static char help[] = "Chemo-taxis Problems from Mathematical Biology.\n";

/*
     Page 18, Chemo-taxis Problems from Mathematical Biology

        rho_t =
        c_t   =

     Further discussion on Page 134 and in 2d on Page 409
*/

/*

   Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
   Include "petscts.h" so that we can use SNES 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
*/

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

typedef struct {
  PetscScalar rho, c;
} Field;

typedef struct {
  PetscScalar epsilon, delta, alpha, beta, gamma, kappa, lambda, mu, cstar;
  PetscBool   upwind;
} AppCtx;

/*
   User-defined routines
*/
extern PetscErrorCode IFunction(TS, PetscReal, Vec, Vec, Vec, void *), InitialConditions(DM, Vec);

int main(int argc, char **argv)
{
  TS     ts; /* nonlinear solver */
  Vec    U;  /* solution, residual vectors */
  DM     da;
  AppCtx appctx;

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Initialize program
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));

  appctx.epsilon = 1.0e-3;
  appctx.delta   = 1.0;
  appctx.alpha   = 10.0;
  appctx.beta    = 4.0;
  appctx.gamma   = 1.0;
  appctx.kappa   = .75;
  appctx.lambda  = 1.0;
  appctx.mu      = 100.;
  appctx.cstar   = .2;
  appctx.upwind  = PETSC_TRUE;

  PetscCall(PetscOptionsGetScalar(NULL, NULL, "-delta", &appctx.delta, NULL));
  PetscCall(PetscOptionsGetBool(NULL, NULL, "-upwind", &appctx.upwind, NULL));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create distributed array (DMDA) to manage parallel grid and vectors
  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, 8, 2, 1, NULL, &da));
  PetscCall(DMSetFromOptions(da));
  PetscCall(DMSetUp(da));
  PetscCall(DMDASetFieldName(da, 0, "rho"));
  PetscCall(DMDASetFieldName(da, 1, "c"));

  /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Extract global vectors from DMDA; then duplicate for remaining
     vectors that are the same types
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(DMCreateGlobalVector(da, &U));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create timestepping solver context
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
  PetscCall(TSSetType(ts, TSROSW));
  PetscCall(TSSetDM(ts, da));
  PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
  PetscCall(TSSetIFunction(ts, NULL, IFunction, &appctx));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set initial conditions
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(InitialConditions(da, U));
  PetscCall(TSSetSolution(ts, U));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set solver options
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSSetTimeStep(ts, .0001));
  PetscCall(TSSetMaxTime(ts, 1.0));
  PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
  PetscCall(TSSetFromOptions(ts));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Solve nonlinear system
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSSolve(ts, U));

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

  PetscCall(PetscFinalize());
  return 0;
}
/* ------------------------------------------------------------------- */
/*
   IFunction - Evaluates nonlinear function, F(U).

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

   Output Parameter:
.  F - function vector
 */
PetscErrorCode IFunction(TS ts, PetscReal ftime, Vec U, Vec Udot, Vec F, void *ptr)
{
  AppCtx     *appctx = (AppCtx *)ptr;
  DM          da;
  PetscInt    i, Mx, xs, xm;
  PetscReal   hx, sx;
  PetscScalar rho, c, rhoxx, cxx, cx, rhox, kcxrhox;
  Field      *u, *f, *udot;
  Vec         localU;

  PetscFunctionBegin;
  PetscCall(TSGetDM(ts, &da));
  PetscCall(DMGetLocalVector(da, &localU));
  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 - 1);
  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, U, INSERT_VALUES, localU));
  PetscCall(DMGlobalToLocalEnd(da, U, INSERT_VALUES, localU));

  /*
     Get pointers to vector data
  */
  PetscCall(DMDAVecGetArrayRead(da, localU, &u));
  PetscCall(DMDAVecGetArrayRead(da, Udot, &udot));
  PetscCall(DMDAVecGetArrayWrite(da, F, &f));

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

  if (!xs) {
    f[0].rho = udot[0].rho; /* u[0].rho - 0.0; */
    f[0].c   = udot[0].c;   /* u[0].c   - 1.0; */
    xs++;
    xm--;
  }
  if (xs + xm == Mx) {
    f[Mx - 1].rho = udot[Mx - 1].rho; /* u[Mx-1].rho - 1.0; */
    f[Mx - 1].c   = udot[Mx - 1].c;   /* u[Mx-1].c   - 0.0;  */
    xm--;
  }

  /*
     Compute function over the locally owned part of the grid
  */
  for (i = xs; i < xs + xm; i++) {
    rho   = u[i].rho;
    rhoxx = (-2.0 * rho + u[i - 1].rho + u[i + 1].rho) * sx;
    c     = u[i].c;
    cxx   = (-2.0 * c + u[i - 1].c + u[i + 1].c) * sx;

    if (!appctx->upwind) {
      rhox    = .5 * (u[i + 1].rho - u[i - 1].rho) / hx;
      cx      = .5 * (u[i + 1].c - u[i - 1].c) / hx;
      kcxrhox = appctx->kappa * (cxx * rho + cx * rhox);
    } else {
      kcxrhox = appctx->kappa * ((u[i + 1].c - u[i].c) * u[i + 1].rho - (u[i].c - u[i - 1].c) * u[i].rho) * sx;
    }

    f[i].rho = udot[i].rho - appctx->epsilon * rhoxx + kcxrhox - appctx->mu * PetscAbsScalar(rho) * (1.0 - rho) * PetscMax(0, PetscRealPart(c - appctx->cstar)) + appctx->beta * rho;
    f[i].c   = udot[i].c - appctx->delta * cxx + appctx->lambda * c + appctx->alpha * rho * c / (appctx->gamma + c);
  }

  /*
     Restore vectors
  */
  PetscCall(DMDAVecRestoreArrayRead(da, localU, &u));
  PetscCall(DMDAVecRestoreArrayRead(da, Udot, &udot));
  PetscCall(DMDAVecRestoreArrayWrite(da, F, &f));
  PetscCall(DMRestoreLocalVector(da, &localU));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* ------------------------------------------------------------------- */
PetscErrorCode InitialConditions(DM da, Vec U)
{
  PetscInt  i, xs, xm, Mx;
  Field    *u;
  PetscReal hx, x;

  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 - 1);

  /*
     Get pointers to vector data
  */
  PetscCall(DMDAVecGetArrayWrite(da, U, &u));

  /*
     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++) {
    x = i * hx;
    if (i < Mx - 1) u[i].rho = 0.0;
    else u[i].rho = 1.0;
    u[i].c = PetscCosReal(.5 * PETSC_PI * x);
  }

  /*
     Restore vectors
  */
  PetscCall(DMDAVecRestoreArrayWrite(da, U, &u));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*TEST

   test:
      args: -pc_type lu -da_refine 2  -ts_view  -ts_monitor -ts_max_time 1
      requires: double

   test:
     suffix: 2
     args:  -pc_type lu -da_refine 2  -ts_view  -ts_monitor_draw_solution -ts_monitor -ts_max_time 1
     requires: x double
     output_file: output/ex4_1.out

TEST*/
