static char help[] = "Model Equations for Advection-Diffusion\n";

/*
    Page 9, Section 1.2 Model Equations for Advection-Diffusion

          u_t = a u_x + d u_xx

   The initial conditions used here different then in the book.

*/

/*
     Helpful runtime linear solver options:
           -pc_type mg -da_refine 2 -snes_monitor -ksp_monitor -ts_view   (geometric multigrid with three levels)

*/

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

/*
   User-defined application context - contains data needed by the
   application-provided call-back routines.
*/
typedef struct {
  PetscScalar a, d; /* advection and diffusion strength */
  PetscBool   upwind;
} AppCtx;

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

int main(int argc, char **argv)
{
  AppCtx    appctx; /* user-defined application context */
  TS        ts;     /* timestepping context */
  Vec       U;      /* approximate solution vector */
  PetscReal dt;
  DM        da;
  PetscInt  M;

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

  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, NULL, help));
  appctx.a = 1.0;
  appctx.d = 0.0;
  PetscCall(PetscOptionsGetScalar(NULL, NULL, "-a", &appctx.a, NULL));
  PetscCall(PetscOptionsGetScalar(NULL, NULL, "-d", &appctx.d, NULL));
  appctx.upwind = PETSC_TRUE;
  PetscCall(PetscOptionsGetBool(NULL, NULL, "-upwind", &appctx.upwind, NULL));

  PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 60, 1, 1, NULL, &da));
  PetscCall(DMSetFromOptions(da));
  PetscCall(DMSetUp(da));
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create vector data structures
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  /*
     Create vector data structures for approximate and exact solutions
  */
  PetscCall(DMCreateGlobalVector(da, &U));

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

  PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
  PetscCall(TSSetDM(ts, da));

  /*
      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, NULL, NULL, RHSMatrixHeat, &appctx));
  PetscCall(TSSetSolutionFunction(ts, (PetscErrorCode (*)(TS, PetscReal, Vec, void *))Solution, &appctx));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Customize timestepping solver:
       - 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(DMDAGetInfo(da, PETSC_IGNORE, &M, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
  dt = .48 / (M * M);
  PetscCall(TSSetTimeStep(ts, dt));
  PetscCall(TSSetMaxSteps(ts, 1000));
  PetscCall(TSSetMaxTime(ts, 100.0));
  PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
  PetscCall(TSSetType(ts, TSARKIMEX));
  PetscCall(TSSetFromOptions(ts));

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

  /*
     Run the timestepping solver
  */
  PetscCall(TSSolve(ts, U));

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

  PetscCall(TSDestroy(&ts));
  PetscCall(VecDestroy(&U));
  PetscCall(DMDestroy(&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(TS ts, Vec U, AppCtx *appctx)
{
  PetscScalar *u, h;
  PetscInt     i, mstart, mend, xm, M;
  DM           da;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &da));
  PetscCall(DMDAGetCorners(da, &mstart, 0, 0, &xm, 0, 0));
  PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &M, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
  h    = 1.0 / M;
  mend = mstart + xm;
  /*
    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(DMDAVecGetArray(da, U, &u));

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

  /*
     Restore vector
  */
  PetscCall(DMDAVecRestoreArray(da, U, &u));
  PetscFunctionReturn(PETSC_SUCCESS);
}
/* --------------------------------------------------------------------- */
/*
   Solution - 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 Solution(TS ts, PetscReal t, Vec U, AppCtx *appctx)
{
  PetscScalar *u, ex1, ex2, sc1, sc2, h;
  PetscInt     i, mstart, mend, xm, M;
  DM           da;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &da));
  PetscCall(DMDAGetCorners(da, &mstart, 0, 0, &xm, 0, 0));
  PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &M, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
  h    = 1.0 / M;
  mend = mstart + xm;
  /*
     Get a pointer to vector data.
  */
  PetscCall(DMDAVecGetArray(da, U, &u));

  /*
     Simply write the solution directly into the array locations.
     Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
  */
  ex1 = PetscExpScalar(-36. * PETSC_PI * PETSC_PI * appctx->d * t);
  ex2 = PetscExpScalar(-4. * PETSC_PI * PETSC_PI * appctx->d * t);
  sc1 = PETSC_PI * 6. * h;
  sc2 = PETSC_PI * 2. * h;
  for (i = mstart; i < mend; i++) u[i] = PetscSinScalar(sc1 * (PetscReal)i + appctx->a * PETSC_PI * 6. * t) * ex1 + 3. * PetscSinScalar(sc2 * (PetscReal)i + appctx->a * PETSC_PI * 2. * t) * ex2;

  /*
     Restore vector
  */
  PetscCall(DMDAVecRestoreArray(da, U, &u));
  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 preconditioning matrix

   Notes:
   Recall that MatSetValues() uses 0-based row and column numbers
   in Fortran as well as in C.
*/
PetscErrorCode RHSMatrixHeat(TS ts, PetscReal t, Vec U, Mat AA, Mat BB, void *ctx)
{
  Mat         A      = AA;            /* Jacobian matrix */
  AppCtx     *appctx = (AppCtx *)ctx; /* user-defined application context */
  PetscInt    mstart, mend;
  PetscInt    i, idx[3], M, xm;
  PetscScalar v[3], h;
  DM          da;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &da));
  PetscCall(DMDAGetInfo(da, 0, &M, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
  PetscCall(DMDAGetCorners(da, &mstart, 0, 0, &xm, 0, 0));
  h    = 1.0 / M;
  mend = mstart + xm;
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Compute entries for the locally owned part of the matrix
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  /*
     Set matrix rows corresponding to boundary data
  */

  /* diffusion */
  v[0] = appctx->d / (h * h);
  v[1] = -2.0 * appctx->d / (h * h);
  v[2] = appctx->d / (h * h);
  if (!mstart) {
    idx[0] = M - 1;
    idx[1] = 0;
    idx[2] = 1;
    PetscCall(MatSetValues(A, 1, &mstart, 3, idx, v, INSERT_VALUES));
    mstart++;
  }

  if (mend == M) {
    mend--;
    idx[0] = M - 2;
    idx[1] = M - 1;
    idx[2] = 0;
    PetscCall(MatSetValues(A, 1, &mend, 3, idx, v, INSERT_VALUES));
  }

  /*
     Set matrix rows corresponding to interior data.  We construct the
     matrix one row at a time.
  */
  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));
  }
  PetscCall(MatAssemblyBegin(A, MAT_FLUSH_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FLUSH_ASSEMBLY));

  PetscCall(DMDAGetCorners(da, &mstart, 0, 0, &xm, 0, 0));
  mend = mstart + xm;
  if (!appctx->upwind) {
    /* advection -- centered differencing */
    v[0] = -.5 * appctx->a / (h);
    v[1] = .5 * appctx->a / (h);
    if (!mstart) {
      idx[0] = M - 1;
      idx[1] = 1;
      PetscCall(MatSetValues(A, 1, &mstart, 2, idx, v, ADD_VALUES));
      mstart++;
    }

    if (mend == M) {
      mend--;
      idx[0] = M - 2;
      idx[1] = 0;
      PetscCall(MatSetValues(A, 1, &mend, 2, idx, v, ADD_VALUES));
    }

    for (i = mstart; i < mend; i++) {
      idx[0] = i - 1;
      idx[1] = i + 1;
      PetscCall(MatSetValues(A, 1, &i, 2, idx, v, ADD_VALUES));
    }
  } else {
    /* advection -- upwinding */
    v[0] = -appctx->a / (h);
    v[1] = appctx->a / (h);
    if (!mstart) {
      idx[0] = 0;
      idx[1] = 1;
      PetscCall(MatSetValues(A, 1, &mstart, 2, idx, v, ADD_VALUES));
      mstart++;
    }

    if (mend == M) {
      mend--;
      idx[0] = M - 1;
      idx[1] = 0;
      PetscCall(MatSetValues(A, 1, &mend, 2, idx, v, ADD_VALUES));
    }

    for (i = mstart; i < mend; i++) {
      idx[0] = i;
      idx[1] = i + 1;
      PetscCall(MatSetValues(A, 1, &i, 2, idx, v, ADD_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);
}

/*TEST

   test:
      args: -pc_type mg -da_refine 2 -ts_view -ts_monitor -ts_max_time .3 -mg_levels_ksp_max_it 3
      requires: double
      filter: grep -v "total number of"

   test:
      suffix: 2
      args: -pc_type mg -da_refine 2 -ts_view -ts_monitor_draw_solution -ts_monitor -ts_max_time .3 -mg_levels_ksp_max_it 3
      requires: x
      output_file: output/ex3_1.out
      requires: double
      filter: grep -v "total number of"

TEST*/