xref: /petsc/src/ts/tutorials/ex2.c (revision b122ec5aa1bd4469eb4e0673542fb7de3f411254)

static char help[] ="Solves a time-dependent nonlinear PDE. Uses implicit\n\
timestepping.  Runtime options include:\n\
  -M <xg>, where <xg> = number of grid points\n\
  -debug : Activate debugging printouts\n\
  -nox   : Deactivate x-window graphics\n\n";

/*
   Concepts: TS^time-dependent nonlinear problems
   Processors: n
*/

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

   This program solves the PDE

               u * u_xx
         u_t = ---------
               2*(t+1)^2

    on the domain 0 <= x <= 1, with boundary conditions
         u(t,0) = t + 1,  u(t,1) = 2*t + 2,
    and initial condition
         u(0,x) = 1 + x*x.

    The exact solution is:
         u(t,x) = (1 + x*x) * (1 + t)

    Note that since the solution is linear in time and quadratic in x,
    the finite difference scheme actually computes the "exact" solution.

    We use by default the backward Euler method.

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

/*
   Include "petscts.h" to use the PETSc timestepping routines. Note that
   this file automatically includes "petscsys.h" and other lower-level
   PETSc include files.

   Include the "petscdmda.h" to allow us to use the distributed array data
   structures to manage the parallel grid.
*/
#include <petscts.h>
#include <petscdm.h>
#include <petscdmda.h>
#include <petscdraw.h>

/*
   User-defined application context - contains data needed by the
   application-provided callback 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) */
} AppCtx;

/*
   User-defined routines, provided below.
*/
extern PetscErrorCode InitialConditions(Vec,AppCtx*);
extern PetscErrorCode RHSFunction(TS,PetscReal,Vec,Vec,void*);
extern PetscErrorCode RHSJacobian(TS,PetscReal,Vec,Mat,Mat,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;                      /* Jacobian matrix data structure */
  Vec            u;                      /* approximate solution vector */
  PetscInt       time_steps_max = 100;  /* default max timesteps */
  PetscReal      dt;
  PetscReal      time_total_max = 100.0; /* default max total time */
  PetscBool      mymonitor      = PETSC_FALSE;
  PetscReal      bounds[]       = {1.0, 3.3};

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

  CHKERRQ(PetscInitialize(&argc,&argv,(char*)0,help));
  CHKERRQ(PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),1,bounds));

  appctx.comm = PETSC_COMM_WORLD;
  appctx.m    = 60;

  CHKERRQ(PetscOptionsGetInt(NULL,NULL,"-M",&appctx.m,NULL));
  CHKERRQ(PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug));
  CHKERRQ(PetscOptionsHasName(NULL,NULL,"-mymonitor",&mymonitor));

  appctx.h    = 1.0/(appctx.m-1.0);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     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.
  */
  CHKERRQ(DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,appctx.m,1,1,NULL,&appctx.da));
  CHKERRQ(DMSetFromOptions(appctx.da));
  CHKERRQ(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.
  */
  CHKERRQ(DMCreateGlobalVector(appctx.da,&u));
  CHKERRQ(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.
  */
  CHKERRQ(VecDuplicate(appctx.u_local,&appctx.localwork));
  CHKERRQ(VecDuplicate(u,&appctx.solution));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create timestepping solver context; set callback routine for
     right-hand-side function evaluation.
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  CHKERRQ(TSCreate(PETSC_COMM_WORLD,&ts));
  CHKERRQ(TSSetProblemType(ts,TS_NONLINEAR));
  CHKERRQ(TSSetRHSFunction(ts,NULL,RHSFunction,&appctx));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set optional user-defined monitoring routine
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  if (mymonitor) {
    CHKERRQ(TSMonitorSet(ts,Monitor,&appctx,NULL));
  }

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     For nonlinear problems, the user can provide a Jacobian evaluation
     routine (or use a finite differencing approximation).

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

  CHKERRQ(MatCreate(PETSC_COMM_WORLD,&A));
  CHKERRQ(MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,appctx.m,appctx.m));
  CHKERRQ(MatSetFromOptions(A));
  CHKERRQ(MatSetUp(A));
  CHKERRQ(TSSetRHSJacobian(ts,A,A,RHSJacobian,&appctx));

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

  dt   = appctx.h/2.0;
  CHKERRQ(TSSetTimeStep(ts,dt));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     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().
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  CHKERRQ(TSSetType(ts,TSBEULER));
  CHKERRQ(TSSetMaxSteps(ts,time_steps_max));
  CHKERRQ(TSSetMaxTime(ts,time_total_max));
  CHKERRQ(TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER));
  CHKERRQ(TSSetFromOptions(ts));

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

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

  /*
     Run the timestepping solver
  */
  CHKERRQ(TSSolve(ts,u));

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

  CHKERRQ(TSDestroy(&ts));
  CHKERRQ(VecDestroy(&u));
  CHKERRQ(MatDestroy(&A));
  CHKERRQ(DMDestroy(&appctx.da));
  CHKERRQ(VecDestroy(&appctx.localwork));
  CHKERRQ(VecDestroy(&appctx.solution));
  CHKERRQ(VecDestroy(&appctx.u_local));

  /*
     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).
  */
  CHKERRQ(PetscFinalize());
  return 0;
}
/* --------------------------------------------------------------------- */
/*
   InitialConditions - Computes the solution at the initial time.

   Input Parameters:
   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,x;
  PetscInt       i,mybase,myend;

  /*
     Determine starting point of each processor's range of
     grid values.
  */
  CHKERRQ(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.
  */
  CHKERRQ(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++) {
    x = h*(PetscReal)i; /* current location in global grid */
    u_localptr[i-mybase] = 1.0 + x*x;
  }

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

  /*
     Print debugging information if desired
  */
  if (appctx->debug) {
    CHKERRQ(PetscPrintf(appctx->comm,"initial guess vector\n"));
    CHKERRQ(VecView(u,PETSC_VIEWER_STDOUT_WORLD));
  }

  return 0;
}
/* --------------------------------------------------------------------- */
/*
   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,x;
  PetscInt       i,mybase,myend;

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

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

  /*
     Simply write the solution directly into the array locations.
     Alternatively, we could use VecSetValues() or VecSetValuesLocal().
  */
  for (i=mybase; i<myend; i++) {
    x = h*(PetscReal)i;
    s_localptr[i-mybase] = (t + 1.0)*(1.0 + x*x);
  }

  /*
     Restore vector
  */
  CHKERRQ(VecRestoreArray(solution,&s_localptr));
  return 0;
}
/* --------------------------------------------------------------------- */
/*
   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      en2,en2s,enmax;
  PetscDraw      draw;

  /*
     We use the default X Windows viewer
             PETSC_VIEWER_DRAW_(appctx->comm)
     that is associated with the current communicator. This saves
     the effort of calling PetscViewerDrawOpen() to create the window.
     Note that if we wished to plot several items in separate windows we
     would create each viewer with PetscViewerDrawOpen() and store them in
     the application context, appctx.

     PetscReal buffering makes graphics look better.
  */
  CHKERRQ(PetscViewerDrawGetDraw(PETSC_VIEWER_DRAW_(appctx->comm),0,&draw));
  CHKERRQ(PetscDrawSetDoubleBuffer(draw));
  CHKERRQ(VecView(u,PETSC_VIEWER_DRAW_(appctx->comm)));

  /*
     Compute the exact solution at this timestep
  */
  CHKERRQ(ExactSolution(time,appctx->solution,appctx));

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

  /*
     Compute the 2-norm and max-norm of the error
  */
  CHKERRQ(VecAXPY(appctx->solution,-1.0,u));
  CHKERRQ(VecNorm(appctx->solution,NORM_2,&en2));
  en2s = PetscSqrtReal(appctx->h)*en2;  /* scale the 2-norm by the grid spacing */
  CHKERRQ(VecNorm(appctx->solution,NORM_MAX,&enmax));

  /*
     PetscPrintf() causes only the first processor in this
     communicator to print the timestep information.
  */
  CHKERRQ(PetscPrintf(appctx->comm,"Timestep %D: time = %g 2-norm error = %g  max norm error = %g\n",step,(double)time,(double)en2s,(double)enmax));

  /*
     Print debugging information if desired
  */
  if (appctx->debug) {
    CHKERRQ(PetscPrintf(appctx->comm,"Error vector\n"));
    CHKERRQ(VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD));
  }
  return 0;
}
/* --------------------------------------------------------------------- */
/*
   RHSFunction - User-provided routine that evalues the right-hand-side
   function of the ODE.  This routine is set in the main program by
   calling TSSetRHSFunction().  We compute:
          global_out = F(global_in)

   Input Parameters:
   ts         - timesteping context
   t          - current time
   global_in  - vector containing the current iterate
   ctx        - (optional) user-provided context for function evaluation.
                In this case we use the appctx defined above.

   Output Parameter:
   global_out - vector containing the newly evaluated function
*/
PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec global_in,Vec global_out,void *ctx)
{
  AppCtx            *appctx   = (AppCtx*) ctx;     /* user-defined application context */
  DM                da        = appctx->da;        /* distributed array */
  Vec               local_in  = appctx->u_local;   /* local ghosted input vector */
  Vec               localwork = appctx->localwork; /* local ghosted work vector */
  PetscInt          i,localsize;
  PetscMPIInt       rank,size;
  PetscScalar       *copyptr,sc;
  const PetscScalar *localptr;

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Get ready for local function computations
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  /*
     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.
  */
  CHKERRQ(DMGlobalToLocalBegin(da,global_in,INSERT_VALUES,local_in));
  CHKERRQ(DMGlobalToLocalEnd(da,global_in,INSERT_VALUES,local_in));

  /*
      Access directly the values in our local INPUT work array
  */
  CHKERRQ(VecGetArrayRead(local_in,&localptr));

  /*
      Access directly the values in our local OUTPUT work array
  */
  CHKERRQ(VecGetArray(localwork,&copyptr));

  sc = 1.0/(appctx->h*appctx->h*2.0*(1.0+t)*(1.0+t));

  /*
      Evaluate our function on the nodes owned by this processor
  */
  CHKERRQ(VecGetLocalSize(local_in,&localsize));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Compute entries for the locally owned part
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  /*
     Handle boundary conditions: This is done by using the boundary condition
        u(t,boundary) = g(t,boundary)
     for some function g. Now take the derivative with respect to t to obtain
        u_{t}(t,boundary) = g_{t}(t,boundary)

     In our case, u(t,0) = t + 1, so that u_{t}(t,0) = 1
             and  u(t,1) = 2t+ 2, so that u_{t}(t,1) = 2
  */
  CHKERRMPI(MPI_Comm_rank(appctx->comm,&rank));
  CHKERRMPI(MPI_Comm_size(appctx->comm,&size));
  if (rank == 0)          copyptr[0]           = 1.0;
  if (rank == size-1) copyptr[localsize-1] = 2.0;

  /*
     Handle the interior nodes where the PDE is replace by finite
     difference operators.
  */
  for (i=1; i<localsize-1; i++) copyptr[i] =  localptr[i] * sc * (localptr[i+1] + localptr[i-1] - 2.0*localptr[i]);

  /*
     Restore vectors
  */
  CHKERRQ(VecRestoreArrayRead(local_in,&localptr));
  CHKERRQ(VecRestoreArray(localwork,&copyptr));

  /*
     Insert values from the local OUTPUT vector into the global
     output vector
  */
  CHKERRQ(DMLocalToGlobalBegin(da,localwork,INSERT_VALUES,global_out));
  CHKERRQ(DMLocalToGlobalEnd(da,localwork,INSERT_VALUES,global_out));

  /* Print debugging information if desired */
  if (appctx->debug) {
    CHKERRQ(PetscPrintf(appctx->comm,"RHS function vector\n"));
    CHKERRQ(VecView(global_out,PETSC_VIEWER_STDOUT_WORLD));
  }

  return 0;
}
/* --------------------------------------------------------------------- */
/*
   RHSJacobian - User-provided routine to compute the Jacobian of
   the nonlinear right-hand-side function of the ODE.

   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
   str - flag indicating matrix structure

  Notes:
  RHSJacobian computes entries for the locally owned part of the Jacobian.
   - 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().
   - 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 RHSJacobian(TS ts,PetscReal t,Vec global_in,Mat AA,Mat BB,void *ctx)
{
  AppCtx            *appctx  = (AppCtx*)ctx;    /* user-defined application context */
  Vec               local_in = appctx->u_local;   /* local ghosted input vector */
  DM                da       = appctx->da;        /* distributed array */
  PetscScalar       v[3],sc;
  const PetscScalar *localptr;
  PetscInt          i,mstart,mend,mstarts,mends,idx[3],is;

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Get ready for local Jacobian computations
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  /*
     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.
  */
  CHKERRQ(DMGlobalToLocalBegin(da,global_in,INSERT_VALUES,local_in));
  CHKERRQ(DMGlobalToLocalEnd(da,global_in,INSERT_VALUES,local_in));

  /*
     Get pointer to vector data
  */
  CHKERRQ(VecGetArrayRead(local_in,&localptr));

  /*
     Get starting and ending locally owned rows of the matrix
  */
  CHKERRQ(MatGetOwnershipRange(BB,&mstarts,&mends));
  mstart = mstarts; mend = mends;

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Compute entries for the locally owned part of the Jacobian.
      - 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).
      - Here, we set all entries for a particular row at once.
      - We can set matrix entries either using either
        MatSetValuesLocal() or MatSetValues().
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  /*
     Set matrix rows corresponding to boundary data
  */
  if (mstart == 0) {
    v[0] = 0.0;
    CHKERRQ(MatSetValues(BB,1,&mstart,1,&mstart,v,INSERT_VALUES));
    mstart++;
  }
  if (mend == appctx->m) {
    mend--;
    v[0] = 0.0;
    CHKERRQ(MatSetValues(BB,1,&mend,1,&mend,v,INSERT_VALUES));
  }

  /*
     Set matrix rows corresponding to interior data.  We construct the
     matrix one row at a time.
  */
  sc = 1.0/(appctx->h*appctx->h*2.0*(1.0+t)*(1.0+t));
  for (i=mstart; i<mend; i++) {
    idx[0] = i-1; idx[1] = i; idx[2] = i+1;
    is     = i - mstart + 1;
    v[0]   = sc*localptr[is];
    v[1]   = sc*(localptr[is+1] + localptr[is-1] - 4.0*localptr[is]);
    v[2]   = sc*localptr[is];
    CHKERRQ(MatSetValues(BB,1,&i,3,idx,v,INSERT_VALUES));
  }

  /*
     Restore vector
  */
  CHKERRQ(VecRestoreArrayRead(local_in,&localptr));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     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.
  */
  CHKERRQ(MatAssemblyBegin(BB,MAT_FINAL_ASSEMBLY));
  CHKERRQ(MatAssemblyEnd(BB,MAT_FINAL_ASSEMBLY));
  if (BB != AA) {
    CHKERRQ(MatAssemblyBegin(AA,MAT_FINAL_ASSEMBLY));
    CHKERRQ(MatAssemblyEnd(AA,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.
  */
  CHKERRQ(MatSetOption(BB,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE));

  return 0;
}

/*TEST

    test:
      args: -nox -ts_dt 10 -mymonitor
      nsize: 2
      requires: !single

    test:
      suffix: tut_1
      nsize: 1
      args: -ts_max_steps 10 -ts_monitor

    test:
      suffix: tut_2
      nsize: 4
      args: -ts_max_steps 10 -ts_monitor -snes_monitor -ksp_monitor

    test:
      suffix: tut_3
      nsize: 4
      args: ./ex2 -ts_max_steps 10 -ts_monitor -M 128

TEST*/