static char help[] = "Basic problem for multi-rate method.\n";

/*F

\begin{eqnarray}
                 ys' = \frac{ys}{a}\\
                 yf' = ys*cos(bt)\\
\end{eqnarray}

F*/

#include <petscts.h>

typedef struct {
  PetscReal a,b,Tf,dt;
} AppCtx;

static PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec U,Vec F,AppCtx *ctx)
{
  PetscErrorCode    ierr;
  const PetscScalar *u;
  PetscScalar       *f;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(U,&u);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = u[0]/ctx->a;
  f[1] = u[0]*PetscCosScalar(t*ctx->b);
  ierr = VecRestoreArrayRead(U,&u);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
 }

static PetscErrorCode RHSFunctionslow(TS ts,PetscReal t,Vec U,Vec F,AppCtx *ctx)
{
  PetscErrorCode    ierr;
  const PetscScalar *u;
  PetscScalar       *f;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(U,&u);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = u[0]/ctx->a;
  ierr = VecRestoreArrayRead(U,&u);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode RHSFunctionfast(TS ts,PetscReal t,Vec U,Vec F,AppCtx *ctx)
{
  PetscErrorCode    ierr;
  const PetscScalar *u;
  PetscScalar       *f;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(U,&u);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = u[0]*PetscCosScalar(t*ctx->b);
  ierr = VecRestoreArrayRead(U,&u);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*
  Define the analytic solution for check method easily
*/
static PetscErrorCode sol_true(PetscReal t, Vec U,AppCtx *ctx)
{
  PetscErrorCode ierr;
  PetscScalar    *u;

  PetscFunctionBegin;
  ierr = VecGetArray(U,&u);CHKERRQ(ierr);
  u[0] = PetscExpScalar(t/ctx->a);
  u[1] = (ctx->a*PetscCosScalar(ctx->b*t)+ctx->a*ctx->a*ctx->b*PetscSinScalar(ctx->b*t))*PetscExpScalar(t/ctx->a)/(1+ctx->a*ctx->a*ctx->b*ctx->b);
  ierr = VecRestoreArray(U,&u);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

int main(int argc,char **argv)
{
  TS             ts; /* ODE integrator */
  Vec            U;  /* solution will be stored here */
  Vec            Utrue;
  PetscErrorCode ierr;
  PetscMPIInt    size;
  AppCtx         ctx;
  PetscScalar    *u;
  IS             iss;
  IS             isf;
  PetscInt       *indicess;
  PetscInt       *indicesf;
  PetscInt       n=2;
  PetscReal      error,tt;

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Initialize program
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
  ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRMPI(ierr);
  PetscCheckFalse(size > 1,PETSC_COMM_WORLD,PETSC_ERR_SUP,"Only for sequential runs");

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    Create index for slow part and fast part
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = PetscMalloc1(1,&indicess);CHKERRQ(ierr);
  indicess[0] = 0;
  ierr = PetscMalloc1(1,&indicesf);CHKERRQ(ierr);
  indicesf[0] = 1;
  ierr = ISCreateGeneral(PETSC_COMM_SELF,1,indicess,PETSC_COPY_VALUES,&iss);CHKERRQ(ierr);
  ierr = ISCreateGeneral(PETSC_COMM_SELF,1,indicesf,PETSC_COPY_VALUES,&isf);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    Create necesary vector
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = VecCreate(PETSC_COMM_WORLD,&U);CHKERRQ(ierr);
  ierr = VecSetSizes(U,n,PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = VecSetFromOptions(U);CHKERRQ(ierr);
  ierr = VecDuplicate(U,&Utrue);CHKERRQ(ierr);
  ierr = VecCopy(U,Utrue);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    Set runtime options
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"ODE options","");CHKERRQ(ierr);
  {
    ctx.a  = 2.0;
    ctx.b  = 25.0;
    ierr   = PetscOptionsReal("-a","","",ctx.a,&ctx.a,NULL);CHKERRQ(ierr);
    ierr   = PetscOptionsReal("-b","","",ctx.b,&ctx.b,NULL);CHKERRQ(ierr);
    ctx.Tf = 2;
    ctx.dt = 0.01;
    ierr   = PetscOptionsReal("-Tf","","",ctx.Tf,&ctx.Tf,NULL);CHKERRQ(ierr);
    ierr   = PetscOptionsReal("-dt","","",ctx.dt,&ctx.dt,NULL);CHKERRQ(ierr);
  }
 ierr = PetscOptionsEnd();CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Initialize the solution
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = VecGetArray(U,&u);CHKERRQ(ierr);
  u[0] = 1.0;
  u[1] = ctx.a/(1+ctx.a*ctx.a*ctx.b*ctx.b);
  ierr = VecRestoreArray(U,&u);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create timestepping solver context
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr);
  ierr = TSSetType(ts,TSMPRK);CHKERRQ(ierr);

  ierr = TSSetRHSFunction(ts,NULL,(TSRHSFunction)RHSFunction,&ctx);CHKERRQ(ierr);
  ierr = TSRHSSplitSetIS(ts,"slow",iss);CHKERRQ(ierr);
  ierr = TSRHSSplitSetIS(ts,"fast",isf);CHKERRQ(ierr);
  ierr = TSRHSSplitSetRHSFunction(ts,"slow",NULL,(TSRHSFunction)RHSFunctionslow,&ctx);CHKERRQ(ierr);
  ierr = TSRHSSplitSetRHSFunction(ts,"fast",NULL,(TSRHSFunction)RHSFunctionfast,&ctx);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set initial conditions
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = TSSetSolution(ts,U);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set solver options
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = TSSetMaxTime(ts,ctx.Tf);CHKERRQ(ierr);
  ierr = TSSetTimeStep(ts,ctx.dt);CHKERRQ(ierr);
  ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_MATCHSTEP);CHKERRQ(ierr);
  ierr = TSSetFromOptions(ts);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Solve linear system
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = TSSolve(ts,U);CHKERRQ(ierr);
  ierr = VecView(U,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Check the error of the Petsc solution
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = TSGetTime(ts,&tt);CHKERRQ(ierr);
  ierr = sol_true(tt,Utrue,&ctx);CHKERRQ(ierr);
  ierr = VecAXPY(Utrue,-1.0,U);CHKERRQ(ierr);
  ierr = VecNorm(Utrue,NORM_2,&error);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Print norm2 error
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = PetscPrintf(PETSC_COMM_WORLD,"l2 error norm = %g\n", error);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Free work space.  All PETSc objects should be destroyed when they are no longer needed.
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  ierr = VecDestroy(&U);CHKERRQ(ierr);
  ierr = VecDestroy(&Utrue);CHKERRQ(ierr);
  ierr = TSDestroy(&ts);CHKERRQ(ierr);
  ierr = ISDestroy(&iss);CHKERRQ(ierr);
  ierr = ISDestroy(&isf);CHKERRQ(ierr);
  ierr = PetscFree(indicess);CHKERRQ(ierr);
  ierr = PetscFree(indicesf);CHKERRQ(ierr);
  ierr = PetscFinalize();
  return ierr;
}

/*TEST
    build:
      requires: !complex

    test:

TEST*/