/*$Id: ex2.c,v 1.30 2000/05/05 22:19:01 balay Exp bsmith $*/ 
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 "petsc.h" and other lower-level 
   PETSc include files. 
 
   Include the "petscda.h" to allow us to use the distributed array data  
   structures to manage the parallel grid. 
*/ 
#include "petscts.h" 
#include "petscda.h" 
 
/*  
   User-defined application context - contains data needed by the  
   application-provided callback routines. 
*/ 
typedef struct { 
  MPI_Comm   comm;          /* communicator */ 
  DA         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 */ 
  int        m;             /* total number of grid points */ 
  double     h;             /* mesh width: h = 1/(m-1) */ 
  PetscTruth debug;         /* flag (1 indicates activation of debugging printouts) */ 
} AppCtx; 
 
/*  
   User-defined routines, provided below. 
*/ 
extern int InitialConditions(Vec,AppCtx*); 
extern int RHSFunction(TS,double,Vec,Vec,void*); 
extern int RHSJacobian(TS,double,Vec,Mat*,Mat*,MatStructure*,void*); 
extern int Monitor(TS,int,double,Vec,void*); 
extern int ExactSolution(double,Vec,AppCtx*); 
 
/* 
   Utility routine for finite difference Jacobian approximation 
*/ 
extern int RHSJacobianFD(TS,double,Vec,Mat*,Mat*,MatStructure*,void*); 
 
#undef __FUNC__ 
#define __FUNC__ "main" 
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 */ 
  int        time_steps_max = 1000;  /* default max timesteps */ 
  int        ierr,steps; 
  double     ftime;                  /* final time */ 
  double     dt; 
  double     time_total_max = 100.0; /* default max total time */ 
  PetscTruth flg; 
 
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     Initialize program and set problem parameters 
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 
  
  PetscInitialize(&argc,&argv,(char*)0,help); 
 
  appctx.comm = PETSC_COMM_WORLD; 
  appctx.m    = 60; 
  ierr = OptionsGetInt(PETSC_NULL,"-M",&appctx.m,PETSC_NULL);CHKERRA(ierr); 
  ierr = OptionsHasName(PETSC_NULL,"-debug",&appctx.debug);CHKERRA(ierr); 
  appctx.h    = 1.0/(appctx.m-1.0); 
 
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     Create vector data structures 
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 
 
  /* 
     Create distributed array (DA) 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. 
  */  
  ierr = DACreate1d(PETSC_COMM_WORLD,DA_NONPERIODIC,appctx.m,1,1,PETSC_NULL, 
                    &appctx.da);CHKERRA(ierr); 
 
  /* 
     Extract global and local vectors from DA; we use these to store the 
     approximate solution.  Then duplicate these for remaining vectors that 
     have the same types. 
  */  
  ierr = DACreateGlobalVector(appctx.da,&u);CHKERRA(ierr); 
  ierr = DACreateLocalVector(appctx.da,&appctx.u_local);CHKERRA(ierr); 
 
  /* 
     Create local work vector for use in evaluating right-hand-side function; 
     create global work vector for storing exact solution. 
  */ 
  ierr = VecDuplicate(appctx.u_local,&appctx.localwork);CHKERRA(ierr); 
  ierr = VecDuplicate(u,&appctx.solution);CHKERRA(ierr); 
 
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     Create timestepping solver context; set callback routine for 
     right-hand-side function evaluation. 
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 
 
  ierr = TSCreate(PETSC_COMM_WORLD,TS_NONLINEAR,&ts);CHKERRA(ierr); 
  ierr = TSSetRHSFunction(ts,RHSFunction,&appctx);CHKERRA(ierr); 
 
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     Set optional user-defined monitoring routine 
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 
 
  ierr = TSSetMonitor(ts,Monitor,&appctx,PETSC_NULL);CHKERRA(ierr); 
 
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     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. 
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 
 
  ierr = MatCreate(PETSC_COMM_WORLD,PETSC_DECIDE,PETSC_DECIDE,appctx.m,appctx.m,&A);CHKERRA(ierr); 
  ierr = OptionsHasName(PETSC_NULL,"-fdjac",&flg);CHKERRA(ierr); 
  if (flg) { 
    ierr = TSSetRHSJacobian(ts,A,A,RHSJacobianFD,&appctx);CHKERRA(ierr); 
  } else { 
    ierr = TSSetRHSJacobian(ts,A,A,RHSJacobian,&appctx);CHKERRA(ierr); 
  } 
 
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     Set solution vector and initial timestep 
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 
 
  dt   = appctx.h/2.0; 
  ierr = TSSetInitialTimeStep(ts,0.0,dt);CHKERRA(ierr); 
  ierr = TSSetSolution(ts,u);CHKERRA(ierr); 
 
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     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 TSSetDuration(). 
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 
 
  ierr = TSSetType(ts,TS_BEULER);CHKERRA(ierr); 
  ierr = TSSetDuration(ts,time_steps_max,time_total_max);CHKERRA(ierr); 
  ierr = TSSetFromOptions(ts);CHKERRA(ierr); 
 
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     Solve the problem 
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 
 
  /* 
     Evaluate initial conditions 
  */ 
  ierr = InitialConditions(u,&appctx);CHKERRA(ierr); 
 
  /* 
     Run the timestepping solver 
  */ 
  ierr = TSStep(ts,&steps,&ftime);CHKERRA(ierr); 
 
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     Free work space.  All PETSc objects should be destroyed when they 
     are no longer needed. 
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 
 
  ierr = TSDestroy(ts);CHKERRA(ierr); 
  ierr = VecDestroy(u);CHKERRA(ierr); 
  ierr = MatDestroy(A);CHKERRA(ierr); 
  ierr = DADestroy(appctx.da);CHKERRA(ierr); 
  ierr = VecDestroy(appctx.localwork);CHKERRA(ierr); 
  ierr = VecDestroy(appctx.solution);CHKERRA(ierr); 
  ierr = VecDestroy(appctx.u_local);CHKERRA(ierr); 
 
  /* 
     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_summary).  
  */ 
  PetscFinalize(); 
  return 0; 
} 
/* --------------------------------------------------------------------- */ 
#undef __FUNC__ 
#define __FUNC__ "InitialConditions" 
/* 
   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) 
*/  
int InitialConditions(Vec u,AppCtx *appctx) 
{ 
  Scalar *u_localptr,h = appctx->h,x; 
  int    i,mybase,myend,ierr; 
 
  /*  
     Determine starting point of each processor's range of 
     grid values. 
  */ 
  ierr = VecGetOwnershipRange(u,&mybase,&myend);CHKERRQ(ierr); 
 
  /*  
    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. 
  */ 
  ierr = VecGetArray(u,&u_localptr);CHKERRQ(ierr); 
 
  /*  
     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*(double)i; /* current location in global grid */ 
    u_localptr[i-mybase] = 1.0 + x*x; 
  } 
 
  /*  
     Restore vector 
  */ 
  ierr = VecRestoreArray(u,&u_localptr);CHKERRQ(ierr); 
 
  /*  
     Print debugging information if desired 
  */ 
  if (appctx->debug) { 
     ierr = PetscPrintf(appctx->comm,"initial guess vector\n");CHKERRQ(ierr); 
     ierr = VecView(u,VIEWER_STDOUT_WORLD);CHKERRQ(ierr); 
  } 
 
  return 0; 
} 
/* --------------------------------------------------------------------- */ 
#undef __FUNC__ 
#define __FUNC__ "ExactSolution" 
/* 
   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 
*/ 
int ExactSolution(double t,Vec solution,AppCtx *appctx) 
{ 
  Scalar *s_localptr,h = appctx->h,x; 
  int    i,mybase,myend,ierr; 
 
  /*  
     Determine starting and ending points of each processor's  
     range of grid values 
  */ 
  ierr = VecGetOwnershipRange(solution,&mybase,&myend);CHKERRQ(ierr); 
 
  /* 
     Get a pointer to vector data. 
  */ 
  ierr = VecGetArray(solution,&s_localptr);CHKERRQ(ierr); 
 
  /*  
     Simply write the solution directly into the array locations. 
     Alternatively, we could use VecSetValues() or VecSetValuesLocal(). 
  */ 
  for (i=mybase; i<myend; i++) { 
    x = h*(double)i; 
    s_localptr[i-mybase] = (t + 1.0)*(1.0 + x*x); 
  } 
 
  /*  
     Restore vector 
  */ 
  ierr = VecRestoreArray(solution,&s_localptr);CHKERRQ(ierr); 
  return 0; 
} 
/* --------------------------------------------------------------------- */ 
#undef __FUNC__ 
#define __FUNC__ "Monitor" 
/* 
   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. 
*/ 
int Monitor(TS ts,int step,double time,Vec u,void *ctx) 
{ 
  AppCtx   *appctx = (AppCtx*) ctx;   /* user-defined application context */ 
  int      ierr; 
  double   en2,en2s,enmax; 
  Scalar   mone = -1.0; 
  Draw     draw; 
 
  /* 
     We use the default X windows viewer 
             VIEWER_DRAW_(appctx->comm) 
     that is associated with the current communicator. This saves 
     the effort of calling ViewerDrawOpen() to create the window. 
     Note that if we wished to plot several items in separate windows we 
     would create each viewer with ViewerDrawOpen() and store them in 
     the application context, appctx. 
 
     Double buffering makes graphics look better. 
  */ 
  ierr = ViewerDrawGetDraw(VIEWER_DRAW_(appctx->comm),0,&draw);CHKERRQ(ierr); 
  ierr = DrawSetDoubleBuffer(draw);CHKERRQ(ierr); 
  ierr = VecView(u,VIEWER_DRAW_(appctx->comm));CHKERRQ(ierr); 
 
  /* 
     Compute the exact solution at this timestep 
  */ 
  ierr = ExactSolution(time,appctx->solution,appctx);CHKERRQ(ierr); 
 
  /* 
     Print debugging information if desired 
  */ 
  if (appctx->debug) { 
     ierr = PetscPrintf(appctx->comm,"Computed solution vector\n");CHKERRQ(ierr); 
     ierr = VecView(u,VIEWER_STDOUT_WORLD);CHKERRQ(ierr); 
     ierr = PetscPrintf(appctx->comm,"Exact solution vector\n");CHKERRQ(ierr); 
     ierr = VecView(appctx->solution,VIEWER_STDOUT_WORLD);CHKERRQ(ierr); 
  } 
 
  /* 
     Compute the 2-norm and max-norm of the error 
  */ 
  ierr = VecAXPY(&mone,u,appctx->solution);CHKERRQ(ierr); 
  ierr = VecNorm(appctx->solution,NORM_2,&en2);CHKERRQ(ierr); 
  en2s  = sqrt(appctx->h)*en2; /* scale the 2-norm by the grid spacing */ 
  ierr = VecNorm(appctx->solution,NORM_MAX,&enmax);CHKERRQ(ierr); 
 
  /* 
     PetscPrintf() causes only the first processor in this  
     communicator to print the timestep information. 
  */ 
  ierr = PetscPrintf(appctx->comm,"Timestep %d: time = %g,2-norm error = %g, max norm error = %g\n", 
              step,time,en2s,enmax);CHKERRQ(ierr); 
 
  /* 
     Print debugging information if desired 
  */ 
  if (appctx->debug) { 
     ierr = PetscPrintf(appctx->comm,"Error vector\n");CHKERRQ(ierr); 
     ierr = VecView(appctx->solution,VIEWER_STDOUT_WORLD);CHKERRQ(ierr); 
  } 
  return 0; 
} 
/* --------------------------------------------------------------------- */ 
#undef __FUNC__ 
#define __FUNC__ "RHSFunction" 
/* 
   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 
*/ 
int RHSFunction(TS ts,double t,Vec global_in,Vec global_out,void *ctx) 
{ 
  AppCtx *appctx = (AppCtx*) ctx;       /* user-defined application context */ 
  DA     da = appctx->da;               /* distributed array */ 
  Vec    local_in = appctx->u_local;    /* local ghosted input vector */ 
  Vec    localwork = appctx->localwork; /* local ghosted work vector */ 
  int    ierr,i,localsize,rank,size;  
  Scalar *copyptr,*localptr,sc; 
 
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     Get ready for local function computations 
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 
  /* 
     Scatter ghost points to local vector, using the 2-step process 
        DAGlobalToLocalBegin(), DAGlobalToLocalEnd(). 
     By placing code between these two statements, computations can be 
     done while messages are in transition. 
  */ 
  ierr = DAGlobalToLocalBegin(da,global_in,INSERT_VALUES,local_in);CHKERRQ(ierr); 
  ierr = DAGlobalToLocalEnd(da,global_in,INSERT_VALUES,local_in);CHKERRQ(ierr); 
 
  /* 
      Access directly the values in our local INPUT work array 
  */ 
  ierr = VecGetArray(local_in,&localptr);CHKERRQ(ierr); 
 
  /* 
      Access directly the values in our local OUTPUT work array 
  */ 
  ierr = VecGetArray(localwork,&copyptr);CHKERRQ(ierr); 
 
  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 
  */ 
  ierr = VecGetLocalSize(local_in,&localsize);CHKERRQ(ierr); 
 
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     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+ 1, so that u_{t}(t,1) = 2 
  */ 
  ierr = MPI_Comm_rank(appctx->comm,&rank);CHKERRQ(ierr); 
  ierr = MPI_Comm_size(appctx->comm,&size);CHKERRQ(ierr); 
  if (!rank)          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 
  */ 
  ierr = VecRestoreArray(local_in,&localptr);CHKERRQ(ierr); 
  ierr = VecRestoreArray(localwork,&copyptr);CHKERRQ(ierr); 
 
  /* 
     Insert values from the local OUTPUT vector into the global  
     output vector 
  */ 
  ierr = DALocalToGlobal(da,localwork,INSERT_VALUES,global_out);CHKERRQ(ierr); 
 
  /* Print debugging information if desired */ 
  if (appctx->debug) { 
     ierr = PetscPrintf(appctx->comm,"RHS function vector\n");CHKERRQ(ierr); 
     ierr = VecView(global_out,VIEWER_STDOUT_WORLD);CHKERRQ(ierr); 
  } 
 
  return 0; 
} 
/* --------------------------------------------------------------------- */ 
#undef __FUNC__ 
#define __FUNC__ "RHSJacobian" 
/* 
   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. 
*/ 
int RHSJacobian(TS ts,double t,Vec global_in,Mat *AA,Mat *BB,MatStructure *str,void *ctx) 
{ 
  Mat    A = *AA;                      /* Jacobian matrix */ 
  AppCtx *appctx = (AppCtx*)ctx;     /* user-defined application context */ 
  Vec    local_in = appctx->u_local;   /* local ghosted input vector */ 
  DA     da = appctx->da;              /* distributed array */ 
  Scalar v[3],*localptr,sc; 
  int    ierr,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 
        DAGlobalToLocalBegin(), DAGlobalToLocalEnd(). 
     By placing code between these two statements, computations can be 
     done while messages are in transition. 
  */ 
  ierr = DAGlobalToLocalBegin(da,global_in,INSERT_VALUES,local_in);CHKERRQ(ierr); 
  ierr = DAGlobalToLocalEnd(da,global_in,INSERT_VALUES,local_in);CHKERRQ(ierr); 
 
  /* 
     Get pointer to vector data 
  */ 
  ierr = VecGetArray(local_in,&localptr);CHKERRQ(ierr); 
 
  /*  
     Get starting and ending locally owned rows of the matrix 
  */ 
  ierr = MatGetOwnershipRange(A,&mstarts,&mends);CHKERRQ(ierr); 
  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; 
    ierr = MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);CHKERRQ(ierr); 
    mstart++; 
  } 
  if (mend == appctx->m) { 
    mend--; 
    v[0] = 0.0; 
    ierr = MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);CHKERRQ(ierr); 
  } 
 
  /* 
     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]; 
    ierr = MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);CHKERRQ(ierr); 
  } 
 
  /*  
     Restore vector 
  */ 
  ierr = VecRestoreArray(local_in,&localptr);CHKERRQ(ierr); 
 
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     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. 
  */ 
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 
  ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 
 
  /* 
     Set flag to indicate that the Jacobian matrix retains an identical 
     nonzero structure throughout all timestepping iterations (although the 
     values of the entries change). Thus, we can save some work in setting 
     up the preconditioner (e.g., no need to redo symbolic factorization for 
     ILU/ICC preconditioners). 
      - If the nonzero structure of the matrix is different during 
        successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN 
        must be used instead.  If you are unsure whether the matrix 
        structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN. 
      - Caution:  If you specify SAME_NONZERO_PATTERN, PETSc 
        believes your assertion and does not check the structure 
        of the matrix.  If you erroneously claim that the structure 
        is the same when it actually is not, the new preconditioner 
        will not function correctly.  Thus, use this optimization 
        feature with caution! 
  */ 
  *str = SAME_NONZERO_PATTERN; 
 
  /* 
     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. 
  */ 
  ierr = MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR);CHKERRQ(ierr); 
 
  return 0; 
}