processor only */
    v[0] = 1.0;
    PetscCall(MatSetValues(A, 1, &mstart, 1, &mstart, v, INSERT_VALUES));
    mstart++;
  }

  if (mend == appctx->m) { /* last processor only */
    mend--;
    v[0] = 1.0;
    PetscCall(MatSetValues(A, 1, &mend, 1, &mend, v, INSERT_VALUES));
  }

  /*
     Set matrix rows corresponding to interior data.  We construct the
     matrix one row at a time.
  */
  v[0] = sone;
  v[1] = stwo;
  v[2] = sone;
  for (i = mstart; i < mend; i++) {
    idx[0] = i - 1;
    idx[1] = i;
    idx[2] = i + 1;
    PetscCall(MatSetValues(A, 1, &i, 3, idx, v, INSERT_VALUES));
  }

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Complete the matrix assembly process and set some options
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  /*
     Assemble matrix, using the 2-step process:
       MatAssemblyBegin(), MatAssemblyEnd()
     Computations can be done while messages are in transition
     by placing code between these two statements.
  */
  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));

  /*
     Set and option to indicate that we will never add a new nonzero location
     to the matrix. If we do, it will generate an error.
  */
  PetscCall(MatSetOption(A, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode RHSFunctionHeat(TS ts, PetscReal t, Vec globalin, Vec globalout, void *ctx)
{
  Mat A;

  PetscFunctionBeginUser;
  PetscCall(TSGetRHSJacobian(ts, &A, NULL, NULL, &ctx));
  PetscCall(RHSMatrixHeat(ts, t, globalin, A, NULL, ctx));
  /* PetscCall(MatView(A,PETSC_VIEWER_STDOUT_WORLD)); */
  PetscCall(MatMult(A, globalin, globalout));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*TEST

    test:
      args: -ts_view -nox

    test:
      suffix: 2
      args: -ts_view -nox
      nsize: 3

    test:
      suffix: 3
      args: -ts_view -nox -nonlinear

    test:
      suffix: 4
      args: -ts_view -nox -nonlinear
      nsize: 3
      timeoutfactor: 3

    test:
      suffix: sundials
      requires: sundials2
      args: -nox -ts_type sundials -ts_max_steps 5 -nonlinear
      nsize: 4

    test:
      suffix: sundials_dense
      requires: sundials2
      args: -nox -ts_type sundials -ts_sundials_use_dense -ts_max_steps 5 -nonlinear
      nsize: 1

TEST*/