Date: 2013-07-02T14:51:19+00:00

Actual source code: ex3.c
petsc-3.4.2 2013-07-02

      2: static char help[] = "Solves 1D heat equation with FEM formulation.\n\
      3: Input arguments are\n\
      4:   -useAlhs: solve Alhs*U' =  (Arhs*U + g) \n\
      5:             otherwise, solve U' = inv(Alhs)*(Arhs*U + g) \n\n";

      7: /*--------------------------------------------------------------------------
      8:   Solves 1D heat equation U_t = U_xx with FEM formulation:
      9:                           Alhs*U' = rhs (= Arhs*U + g)
     10:   We thank Chris Cox <clcox@clemson.edu> for contributing the original code
     11: ----------------------------------------------------------------------------*/

     13: #include <petscksp.h>
     14: #include <petscts.h>

     16: /* special variable - max size of all arrays  */
     17: #define num_z 60

     19: /*
     20:    User-defined application context - contains data needed by the
     21:    application-provided call-back routines.
     22: */
     23: typedef struct {
     24:   Mat         Amat;               /* left hand side matrix */
     25:   Vec         ksp_rhs,ksp_sol;    /* working vectors for formulating inv(Alhs)*(Arhs*U+g) */
     26:   int         max_probsz;         /* max size of the problem */
     27:   PetscBool   useAlhs;            /* flag (1 indicates solving Alhs*U' = Arhs*U+g */
     28:   int         nz;                 /* total number of grid points */
     29:   PetscInt    m;                  /* total number of interio grid points */
     30:   Vec         solution;           /* global exact ts solution vector */
     31:   PetscScalar *z;                 /* array of grid points */
     32:   PetscBool   debug;              /* flag (1 indicates activation of debugging printouts) */
     33: } AppCtx;

     35: extern PetscScalar exact(PetscScalar,PetscReal);
     36: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
     37: extern PetscErrorCode Petsc_KSPSolve(AppCtx*);
     38: extern PetscScalar bspl(PetscScalar*,PetscScalar,PetscInt,PetscInt,PetscInt[][2],PetscInt);
     39: extern void femBg(PetscScalar[][3],PetscScalar*,PetscInt,PetscScalar*,PetscReal);
     40: extern void femA(AppCtx*,PetscInt,PetscScalar*);
     41: extern void rhs(AppCtx*,PetscScalar*, PetscInt, PetscScalar*,PetscReal);
     42: extern PetscErrorCode RHSfunction(TS,PetscReal,Vec,Vec,void*);

     46: int main(int argc,char **argv)
     47: {
     48:   PetscInt       i,m,nz,steps,max_steps,k,nphase=1;
     49:   PetscScalar    zInitial,zFinal,val,*z;
     50:   PetscReal      stepsz[4],T,ftime;
     52:   TS             ts;
     53:   SNES           snes;
     54:   Mat            Jmat;
     55:   AppCtx         appctx;   /* user-defined application context */
     56:   Vec            init_sol; /* ts solution vector */
     57:   PetscMPIInt    size;

     59:   PetscInitialize(&argc,&argv,(char*)0,help);
     60:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
     61:   if (size != 1) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"This is a uniprocessor example only");

     63:   PetscOptionsHasName(NULL,"-debug",&appctx.debug);
     64:   PetscOptionsHasName(NULL,"-useAlhs",&appctx.useAlhs);
     65:   PetscOptionsGetInt(NULL,"-nphase",&nphase,NULL);
     66:   if (nphase > 3) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"nphase must be an integer between 1 and 3");

     68:   /* initializations */
     69:   zInitial  = 0.0;
     70:   zFinal    = 1.0;
     71:   T         = 0.014/nphase;
     72:   nz        = num_z;
     73:   m         = nz-2;
     74:   appctx.nz = nz;
     75:   max_steps = (PetscInt)10000;

     77:   appctx.m          = m;
     78:   appctx.max_probsz = nz;
     79:   appctx.debug      = PETSC_FALSE;
     80:   appctx.useAlhs    = PETSC_FALSE;

     82:   /* create vector to hold ts solution */
     83:   /*-----------------------------------*/
     84:   VecCreate(PETSC_COMM_WORLD, &init_sol);
     85:   VecSetSizes(init_sol, PETSC_DECIDE, m);
     86:   VecSetFromOptions(init_sol);

     88:   /* create vector to hold true ts soln for comparison */
     89:   VecDuplicate(init_sol, &appctx.solution);

     91:   /* create LHS matrix Amat */
     92:   /*------------------------*/
     93:   MatCreateSeqAIJ(PETSC_COMM_WORLD, m, m, 3, NULL, &appctx.Amat);
     94:   MatSetFromOptions(appctx.Amat);
     95:   /* set space grid points - interio points only! */
     96:   PetscMalloc((nz+1)*sizeof(PetscScalar),&z);
     97:   for (i=0; i<nz; i++) z[i]=(i)*((zFinal-zInitial)/(nz-1));
     98:   appctx.z = z;
     99:   femA(&appctx,nz,z);

    101:   /* create the jacobian matrix */
    102:   /*----------------------------*/
    103:   MatCreate(PETSC_COMM_WORLD, &Jmat);
    104:   MatSetSizes(Jmat,PETSC_DECIDE,PETSC_DECIDE,m,m);
    105:   MatSetFromOptions(Jmat);

    107:   /* create working vectors for formulating rhs=inv(Alhs)*(Arhs*U + g) */
    108:   VecDuplicate(init_sol,&appctx.ksp_rhs);
    109:   VecDuplicate(init_sol,&appctx.ksp_sol);

    111:   /* set intial guess */
    112:   /*------------------*/
    113:   for (i=0; i<nz-2; i++) {
    114:     val  = exact(z[i+1], 0.0);
    115:     VecSetValue(init_sol,i,(PetscScalar)val,INSERT_VALUES);
    116:   }
    117:   VecAssemblyBegin(init_sol);
    118:   VecAssemblyEnd(init_sol);

    120:   /*create a time-stepping context and set the problem type */
    121:   /*--------------------------------------------------------*/
    122:   TSCreate(PETSC_COMM_WORLD, &ts);
    123:   TSSetProblemType(ts,TS_NONLINEAR);

    125:   /* set time-step method */
    126:   TSSetType(ts,TSCN);

    128:   /* Set optional user-defined monitoring routine */
    129:   TSMonitorSet(ts,Monitor,&appctx,NULL);
    130:   /* set the right hand side of U_t = RHSfunction(U,t) */
    131:   TSSetRHSFunction(ts,NULL,(PetscErrorCode (*)(TS,PetscScalar,Vec,Vec,void*))RHSfunction,&appctx);

    133:   if (appctx.useAlhs) {
    134:     /* set the left hand side matrix of Amat*U_t = rhs(U,t) */
    135:     TSSetIFunction(ts,NULL,TSComputeIFunctionLinear,&appctx);
    136:     TSSetIJacobian(ts,appctx.Amat,appctx.Amat,TSComputeIJacobianConstant,&appctx);
    137:   }

    139:   /* use petsc to compute the jacobian by finite differences */
    140:   TSGetSNES(ts,&snes);
    141:   SNESSetJacobian(snes,Jmat,Jmat,SNESComputeJacobianDefault,NULL);

    143:   /* get the command line options if there are any and set them */
    144:   TSSetFromOptions(ts);

    146: #if defined(PETSC_HAVE_SUNDIALS)
    147:   {
    148:     TSType    type;
    149:     PetscBool sundialstype=PETSC_FALSE;
    150:     TSGetType(ts,&type);
    151:     PetscObjectTypeCompare((PetscObject)ts,TSSUNDIALS,&sundialstype);
    152:     if (sundialstype && appctx.useAlhs) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Cannot use Alhs formulation for TSSUNDIALS type");
    153:   }
    154: #endif
    155:   /* Sets the initial solution */
    156:   TSSetSolution(ts,init_sol);

    158:   stepsz[0] = 1.0/(2.0*(nz-1)*(nz-1)); /* (mesh_size)^2/2.0 */
    159:   ftime     = 0.0;
    160:   for (k=0; k<nphase; k++) {
    161:     if (nphase > 1) printf("Phase %d: initial time %g, stepsz %g, duration: %g\n",k,ftime,stepsz[k],(k+1)*T);
    162:     TSSetInitialTimeStep(ts,ftime,stepsz[k]);
    163:     TSSetDuration(ts,max_steps,(k+1)*T);

    165:     /* loop over time steps */
    166:     /*----------------------*/
    167:     TSSolve(ts,init_sol);
    168:     TSGetSolveTime(ts,&ftime);
    169:     TSGetTimeStepNumber(ts,&steps);
    170:     stepsz[k+1] = stepsz[k]*1.5; /* change step size for the next phase */
    171:   }

    173:   /* free space */
    174:   TSDestroy(&ts);
    175:   MatDestroy(&appctx.Amat);
    176:   MatDestroy(&Jmat);
    177:   VecDestroy(&appctx.ksp_rhs);
    178:   VecDestroy(&appctx.ksp_sol);
    179:   VecDestroy(&init_sol);
    180:   VecDestroy(&appctx.solution);
    181:   PetscFree(z);

    183:   PetscFinalize();
    184:   return 0;
    185: }

    187: /*------------------------------------------------------------------------
    188:   Set exact solution
    189:   u(z,t) = sin(6*PI*z)*exp(-36.*PI*PI*t) + 3.*sin(2*PI*z)*exp(-4.*PI*PI*t)
    190: --------------------------------------------------------------------------*/
    191: PetscScalar exact(PetscScalar z,PetscReal t)
    192: {
    193:   PetscScalar val, ex1, ex2;

    195:   ex1 = exp(-36.*PETSC_PI*PETSC_PI*t);
    196:   ex2 = exp(-4.*PETSC_PI*PETSC_PI*t);
    197:   val = sin(6*PETSC_PI*z)*ex1 + 3.*sin(2*PETSC_PI*z)*ex2;
    198:   return val;
    199: }

    203: /*
    204:    Monitor - User-provided routine to monitor the solution computed at
    205:    each timestep.  This example plots the solution and computes the
    206:    error in two different norms.

    208:    Input Parameters:
    209:    ts     - the timestep context
    210:    step   - the count of the current step (with 0 meaning the
    211:              initial condition)
    212:    time   - the current time
    213:    u      - the solution at this timestep
    214:    ctx    - the user-provided context for this monitoring routine.
    215:             In this case we use the application context which contains
    216:             information about the problem size, workspace and the exact
    217:             solution.
    218: */
    219: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
    220: {
    221:   AppCtx         *appctx = (AppCtx*)ctx;
    223:   PetscInt       i,m=appctx->m;
    224:   PetscReal      norm_2,norm_max,h=1.0/(m+1);
    225:   PetscScalar    *u_exact;

    227:   /* Compute the exact solution */
    228:   VecGetArray(appctx->solution,&u_exact);
    229:   for (i=0; i<m; i++) u_exact[i] = exact(appctx->z[i+1],time);
    230:   VecRestoreArray(appctx->solution,&u_exact);

    232:   /* Print debugging information if desired */
    233:   if (appctx->debug) {
    234:     PetscPrintf(PETSC_COMM_SELF,"Computed solution vector at time %g\n",time);
    235:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
    236:     PetscPrintf(PETSC_COMM_SELF,"Exact solution vector\n");
    237:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    238:   }

    240:   /* Compute the 2-norm and max-norm of the error */
    241:   VecAXPY(appctx->solution,-1.0,u);
    242:   VecNorm(appctx->solution,NORM_2,&norm_2);

    244:   norm_2 = PetscSqrtReal(h)*norm_2;
    245:   VecNorm(appctx->solution,NORM_MAX,&norm_max);

    247:   PetscPrintf(PETSC_COMM_SELF,"Timestep %D: time = %G, 2-norm error = %6.4f, max norm error = %6.4f\n",
    248:                      step,time,norm_2,norm_max);

    250:   /*
    251:      Print debugging information if desired
    252:   */
    253:   if (appctx->debug) {
    254:     PetscPrintf(PETSC_COMM_SELF,"Error vector\n");
    255:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    256:   }
    257:   return 0;
    258: }

    260: /*%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    261: %%      Function to solve a linear system using KSP                                           %%
    262: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%*/

    264: PetscErrorCode Petsc_KSPSolve(AppCtx *obj)
    265: {
    267:   KSP            ksp;
    268:   PC             pc;

    270:   /*create the ksp context and set the operators,that is, associate the system matrix with it*/
    271:   KSPCreate(PETSC_COMM_WORLD,&ksp);
    272:   KSPSetOperators(ksp,obj->Amat,obj->Amat,DIFFERENT_NONZERO_PATTERN);

    274:   /*get the preconditioner context, set its type and the tolerances*/
    275:   KSPGetPC(ksp,&pc);
    276:   PCSetType(pc,PCLU);
    277:   KSPSetTolerances(ksp,1.e-7,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT);

    279:   /*get the command line options if there are any and set them*/
    280:   KSPSetFromOptions(ksp);

    282:   /*get the linear system (ksp) solve*/
    283:   KSPSolve(ksp,obj->ksp_rhs,obj->ksp_sol);

    285:   KSPDestroy(&ksp);
    286:   return 0;
    287: }

    289: /***********************************************************************
    290:  * Function to return value of basis function or derivative of basis   *
    291:  *              function.                                              *
    292:  ***********************************************************************
    293:  *                                                                     *
    294:  *       Arguments:                                                    *
    295:  *         x       = array of xpoints or nodal values                  *
    296:  *         xx      = point at which the basis function is to be        *
    297:  *                     evaluated.                                      *
    298:  *         il      = interval containing xx.                           *
    299:  *         iq      = indicates which of the two basis functions in     *
    300:  *                     interval intrvl should be used                  *
    301:  *         nll     = array containing the endpoints of each interval.  *
    302:  *         id      = If id ~= 2, the value of the basis function       *
    303:  *                     is calculated; if id = 2, the value of the      *
    304:  *                     derivative of the basis function is returned.   *
    305:  ***********************************************************************/

    307: PetscScalar bspl(PetscScalar *x, PetscScalar xx,PetscInt il,PetscInt iq,PetscInt nll[][2],PetscInt id)
    308: {
    309:   PetscScalar x1,x2,bfcn;
    310:   PetscInt    i1,i2,iq1,iq2;

    312:   /*** Determine which basis function in interval intrvl is to be used in ***/
    313:   iq1 = iq;
    314:   if (iq1==0) iq2 = 1;
    315:   else iq2 = 0;

    317:   /***  Determine endpoint of the interval intrvl ***/
    318:   i1=nll[il][iq1];
    319:   i2=nll[il][iq2];

    321:   /*** Determine nodal values at the endpoints of the interval intrvl ***/
    322:   x1=x[i1];
    323:   x2=x[i2];
    324:   /* printf("x1=%g\tx2=%g\txx=%g\n",x1,x2,xx); */
    325:   /*** Evaluate basis function ***/
    326:   if (id == 2) bfcn=(1.0)/(x1-x2);
    327:   else bfcn=(xx-x2)/(x1-x2);
    328:   /* printf("bfcn=%g\n",bfcn); */
    329:   return bfcn;
    330: }

    332: /*---------------------------------------------------------
    333:   Function called by rhs function to get B and g
    334: ---------------------------------------------------------*/
    335: void femBg(PetscScalar btri[][3],PetscScalar *f,PetscInt nz,PetscScalar *z, PetscReal t)
    336: {
    337:   PetscInt    i,j,jj,il,ip,ipp,ipq,iq,iquad,iqq;
    338:   PetscInt    nli[num_z][2],indx[num_z];
    339:   PetscScalar dd,dl,zip,zipq,zz,bb,b_z,bbb,bb_z,bij;
    340:   PetscScalar zquad[num_z][3],dlen[num_z],qdwt[3];

    342:   /*  initializing everything - btri and f are initialized in rhs.c  */
    343:   for (i=0; i < nz; i++) {
    344:     nli[i][0]   = 0;
    345:     nli[i][1]   = 0;
    346:     indx[i]     = 0;
    347:     zquad[i][0] = 0.0;
    348:     zquad[i][1] = 0.0;
    349:     zquad[i][2] = 0.0;
    350:     dlen[i]     = 0.0;
    351:   } /*end for (i)*/

    353:   /*  quadrature weights  */
    354:   qdwt[0] = 1.0/6.0;
    355:   qdwt[1] = 4.0/6.0;
    356:   qdwt[2] = 1.0/6.0;

    358:   /* 1st and last nodes have Dirichlet boundary condition -
    359:      set indices there to -1 */

    361:   for (i=0; i < nz-1; i++) indx[i] = i-1;
    362:   indx[nz-1] = -1;

    364:   ipq = 0;
    365:   for (il=0; il < nz-1; il++) {
    366:     ip           = ipq;
    367:     ipq          = ip+1;
    368:     zip          = z[ip];
    369:     zipq         = z[ipq];
    370:     dl           = zipq-zip;
    371:     zquad[il][0] = zip;
    372:     zquad[il][1] = (0.5)*(zip+zipq);
    373:     zquad[il][2] = zipq;
    374:     dlen[il]     = fabs(dl);
    375:     nli[il][0]   = ip;
    376:     nli[il][1]   = ipq;
    377:   }

    379:   for (il=0; il < nz-1; il++) {
    380:     for (iquad=0; iquad < 3; iquad++) {
    381:       dd = (dlen[il])*(qdwt[iquad]);
    382:       zz = zquad[il][iquad];

    384:       for (iq=0; iq < 2; iq++) {
    385:         ip  = nli[il][iq];
    386:         bb  = bspl(z,zz,il,iq,nli,1);
    387:         b_z = bspl(z,zz,il,iq,nli,2);
    388:         i   = indx[ip];

    390:         if (i > -1) {
    391:           for (iqq=0; iqq < 2; iqq++) {
    392:             ipp  = nli[il][iqq];
    393:             bbb  = bspl(z,zz,il,iqq,nli,1);
    394:             bb_z = bspl(z,zz,il,iqq,nli,2);
    395:             j    = indx[ipp];
    396:             bij  = -b_z*bb_z;

    398:             if (j > -1) {
    399:               jj = 1+j-i;
    400:               btri[i][jj] += bij*dd;
    401:             } else {
    402:               f[i] += bij*dd*exact(z[ipp], t);
    403:               /* f[i] += 0.0; */
    404:               /* if (il==0 && j==-1) { */
    405:               /* f[i] += bij*dd*exact(zz,t); */
    406:               /* }*/ /*end if*/
    407:             } /*end else*/
    408:           } /*end for (iqq)*/
    409:         } /*end if (i>0)*/
    410:       } /*end for (iq)*/
    411:     } /*end for (iquad)*/
    412:   } /*end for (il)*/
    413:   return;
    414: }

    416: void femA(AppCtx *obj,PetscInt nz,PetscScalar *z)
    417: {
    418:   PetscInt       i,j,il,ip,ipp,ipq,iq,iquad,iqq;
    419:   PetscInt       nli[num_z][2],indx[num_z];
    420:   PetscScalar    dd,dl,zip,zipq,zz,bb,bbb,aij;
    421:   PetscScalar    rquad[num_z][3],dlen[num_z],qdwt[3],add_term;

    424:   /*  initializing everything  */

    426:   for (i=0; i < nz; i++) {
    427:     nli[i][0]   = 0;
    428:     nli[i][1]   = 0;
    429:     indx[i]     = 0;
    430:     rquad[i][0] = 0.0;
    431:     rquad[i][1] = 0.0;
    432:     rquad[i][2] = 0.0;
    433:     dlen[i]     = 0.0;
    434:   } /*end for (i)*/

    436:   /*  quadrature weights  */
    437:   qdwt[0] = 1.0/6.0;
    438:   qdwt[1] = 4.0/6.0;
    439:   qdwt[2] = 1.0/6.0;

    441:   /* 1st and last nodes have Dirichlet boundary condition -
    442:      set indices there to -1 */

    444:   for (i=0; i < nz-1; i++) indx[i]=i-1;
    445:   indx[nz-1]=-1;

    447:   ipq = 0;

    449:   for (il=0; il < nz-1; il++) {
    450:     ip           = ipq;
    451:     ipq          = ip+1;
    452:     zip          = z[ip];
    453:     zipq         = z[ipq];
    454:     dl           = zipq-zip;
    455:     rquad[il][0] = zip;
    456:     rquad[il][1] = (0.5)*(zip+zipq);
    457:     rquad[il][2] = zipq;
    458:     dlen[il]     = fabs(dl);
    459:     nli[il][0]   = ip;
    460:     nli[il][1]   = ipq;
    461:   } /*end for (il)*/

    463:   for (il=0; il < nz-1; il++) {
    464:     for (iquad=0; iquad < 3; iquad++) {
    465:       dd = (dlen[il])*(qdwt[iquad]);
    466:       zz = rquad[il][iquad];

    468:       for (iq=0; iq < 2; iq++) {
    469:         ip = nli[il][iq];
    470:         bb = bspl(z,zz,il,iq,nli,1);
    471:         i = indx[ip];
    472:         if (i > -1) {
    473:           for (iqq=0; iqq < 2; iqq++) {
    474:             ipp = nli[il][iqq];
    475:             bbb = bspl(z,zz,il,iqq,nli,1);
    476:             j = indx[ipp];
    477:             aij = bb*bbb;
    478:             if (j > -1) {
    479:               add_term = aij*dd;
    480:               MatSetValue(obj->Amat,i,j,add_term,ADD_VALUES);
    481:             }/*endif*/
    482:           } /*end for (iqq)*/
    483:         } /*end if (i>0)*/
    484:       } /*end for (iq)*/
    485:     } /*end for (iquad)*/
    486:   } /*end for (il)*/
    487:   MatAssemblyBegin(obj->Amat,MAT_FINAL_ASSEMBLY);
    488:   MatAssemblyEnd(obj->Amat,MAT_FINAL_ASSEMBLY);
    489:   return;
    490: }

    492: /*---------------------------------------------------------
    493:         Function to fill the rhs vector with
    494:         By + g values ****
    495: ---------------------------------------------------------*/
    496: void rhs(AppCtx *obj,PetscScalar *y, PetscInt nz, PetscScalar *z, PetscReal t)
    497: {
    498:   PetscInt       i,j,js,je,jj;
    499:   PetscScalar    val,g[num_z],btri[num_z][3],add_term;

    502:   for (i=0; i < nz-2; i++) {
    503:     for (j=0; j <= 2; j++) btri[i][j]=0.0;
    504:     g[i] = 0.0;
    505:   }

    507:   /*  call femBg to set the tri-diagonal b matrix and vector g  */
    508:   femBg(btri,g,nz,z,t);

    510:   /*  setting the entries of the right hand side vector  */
    511:   for (i=0; i < nz-2; i++) {
    512:     val = 0.0;
    513:     js  = 0;
    514:     if (i == 0) js = 1;
    515:     je = 2;
    516:     if (i == nz-2) je = 1;

    518:     for (jj=js; jj <= je; jj++) {
    519:       j    = i+jj-1;
    520:       val += (btri[i][jj])*(y[j]);
    521:     }
    522:     add_term = val + g[i];
    523:     VecSetValue(obj->ksp_rhs,(PetscInt)i,(PetscScalar)add_term,INSERT_VALUES);
    524:   }
    525:   VecAssemblyBegin(obj->ksp_rhs);
    526:   VecAssemblyEnd(obj->ksp_rhs);

    528:   /*  return to main driver function  */
    529:   return;
    530: }

    532: /*%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    533: %%   Function to form the right hand side of the time-stepping problem.                       %%
    534: %% -------------------------------------------------------------------------------------------%%
    535:   if (useAlhs):
    536:     globalout = By+g
    537:   else if (!useAlhs):
    538:     globalout = f(y,t)=Ainv(By+g),
    539:       in which the ksp solver to transform the problem A*ydot=By+g
    540:       to the problem ydot=f(y,t)=inv(A)*(By+g)
    541: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%*/

    543: PetscErrorCode RHSfunction(TS ts,PetscReal t,Vec globalin,Vec globalout,void *ctx)
    544: {
    546:   AppCtx         *obj = (AppCtx*)ctx;
    547:   PetscScalar    *soln_ptr,soln[num_z-2];
    548:   PetscInt       i,nz=obj->nz;
    549:   PetscReal      time;

    551:   /* get the previous solution to compute updated system */
    552:   VecGetArray(globalin,&soln_ptr);
    553:   for (i=0; i < num_z-2; i++) soln[i] = soln_ptr[i];
    554:   VecRestoreArray(globalin,&soln_ptr);

    556:   /* clear out the matrix and rhs for ksp to keep things straight */
    557:   VecSet(obj->ksp_rhs,(PetscScalar)0.0);

    559:   time = t;
    560:   /* get the updated system */
    561:   rhs(obj,soln,nz,obj->z,time); /* setup of the By+g rhs */

    563:   /* do a ksp solve to get the rhs for the ts problem */
    564:   if (obj->useAlhs) {
    565:     /* ksp_sol = ksp_rhs */
    566:     VecCopy(obj->ksp_rhs,globalout);
    567:   } else {
    568:     /* ksp_sol = inv(Amat)*ksp_rhs */
    569:     Petsc_KSPSolve(obj);
    570:     VecCopy(obj->ksp_sol,globalout);
    571:   }
    572:   return 0;
    573: }