Actual source code: irk.c

      1: /*
      2:   Code for timestepping with implicit Runge-Kutta method

      4:   Notes:
      5:   The general system is written as

      7:   F(t,U,Udot) = 0

      9: */
     10: #include <petsc/private/tsimpl.h>
     11: #include <petscdm.h>
     12: #include <petscdt.h>

     14: static TSIRKType         TSIRKDefault = TSIRKGAUSS;
     15: static PetscBool         TSIRKRegisterAllCalled;
     16: static PetscBool         TSIRKPackageInitialized;
     17: static PetscFunctionList TSIRKList;

     19: struct _IRKTableau {
     20:   PetscReal   *A, *b, *c;
     21:   PetscScalar *A_inv, *A_inv_rowsum, *I_s;
     22:   PetscReal   *binterp; /* Dense output formula */
     23: };

     25: typedef struct _IRKTableau *IRKTableau;

     27: typedef struct {
     28:   char        *method_name;
     29:   PetscInt     order;   /* Classical approximation order of the method */
     30:   PetscInt     nstages; /* Number of stages */
     31:   PetscBool    stiffly_accurate;
     32:   PetscInt     pinterp; /* Interpolation order */
     33:   IRKTableau   tableau;
     34:   Vec          U0;    /* Backup vector */
     35:   Vec          Z;     /* Combined stage vector */
     36:   Vec         *Y;     /* States computed during the step */
     37:   Vec          Ydot;  /* Work vector holding time derivatives during residual evaluation */
     38:   Vec          U;     /* U is used to compute Ydot = shift(Y-U) */
     39:   Vec         *YdotI; /* Work vectors to hold the residual evaluation */
     40:   Mat          TJ;    /* KAIJ matrix for the Jacobian of the combined system */
     41:   PetscScalar *work;  /* Scalar work */
     42:   TSStepStatus status;
     43:   PetscBool    rebuild_completion;
     44:   PetscReal    ccfl;
     45: } TS_IRK;

     47: /*@C
     48:   TSIRKTableauCreate - create the tableau for `TSIRK` and provide the entries

     50:   Not Collective

     52:   Input Parameters:
     53: + ts           - timestepping context
     54: . nstages      - number of stages, this is the dimension of the matrices below
     55: . A            - stage coefficients (dimension nstages*nstages, row-major)
     56: . b            - step completion table (dimension nstages)
     57: . c            - abscissa (dimension nstages)
     58: . binterp      - coefficients of the interpolation formula (dimension nstages)
     59: . A_inv        - inverse of A (dimension nstages*nstages, row-major)
     60: . A_inv_rowsum - row sum of the inverse of A (dimension nstages)
     61: - I_s          - identity matrix (dimension nstages*nstages)

     63:   Level: advanced

     65: .seealso: [](ch_ts), `TSIRK`, `TSIRKRegister()`
     66: @*/
     67: PetscErrorCode TSIRKTableauCreate(TS ts, PetscInt nstages, const PetscReal *A, const PetscReal *b, const PetscReal *c, const PetscReal *binterp, const PetscScalar *A_inv, const PetscScalar *A_inv_rowsum, const PetscScalar *I_s)
     68: {
     69:   TS_IRK    *irk = (TS_IRK *)ts->data;
     70:   IRKTableau tab = irk->tableau;

     72:   PetscFunctionBegin;
     73:   irk->order = nstages;
     74:   PetscCall(PetscMalloc3(PetscSqr(nstages), &tab->A, PetscSqr(nstages), &tab->A_inv, PetscSqr(nstages), &tab->I_s));
     75:   PetscCall(PetscMalloc4(nstages, &tab->b, nstages, &tab->c, nstages, &tab->binterp, nstages, &tab->A_inv_rowsum));
     76:   PetscCall(PetscArraycpy(tab->A, A, PetscSqr(nstages)));
     77:   PetscCall(PetscArraycpy(tab->b, b, nstages));
     78:   PetscCall(PetscArraycpy(tab->c, c, nstages));
     79:   /* optional coefficient arrays */
     80:   if (binterp) PetscCall(PetscArraycpy(tab->binterp, binterp, nstages));
     81:   if (A_inv) PetscCall(PetscArraycpy(tab->A_inv, A_inv, PetscSqr(nstages)));
     82:   if (A_inv_rowsum) PetscCall(PetscArraycpy(tab->A_inv_rowsum, A_inv_rowsum, nstages));
     83:   if (I_s) PetscCall(PetscArraycpy(tab->I_s, I_s, PetscSqr(nstages)));
     84:   PetscFunctionReturn(PETSC_SUCCESS);
     85: }

     87: /* Arrays should be freed with PetscFree3(A,b,c) */
     88: static PetscErrorCode TSIRKCreate_Gauss(TS ts)
     89: {
     90:   PetscInt     nstages;
     91:   PetscReal   *gauss_A_real, *gauss_b, *b, *gauss_c;
     92:   PetscScalar *gauss_A, *gauss_A_inv, *gauss_A_inv_rowsum, *I_s;
     93:   PetscScalar *G0, *G1;
     94:   PetscInt     i, j;
     95:   Mat          G0mat, G1mat, Amat;

     97:   PetscFunctionBegin;
     98:   PetscCall(TSIRKGetNumStages(ts, &nstages));
     99:   PetscCall(PetscMalloc3(PetscSqr(nstages), &gauss_A_real, nstages, &gauss_b, nstages, &gauss_c));
    100:   PetscCall(PetscMalloc4(PetscSqr(nstages), &gauss_A, PetscSqr(nstages), &gauss_A_inv, nstages, &gauss_A_inv_rowsum, PetscSqr(nstages), &I_s));
    101:   PetscCall(PetscMalloc3(nstages, &b, PetscSqr(nstages), &G0, PetscSqr(nstages), &G1));
    102:   PetscCall(PetscDTGaussQuadrature(nstages, 0., 1., gauss_c, b));
    103:   for (i = 0; i < nstages; i++) gauss_b[i] = b[i]; /* copy to possibly-complex array */

    105:   /* A^T = G0^{-1} G1 */
    106:   for (i = 0; i < nstages; i++) {
    107:     for (j = 0; j < nstages; j++) {
    108:       G0[i * nstages + j] = PetscPowRealInt(gauss_c[i], j);
    109:       G1[i * nstages + j] = PetscPowRealInt(gauss_c[i], j + 1) / (j + 1);
    110:     }
    111:   }
    112:   /* The arrays above are row-aligned, but we create dense matrices as the transpose */
    113:   PetscCall(MatCreateSeqDense(PETSC_COMM_SELF, nstages, nstages, G0, &G0mat));
    114:   PetscCall(MatCreateSeqDense(PETSC_COMM_SELF, nstages, nstages, G1, &G1mat));
    115:   PetscCall(MatCreateSeqDense(PETSC_COMM_SELF, nstages, nstages, gauss_A, &Amat));
    116:   PetscCall(MatLUFactor(G0mat, NULL, NULL, NULL));
    117:   PetscCall(MatMatSolve(G0mat, G1mat, Amat));
    118:   PetscCall(MatTranspose(Amat, MAT_INPLACE_MATRIX, &Amat));
    119:   for (i = 0; i < nstages; i++)
    120:     for (j = 0; j < nstages; j++) gauss_A_real[i * nstages + j] = PetscRealPart(gauss_A[i * nstages + j]);

    122:   PetscCall(MatDestroy(&G0mat));
    123:   PetscCall(MatDestroy(&G1mat));
    124:   PetscCall(MatDestroy(&Amat));
    125:   PetscCall(PetscFree3(b, G0, G1));

    127:   { /* Invert A */
    128:     /* PETSc does not provide a routine to calculate the inverse of a general matrix.
    129:      * To get the inverse of A, we form a sequential BAIJ matrix from it, consisting of a single block with block size
    130:      * equal to the dimension of A, and then use MatInvertBlockDiagonal(). */
    131:     Mat                A_baij;
    132:     PetscInt           idxm[1] = {0}, idxn[1] = {0};
    133:     const PetscScalar *A_inv;

    135:     PetscCall(MatCreateSeqBAIJ(PETSC_COMM_SELF, nstages, nstages, nstages, 1, NULL, &A_baij));
    136:     PetscCall(MatSetOption(A_baij, MAT_ROW_ORIENTED, PETSC_FALSE));
    137:     PetscCall(MatSetValuesBlocked(A_baij, 1, idxm, 1, idxn, gauss_A, INSERT_VALUES));
    138:     PetscCall(MatAssemblyBegin(A_baij, MAT_FINAL_ASSEMBLY));
    139:     PetscCall(MatAssemblyEnd(A_baij, MAT_FINAL_ASSEMBLY));
    140:     PetscCall(MatInvertBlockDiagonal(A_baij, &A_inv));
    141:     PetscCall(PetscMemcpy(gauss_A_inv, A_inv, nstages * nstages * sizeof(PetscScalar)));
    142:     PetscCall(MatDestroy(&A_baij));
    143:   }

    145:   /* Compute row sums A_inv_rowsum and identity I_s */
    146:   for (i = 0; i < nstages; i++) {
    147:     gauss_A_inv_rowsum[i] = 0;
    148:     for (j = 0; j < nstages; j++) {
    149:       gauss_A_inv_rowsum[i] += gauss_A_inv[i + nstages * j];
    150:       I_s[i + nstages * j] = 1. * (i == j);
    151:     }
    152:   }
    153:   PetscCall(TSIRKTableauCreate(ts, nstages, gauss_A_real, gauss_b, gauss_c, NULL, gauss_A_inv, gauss_A_inv_rowsum, I_s));
    154:   PetscCall(PetscFree3(gauss_A_real, gauss_b, gauss_c));
    155:   PetscCall(PetscFree4(gauss_A, gauss_A_inv, gauss_A_inv_rowsum, I_s));
    156:   PetscFunctionReturn(PETSC_SUCCESS);
    157: }

    159: /*@C
    160:   TSIRKRegister -  adds a `TSIRK` implementation

    162:   Not Collective, No Fortran Support

    164:   Input Parameters:
    165: + sname    - name of user-defined IRK scheme
    166: - function - function to create method context

    168:   Level: advanced

    170:   Note:
    171:   `TSIRKRegister()` may be called multiple times to add several user-defined families.

    173:   Example Usage:
    174: .vb
    175:    TSIRKRegister("my_scheme", MySchemeCreate);
    176: .ve

    178:   Then, your scheme can be chosen with the procedural interface via
    179: $     TSIRKSetType(ts, "my_scheme")
    180:   or at runtime via the option
    181: $     -ts_irk_type my_scheme

    183: .seealso: [](ch_ts), `TSIRK`, `TSIRKRegisterAll()`
    184: @*/
    185: PetscErrorCode TSIRKRegister(const char sname[], PetscErrorCode (*function)(TS))
    186: {
    187:   PetscFunctionBegin;
    188:   PetscCall(TSIRKInitializePackage());
    189:   PetscCall(PetscFunctionListAdd(&TSIRKList, sname, function));
    190:   PetscFunctionReturn(PETSC_SUCCESS);
    191: }

    193: /*@C
    194:   TSIRKRegisterAll - Registers all of the implicit Runge-Kutta methods in `TSIRK`

    196:   Not Collective, but should be called by all processes which will need the schemes to be registered

    198:   Level: advanced

    200: .seealso: [](ch_ts), `TSIRK`, `TSIRKRegisterDestroy()`
    201: @*/
    202: PetscErrorCode TSIRKRegisterAll(void)
    203: {
    204:   PetscFunctionBegin;
    205:   if (TSIRKRegisterAllCalled) PetscFunctionReturn(PETSC_SUCCESS);
    206:   TSIRKRegisterAllCalled = PETSC_TRUE;

    208:   PetscCall(TSIRKRegister(TSIRKGAUSS, TSIRKCreate_Gauss));
    209:   PetscFunctionReturn(PETSC_SUCCESS);
    210: }

    212: /*@C
    213:   TSIRKRegisterDestroy - Frees the list of schemes that were registered by `TSIRKRegister()`.

    215:   Not Collective

    217:   Level: advanced

    219: .seealso: [](ch_ts), `TSIRK`, `TSIRKRegister()`, `TSIRKRegisterAll()`
    220: @*/
    221: PetscErrorCode TSIRKRegisterDestroy(void)
    222: {
    223:   PetscFunctionBegin;
    224:   TSIRKRegisterAllCalled = PETSC_FALSE;
    225:   PetscFunctionReturn(PETSC_SUCCESS);
    226: }

    228: /*@C
    229:   TSIRKInitializePackage - This function initializes everything in the `TSIRK` package. It is called
    230:   from `TSInitializePackage()`.

    232:   Level: developer

    234: .seealso: [](ch_ts), `TSIRK`, `PetscInitialize()`, `TSIRKFinalizePackage()`, `TSInitializePackage()`
    235: @*/
    236: PetscErrorCode TSIRKInitializePackage(void)
    237: {
    238:   PetscFunctionBegin;
    239:   if (TSIRKPackageInitialized) PetscFunctionReturn(PETSC_SUCCESS);
    240:   TSIRKPackageInitialized = PETSC_TRUE;
    241:   PetscCall(TSIRKRegisterAll());
    242:   PetscCall(PetscRegisterFinalize(TSIRKFinalizePackage));
    243:   PetscFunctionReturn(PETSC_SUCCESS);
    244: }

    246: /*@C
    247:   TSIRKFinalizePackage - This function destroys everything in the `TSIRK` package. It is
    248:   called from `PetscFinalize()`.

    250:   Level: developer

    252: .seealso: [](ch_ts), `TSIRK`, `PetscFinalize()`, `TSInitializePackage()`
    253: @*/
    254: PetscErrorCode TSIRKFinalizePackage(void)
    255: {
    256:   PetscFunctionBegin;
    257:   PetscCall(PetscFunctionListDestroy(&TSIRKList));
    258:   TSIRKPackageInitialized = PETSC_FALSE;
    259:   PetscFunctionReturn(PETSC_SUCCESS);
    260: }

    262: /*
    263:  This function can be called before or after ts->vec_sol has been updated.
    264: */
    265: static PetscErrorCode TSEvaluateStep_IRK(TS ts, PetscInt order, Vec U, PetscBool *done)
    266: {
    267:   TS_IRK      *irk   = (TS_IRK *)ts->data;
    268:   IRKTableau   tab   = irk->tableau;
    269:   Vec         *YdotI = irk->YdotI;
    270:   PetscScalar *w     = irk->work;
    271:   PetscReal    h;
    272:   PetscInt     j;

    274:   PetscFunctionBegin;
    275:   switch (irk->status) {
    276:   case TS_STEP_INCOMPLETE:
    277:   case TS_STEP_PENDING:
    278:     h = ts->time_step;
    279:     break;
    280:   case TS_STEP_COMPLETE:
    281:     h = ts->ptime - ts->ptime_prev;
    282:     break;
    283:   default:
    284:     SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_PLIB, "Invalid TSStepStatus");
    285:   }

    287:   PetscCall(VecCopy(ts->vec_sol, U));
    288:   for (j = 0; j < irk->nstages; j++) w[j] = h * tab->b[j];
    289:   PetscCall(VecMAXPY(U, irk->nstages, w, YdotI));
    290:   PetscFunctionReturn(PETSC_SUCCESS);
    291: }

    293: static PetscErrorCode TSRollBack_IRK(TS ts)
    294: {
    295:   TS_IRK *irk = (TS_IRK *)ts->data;

    297:   PetscFunctionBegin;
    298:   PetscCall(VecCopy(irk->U0, ts->vec_sol));
    299:   PetscFunctionReturn(PETSC_SUCCESS);
    300: }

    302: static PetscErrorCode TSStep_IRK(TS ts)
    303: {
    304:   TS_IRK        *irk   = (TS_IRK *)ts->data;
    305:   IRKTableau     tab   = irk->tableau;
    306:   PetscScalar   *A_inv = tab->A_inv, *A_inv_rowsum = tab->A_inv_rowsum;
    307:   const PetscInt nstages = irk->nstages;
    308:   SNES           snes;
    309:   PetscInt       i, j, its, lits, bs;
    310:   TSAdapt        adapt;
    311:   PetscInt       rejections     = 0;
    312:   PetscBool      accept         = PETSC_TRUE;
    313:   PetscReal      next_time_step = ts->time_step;

    315:   PetscFunctionBegin;
    316:   if (!ts->steprollback) PetscCall(VecCopy(ts->vec_sol, irk->U0));
    317:   PetscCall(VecGetBlockSize(ts->vec_sol, &bs));
    318:   for (i = 0; i < nstages; i++) PetscCall(VecStrideScatter(ts->vec_sol, i * bs, irk->Z, INSERT_VALUES));

    320:   irk->status = TS_STEP_INCOMPLETE;
    321:   while (!ts->reason && irk->status != TS_STEP_COMPLETE) {
    322:     PetscCall(VecCopy(ts->vec_sol, irk->U));
    323:     PetscCall(TSGetSNES(ts, &snes));
    324:     PetscCall(SNESSolve(snes, NULL, irk->Z));
    325:     PetscCall(SNESGetIterationNumber(snes, &its));
    326:     PetscCall(SNESGetLinearSolveIterations(snes, &lits));
    327:     ts->snes_its += its;
    328:     ts->ksp_its += lits;
    329:     PetscCall(VecStrideGatherAll(irk->Z, irk->Y, INSERT_VALUES));
    330:     for (i = 0; i < nstages; i++) {
    331:       PetscCall(VecZeroEntries(irk->YdotI[i]));
    332:       for (j = 0; j < nstages; j++) PetscCall(VecAXPY(irk->YdotI[i], A_inv[i + j * nstages] / ts->time_step, irk->Y[j]));
    333:       PetscCall(VecAXPY(irk->YdotI[i], -A_inv_rowsum[i] / ts->time_step, irk->U));
    334:     }
    335:     irk->status = TS_STEP_INCOMPLETE;
    336:     PetscCall(TSEvaluateStep_IRK(ts, irk->order, ts->vec_sol, NULL));
    337:     irk->status = TS_STEP_PENDING;
    338:     PetscCall(TSGetAdapt(ts, &adapt));
    339:     PetscCall(TSAdaptChoose(adapt, ts, ts->time_step, NULL, &next_time_step, &accept));
    340:     irk->status = accept ? TS_STEP_COMPLETE : TS_STEP_INCOMPLETE;
    341:     if (!accept) {
    342:       PetscCall(TSRollBack_IRK(ts));
    343:       ts->time_step = next_time_step;
    344:       goto reject_step;
    345:     }

    347:     ts->ptime += ts->time_step;
    348:     ts->time_step = next_time_step;
    349:     break;
    350:   reject_step:
    351:     ts->reject++;
    352:     accept = PETSC_FALSE;
    353:     if (!ts->reason && ++rejections > ts->max_reject && ts->max_reject >= 0) {
    354:       ts->reason = TS_DIVERGED_STEP_REJECTED;
    355:       PetscCall(PetscInfo(ts, "Step=%" PetscInt_FMT ", step rejections %" PetscInt_FMT " greater than current TS allowed, stopping solve\n", ts->steps, rejections));
    356:     }
    357:   }
    358:   PetscFunctionReturn(PETSC_SUCCESS);
    359: }

    361: static PetscErrorCode TSInterpolate_IRK(TS ts, PetscReal itime, Vec U)
    362: {
    363:   TS_IRK          *irk     = (TS_IRK *)ts->data;
    364:   PetscInt         nstages = irk->nstages, pinterp = irk->pinterp, i, j;
    365:   PetscReal        h;
    366:   PetscReal        tt, t;
    367:   PetscScalar     *bt;
    368:   const PetscReal *B = irk->tableau->binterp;

    370:   PetscFunctionBegin;
    371:   PetscCheck(B, PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "TSIRK %s does not have an interpolation formula", irk->method_name);
    372:   switch (irk->status) {
    373:   case TS_STEP_INCOMPLETE:
    374:   case TS_STEP_PENDING:
    375:     h = ts->time_step;
    376:     t = (itime - ts->ptime) / h;
    377:     break;
    378:   case TS_STEP_COMPLETE:
    379:     h = ts->ptime - ts->ptime_prev;
    380:     t = (itime - ts->ptime) / h + 1; /* In the interval [0,1] */
    381:     break;
    382:   default:
    383:     SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_PLIB, "Invalid TSStepStatus");
    384:   }
    385:   PetscCall(PetscMalloc1(nstages, &bt));
    386:   for (i = 0; i < nstages; i++) bt[i] = 0;
    387:   for (j = 0, tt = t; j < pinterp; j++, tt *= t) {
    388:     for (i = 0; i < nstages; i++) bt[i] += h * B[i * pinterp + j] * tt;
    389:   }
    390:   PetscCall(VecMAXPY(U, nstages, bt, irk->YdotI));
    391:   PetscFunctionReturn(PETSC_SUCCESS);
    392: }

    394: static PetscErrorCode TSIRKTableauReset(TS ts)
    395: {
    396:   TS_IRK    *irk = (TS_IRK *)ts->data;
    397:   IRKTableau tab = irk->tableau;

    399:   PetscFunctionBegin;
    400:   if (!tab) PetscFunctionReturn(PETSC_SUCCESS);
    401:   PetscCall(PetscFree3(tab->A, tab->A_inv, tab->I_s));
    402:   PetscCall(PetscFree4(tab->b, tab->c, tab->binterp, tab->A_inv_rowsum));
    403:   PetscFunctionReturn(PETSC_SUCCESS);
    404: }

    406: static PetscErrorCode TSReset_IRK(TS ts)
    407: {
    408:   TS_IRK *irk = (TS_IRK *)ts->data;

    410:   PetscFunctionBegin;
    411:   PetscCall(TSIRKTableauReset(ts));
    412:   if (irk->tableau) PetscCall(PetscFree(irk->tableau));
    413:   if (irk->method_name) PetscCall(PetscFree(irk->method_name));
    414:   if (irk->work) PetscCall(PetscFree(irk->work));
    415:   PetscCall(VecDestroyVecs(irk->nstages, &irk->Y));
    416:   PetscCall(VecDestroyVecs(irk->nstages, &irk->YdotI));
    417:   PetscCall(VecDestroy(&irk->Ydot));
    418:   PetscCall(VecDestroy(&irk->Z));
    419:   PetscCall(VecDestroy(&irk->U));
    420:   PetscCall(VecDestroy(&irk->U0));
    421:   PetscCall(MatDestroy(&irk->TJ));
    422:   PetscFunctionReturn(PETSC_SUCCESS);
    423: }

    425: static PetscErrorCode TSIRKGetVecs(TS ts, DM dm, Vec *U)
    426: {
    427:   TS_IRK *irk = (TS_IRK *)ts->data;

    429:   PetscFunctionBegin;
    430:   if (U) {
    431:     if (dm && dm != ts->dm) {
    432:       PetscCall(DMGetNamedGlobalVector(dm, "TSIRK_U", U));
    433:     } else *U = irk->U;
    434:   }
    435:   PetscFunctionReturn(PETSC_SUCCESS);
    436: }

    438: static PetscErrorCode TSIRKRestoreVecs(TS ts, DM dm, Vec *U)
    439: {
    440:   PetscFunctionBegin;
    441:   if (U) {
    442:     if (dm && dm != ts->dm) PetscCall(DMRestoreNamedGlobalVector(dm, "TSIRK_U", U));
    443:   }
    444:   PetscFunctionReturn(PETSC_SUCCESS);
    445: }

    447: /*
    448:   This defines the nonlinear equations that is to be solved with SNES
    449:     G[e\otimes t + C*dt, Z, Zdot] = 0
    450:     Zdot = (In \otimes S)*Z - (In \otimes Se) U
    451:   where S = 1/(dt*A)
    452: */
    453: static PetscErrorCode SNESTSFormFunction_IRK(SNES snes, Vec ZC, Vec FC, TS ts)
    454: {
    455:   TS_IRK            *irk     = (TS_IRK *)ts->data;
    456:   IRKTableau         tab     = irk->tableau;
    457:   const PetscInt     nstages = irk->nstages;
    458:   const PetscReal   *c       = tab->c;
    459:   const PetscScalar *A_inv = tab->A_inv, *A_inv_rowsum = tab->A_inv_rowsum;
    460:   DM                 dm, dmsave;
    461:   Vec                U, *YdotI = irk->YdotI, Ydot = irk->Ydot, *Y = irk->Y;
    462:   PetscReal          h = ts->time_step;
    463:   PetscInt           i, j;

    465:   PetscFunctionBegin;
    466:   PetscCall(SNESGetDM(snes, &dm));
    467:   PetscCall(TSIRKGetVecs(ts, dm, &U));
    468:   PetscCall(VecStrideGatherAll(ZC, Y, INSERT_VALUES));
    469:   dmsave = ts->dm;
    470:   ts->dm = dm;
    471:   for (i = 0; i < nstages; i++) {
    472:     PetscCall(VecZeroEntries(Ydot));
    473:     for (j = 0; j < nstages; j++) PetscCall(VecAXPY(Ydot, A_inv[j * nstages + i] / h, Y[j]));
    474:     PetscCall(VecAXPY(Ydot, -A_inv_rowsum[i] / h, U)); /* Ydot = (S \otimes In)*Z - (Se \otimes In) U */
    475:     PetscCall(TSComputeIFunction(ts, ts->ptime + ts->time_step * c[i], Y[i], Ydot, YdotI[i], PETSC_FALSE));
    476:   }
    477:   PetscCall(VecStrideScatterAll(YdotI, FC, INSERT_VALUES));
    478:   ts->dm = dmsave;
    479:   PetscCall(TSIRKRestoreVecs(ts, dm, &U));
    480:   PetscFunctionReturn(PETSC_SUCCESS);
    481: }

    483: /*
    484:    For explicit ODE, the Jacobian is
    485:      JC = I_n \otimes S - J \otimes I_s
    486:    For DAE, the Jacobian is
    487:      JC = M_n \otimes S - J \otimes I_s
    488: */
    489: static PetscErrorCode SNESTSFormJacobian_IRK(SNES snes, Vec ZC, Mat JC, Mat JCpre, TS ts)
    490: {
    491:   TS_IRK          *irk     = (TS_IRK *)ts->data;
    492:   IRKTableau       tab     = irk->tableau;
    493:   const PetscInt   nstages = irk->nstages;
    494:   const PetscReal *c       = tab->c;
    495:   DM               dm, dmsave;
    496:   Vec             *Y = irk->Y, Ydot = irk->Ydot;
    497:   Mat              J;
    498:   PetscScalar     *S;
    499:   PetscInt         i, j, bs;

    501:   PetscFunctionBegin;
    502:   PetscCall(SNESGetDM(snes, &dm));
    503:   /* irk->Ydot has already been computed in SNESTSFormFunction_IRK (SNES guarantees this) */
    504:   dmsave = ts->dm;
    505:   ts->dm = dm;
    506:   PetscCall(VecGetBlockSize(Y[nstages - 1], &bs));
    507:   if (ts->equation_type <= TS_EQ_ODE_EXPLICIT) { /* Support explicit formulas only */
    508:     PetscCall(VecStrideGather(ZC, (nstages - 1) * bs, Y[nstages - 1], INSERT_VALUES));
    509:     PetscCall(MatKAIJGetAIJ(JC, &J));
    510:     PetscCall(TSComputeIJacobian(ts, ts->ptime + ts->time_step * c[nstages - 1], Y[nstages - 1], Ydot, 0, J, J, PETSC_FALSE));
    511:     PetscCall(MatKAIJGetS(JC, NULL, NULL, &S));
    512:     for (i = 0; i < nstages; i++)
    513:       for (j = 0; j < nstages; j++) S[i + nstages * j] = tab->A_inv[i + nstages * j] / ts->time_step;
    514:     PetscCall(MatKAIJRestoreS(JC, &S));
    515:   } else SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "TSIRK %s does not support implicit formula", irk->method_name); /* TODO: need the mass matrix for DAE  */
    516:   ts->dm = dmsave;
    517:   PetscFunctionReturn(PETSC_SUCCESS);
    518: }

    520: static PetscErrorCode DMCoarsenHook_TSIRK(DM fine, DM coarse, void *ctx)
    521: {
    522:   PetscFunctionBegin;
    523:   PetscFunctionReturn(PETSC_SUCCESS);
    524: }

    526: static PetscErrorCode DMRestrictHook_TSIRK(DM fine, Mat restrct, Vec rscale, Mat inject, DM coarse, void *ctx)
    527: {
    528:   TS  ts = (TS)ctx;
    529:   Vec U, U_c;

    531:   PetscFunctionBegin;
    532:   PetscCall(TSIRKGetVecs(ts, fine, &U));
    533:   PetscCall(TSIRKGetVecs(ts, coarse, &U_c));
    534:   PetscCall(MatRestrict(restrct, U, U_c));
    535:   PetscCall(VecPointwiseMult(U_c, rscale, U_c));
    536:   PetscCall(TSIRKRestoreVecs(ts, fine, &U));
    537:   PetscCall(TSIRKRestoreVecs(ts, coarse, &U_c));
    538:   PetscFunctionReturn(PETSC_SUCCESS);
    539: }

    541: static PetscErrorCode DMSubDomainHook_TSIRK(DM dm, DM subdm, void *ctx)
    542: {
    543:   PetscFunctionBegin;
    544:   PetscFunctionReturn(PETSC_SUCCESS);
    545: }

    547: static PetscErrorCode DMSubDomainRestrictHook_TSIRK(DM dm, VecScatter gscat, VecScatter lscat, DM subdm, void *ctx)
    548: {
    549:   TS  ts = (TS)ctx;
    550:   Vec U, U_c;

    552:   PetscFunctionBegin;
    553:   PetscCall(TSIRKGetVecs(ts, dm, &U));
    554:   PetscCall(TSIRKGetVecs(ts, subdm, &U_c));

    556:   PetscCall(VecScatterBegin(gscat, U, U_c, INSERT_VALUES, SCATTER_FORWARD));
    557:   PetscCall(VecScatterEnd(gscat, U, U_c, INSERT_VALUES, SCATTER_FORWARD));

    559:   PetscCall(TSIRKRestoreVecs(ts, dm, &U));
    560:   PetscCall(TSIRKRestoreVecs(ts, subdm, &U_c));
    561:   PetscFunctionReturn(PETSC_SUCCESS);
    562: }

    564: static PetscErrorCode TSSetUp_IRK(TS ts)
    565: {
    566:   TS_IRK        *irk = (TS_IRK *)ts->data;
    567:   IRKTableau     tab = irk->tableau;
    568:   DM             dm;
    569:   Mat            J;
    570:   Vec            R;
    571:   const PetscInt nstages = irk->nstages;
    572:   PetscInt       vsize, bs;

    574:   PetscFunctionBegin;
    575:   if (!irk->work) PetscCall(PetscMalloc1(irk->nstages, &irk->work));
    576:   if (!irk->Y) PetscCall(VecDuplicateVecs(ts->vec_sol, irk->nstages, &irk->Y));
    577:   if (!irk->YdotI) PetscCall(VecDuplicateVecs(ts->vec_sol, irk->nstages, &irk->YdotI));
    578:   if (!irk->Ydot) PetscCall(VecDuplicate(ts->vec_sol, &irk->Ydot));
    579:   if (!irk->U) PetscCall(VecDuplicate(ts->vec_sol, &irk->U));
    580:   if (!irk->U0) PetscCall(VecDuplicate(ts->vec_sol, &irk->U0));
    581:   if (!irk->Z) {
    582:     PetscCall(VecCreate(PetscObjectComm((PetscObject)ts->vec_sol), &irk->Z));
    583:     PetscCall(VecGetSize(ts->vec_sol, &vsize));
    584:     PetscCall(VecSetSizes(irk->Z, PETSC_DECIDE, vsize * irk->nstages));
    585:     PetscCall(VecGetBlockSize(ts->vec_sol, &bs));
    586:     PetscCall(VecSetBlockSize(irk->Z, irk->nstages * bs));
    587:     PetscCall(VecSetFromOptions(irk->Z));
    588:   }
    589:   PetscCall(TSGetDM(ts, &dm));
    590:   PetscCall(DMCoarsenHookAdd(dm, DMCoarsenHook_TSIRK, DMRestrictHook_TSIRK, ts));
    591:   PetscCall(DMSubDomainHookAdd(dm, DMSubDomainHook_TSIRK, DMSubDomainRestrictHook_TSIRK, ts));

    593:   PetscCall(TSGetSNES(ts, &ts->snes));
    594:   PetscCall(VecDuplicate(irk->Z, &R));
    595:   PetscCall(SNESSetFunction(ts->snes, R, SNESTSFormFunction, ts));
    596:   PetscCall(TSGetIJacobian(ts, &J, NULL, NULL, NULL));
    597:   if (!irk->TJ) {
    598:     /* Create the KAIJ matrix for solving the stages */
    599:     PetscCall(MatCreateKAIJ(J, nstages, nstages, tab->A_inv, tab->I_s, &irk->TJ));
    600:   }
    601:   PetscCall(SNESSetJacobian(ts->snes, irk->TJ, irk->TJ, SNESTSFormJacobian, ts));
    602:   PetscCall(VecDestroy(&R));
    603:   PetscFunctionReturn(PETSC_SUCCESS);
    604: }

    606: static PetscErrorCode TSSetFromOptions_IRK(TS ts, PetscOptionItems *PetscOptionsObject)
    607: {
    608:   TS_IRK *irk        = (TS_IRK *)ts->data;
    609:   char    tname[256] = TSIRKGAUSS;

    611:   PetscFunctionBegin;
    612:   PetscOptionsHeadBegin(PetscOptionsObject, "IRK ODE solver options");
    613:   {
    614:     PetscBool flg1, flg2;
    615:     PetscCall(PetscOptionsInt("-ts_irk_nstages", "Stages of the IRK method", "TSIRKSetNumStages", irk->nstages, &irk->nstages, &flg1));
    616:     PetscCall(PetscOptionsFList("-ts_irk_type", "Type of IRK method", "TSIRKSetType", TSIRKList, irk->method_name[0] ? irk->method_name : tname, tname, sizeof(tname), &flg2));
    617:     if (flg1 || flg2 || !irk->method_name[0]) { /* Create the method tableau after nstages or method is set */
    618:       PetscCall(TSIRKSetType(ts, tname));
    619:     }
    620:   }
    621:   PetscOptionsHeadEnd();
    622:   PetscFunctionReturn(PETSC_SUCCESS);
    623: }

    625: static PetscErrorCode TSView_IRK(TS ts, PetscViewer viewer)
    626: {
    627:   TS_IRK   *irk = (TS_IRK *)ts->data;
    628:   PetscBool iascii;

    630:   PetscFunctionBegin;
    631:   PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &iascii));
    632:   if (iascii) {
    633:     IRKTableau tab = irk->tableau;
    634:     TSIRKType  irktype;
    635:     char       buf[512];

    637:     PetscCall(TSIRKGetType(ts, &irktype));
    638:     PetscCall(PetscViewerASCIIPrintf(viewer, "  IRK type %s\n", irktype));
    639:     PetscCall(PetscFormatRealArray(buf, sizeof(buf), "% 8.6f", irk->nstages, tab->c));
    640:     PetscCall(PetscViewerASCIIPrintf(viewer, "  Abscissa       c = %s\n", buf));
    641:     PetscCall(PetscViewerASCIIPrintf(viewer, "Stiffly accurate: %s\n", irk->stiffly_accurate ? "yes" : "no"));
    642:     PetscCall(PetscFormatRealArray(buf, sizeof(buf), "% 8.6f", PetscSqr(irk->nstages), tab->A));
    643:     PetscCall(PetscViewerASCIIPrintf(viewer, "  A coefficients       A = %s\n", buf));
    644:   }
    645:   PetscFunctionReturn(PETSC_SUCCESS);
    646: }

    648: static PetscErrorCode TSLoad_IRK(TS ts, PetscViewer viewer)
    649: {
    650:   SNES    snes;
    651:   TSAdapt adapt;

    653:   PetscFunctionBegin;
    654:   PetscCall(TSGetAdapt(ts, &adapt));
    655:   PetscCall(TSAdaptLoad(adapt, viewer));
    656:   PetscCall(TSGetSNES(ts, &snes));
    657:   PetscCall(SNESLoad(snes, viewer));
    658:   /* function and Jacobian context for SNES when used with TS is always ts object */
    659:   PetscCall(SNESSetFunction(snes, NULL, NULL, ts));
    660:   PetscCall(SNESSetJacobian(snes, NULL, NULL, NULL, ts));
    661:   PetscFunctionReturn(PETSC_SUCCESS);
    662: }

    664: /*@
    665:   TSIRKSetType - Set the type of `TSIRK` scheme to use

    667:   Logically Collective

    669:   Input Parameters:
    670: + ts      - timestepping context
    671: - irktype - type of `TSIRK` scheme

    673:   Options Database Key:
    674: . -ts_irk_type <gauss> - set irk type

    676:   Level: intermediate

    678: .seealso: [](ch_ts), `TSIRKGetType()`, `TSIRK`, `TSIRKType`, `TSIRKGAUSS`
    679: @*/
    680: PetscErrorCode TSIRKSetType(TS ts, TSIRKType irktype)
    681: {
    682:   PetscFunctionBegin;
    684:   PetscAssertPointer(irktype, 2);
    685:   PetscTryMethod(ts, "TSIRKSetType_C", (TS, TSIRKType), (ts, irktype));
    686:   PetscFunctionReturn(PETSC_SUCCESS);
    687: }

    689: /*@
    690:   TSIRKGetType - Get the type of `TSIRK` IMEX scheme being used

    692:   Logically Collective

    694:   Input Parameter:
    695: . ts - timestepping context

    697:   Output Parameter:
    698: . irktype - type of `TSIRK` IMEX scheme

    700:   Level: intermediate

    702: .seealso: [](ch_ts), `TSIRK`, `TSIRKType`, `TSIRKGAUSS`
    703: @*/
    704: PetscErrorCode TSIRKGetType(TS ts, TSIRKType *irktype)
    705: {
    706:   PetscFunctionBegin;
    708:   PetscUseMethod(ts, "TSIRKGetType_C", (TS, TSIRKType *), (ts, irktype));
    709:   PetscFunctionReturn(PETSC_SUCCESS);
    710: }

    712: /*@
    713:   TSIRKSetNumStages - Set the number of stages of `TSIRK` scheme to use

    715:   Logically Collective

    717:   Input Parameters:
    718: + ts      - timestepping context
    719: - nstages - number of stages of `TSIRK` scheme

    721:   Options Database Key:
    722: . -ts_irk_nstages <int> - set number of stages

    724:   Level: intermediate

    726: .seealso: [](ch_ts), `TSIRKGetNumStages()`, `TSIRK`
    727: @*/
    728: PetscErrorCode TSIRKSetNumStages(TS ts, PetscInt nstages)
    729: {
    730:   PetscFunctionBegin;
    732:   PetscTryMethod(ts, "TSIRKSetNumStages_C", (TS, PetscInt), (ts, nstages));
    733:   PetscFunctionReturn(PETSC_SUCCESS);
    734: }

    736: /*@
    737:   TSIRKGetNumStages - Get the number of stages of `TSIRK` scheme

    739:   Logically Collective

    741:   Input Parameters:
    742: + ts      - timestepping context
    743: - nstages - number of stages of `TSIRK` scheme

    745:   Level: intermediate

    747: .seealso: [](ch_ts), `TSIRKSetNumStages()`, `TSIRK`
    748: @*/
    749: PetscErrorCode TSIRKGetNumStages(TS ts, PetscInt *nstages)
    750: {
    751:   PetscFunctionBegin;
    753:   PetscAssertPointer(nstages, 2);
    754:   PetscTryMethod(ts, "TSIRKGetNumStages_C", (TS, PetscInt *), (ts, nstages));
    755:   PetscFunctionReturn(PETSC_SUCCESS);
    756: }

    758: static PetscErrorCode TSIRKGetType_IRK(TS ts, TSIRKType *irktype)
    759: {
    760:   TS_IRK *irk = (TS_IRK *)ts->data;

    762:   PetscFunctionBegin;
    763:   *irktype = irk->method_name;
    764:   PetscFunctionReturn(PETSC_SUCCESS);
    765: }

    767: static PetscErrorCode TSIRKSetType_IRK(TS ts, TSIRKType irktype)
    768: {
    769:   TS_IRK *irk = (TS_IRK *)ts->data;
    770:   PetscErrorCode (*irkcreate)(TS);

    772:   PetscFunctionBegin;
    773:   if (irk->method_name) {
    774:     PetscCall(PetscFree(irk->method_name));
    775:     PetscCall(TSIRKTableauReset(ts));
    776:   }
    777:   PetscCall(PetscFunctionListFind(TSIRKList, irktype, &irkcreate));
    778:   PetscCheck(irkcreate, PetscObjectComm((PetscObject)ts), PETSC_ERR_ARG_UNKNOWN_TYPE, "Unknown TSIRK type \"%s\" given", irktype);
    779:   PetscCall((*irkcreate)(ts));
    780:   PetscCall(PetscStrallocpy(irktype, &irk->method_name));
    781:   PetscFunctionReturn(PETSC_SUCCESS);
    782: }

    784: static PetscErrorCode TSIRKSetNumStages_IRK(TS ts, PetscInt nstages)
    785: {
    786:   TS_IRK *irk = (TS_IRK *)ts->data;

    788:   PetscFunctionBegin;
    789:   PetscCheck(nstages > 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "input argument, %" PetscInt_FMT ", out of range", nstages);
    790:   irk->nstages = nstages;
    791:   PetscFunctionReturn(PETSC_SUCCESS);
    792: }

    794: static PetscErrorCode TSIRKGetNumStages_IRK(TS ts, PetscInt *nstages)
    795: {
    796:   TS_IRK *irk = (TS_IRK *)ts->data;

    798:   PetscFunctionBegin;
    799:   PetscAssertPointer(nstages, 2);
    800:   *nstages = irk->nstages;
    801:   PetscFunctionReturn(PETSC_SUCCESS);
    802: }

    804: static PetscErrorCode TSDestroy_IRK(TS ts)
    805: {
    806:   PetscFunctionBegin;
    807:   PetscCall(TSReset_IRK(ts));
    808:   if (ts->dm) {
    809:     PetscCall(DMCoarsenHookRemove(ts->dm, DMCoarsenHook_TSIRK, DMRestrictHook_TSIRK, ts));
    810:     PetscCall(DMSubDomainHookRemove(ts->dm, DMSubDomainHook_TSIRK, DMSubDomainRestrictHook_TSIRK, ts));
    811:   }
    812:   PetscCall(PetscFree(ts->data));
    813:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSIRKSetType_C", NULL));
    814:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSIRKGetType_C", NULL));
    815:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSIRKSetNumStages_C", NULL));
    816:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSIRKGetNumStages_C", NULL));
    817:   PetscFunctionReturn(PETSC_SUCCESS);
    818: }

    820: /*MC
    821:       TSIRK - ODE and DAE solver using Implicit Runge-Kutta schemes

    823:   Level: beginner

    825:   Notes:
    826:   `TSIRK` uses the sparse Kronecker product matrix implementation of `MATKAIJ` to achieve good arithmetic intensity.

    828:   Gauss-Legrendre methods are currently supported. These are A-stable symplectic methods with an arbitrary number of stages. The order of accuracy is 2s
    829:   when using s stages. The default method uses three stages and thus has an order of six. The number of stages (thus order) can be set with
    830:   -ts_irk_nstages or `TSIRKSetNumStages()`.

    832: .seealso: [](ch_ts), `TSCreate()`, `TS`, `TSSetType()`, `TSIRKSetType()`, `TSIRKGetType()`, `TSIRKGAUSS`, `TSIRKRegister()`, `TSIRKSetNumStages()`, `TSType`
    833: M*/
    834: PETSC_EXTERN PetscErrorCode TSCreate_IRK(TS ts)
    835: {
    836:   TS_IRK *irk;

    838:   PetscFunctionBegin;
    839:   PetscCall(TSIRKInitializePackage());

    841:   ts->ops->reset          = TSReset_IRK;
    842:   ts->ops->destroy        = TSDestroy_IRK;
    843:   ts->ops->view           = TSView_IRK;
    844:   ts->ops->load           = TSLoad_IRK;
    845:   ts->ops->setup          = TSSetUp_IRK;
    846:   ts->ops->step           = TSStep_IRK;
    847:   ts->ops->interpolate    = TSInterpolate_IRK;
    848:   ts->ops->evaluatestep   = TSEvaluateStep_IRK;
    849:   ts->ops->rollback       = TSRollBack_IRK;
    850:   ts->ops->setfromoptions = TSSetFromOptions_IRK;
    851:   ts->ops->snesfunction   = SNESTSFormFunction_IRK;
    852:   ts->ops->snesjacobian   = SNESTSFormJacobian_IRK;

    854:   ts->usessnes = PETSC_TRUE;

    856:   PetscCall(PetscNew(&irk));
    857:   ts->data = (void *)irk;

    859:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSIRKSetType_C", TSIRKSetType_IRK));
    860:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSIRKGetType_C", TSIRKGetType_IRK));
    861:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSIRKSetNumStages_C", TSIRKSetNumStages_IRK));
    862:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSIRKGetNumStages_C", TSIRKGetNumStages_IRK));
    863:   /* 3-stage IRK_Gauss is the default */
    864:   PetscCall(PetscNew(&irk->tableau));
    865:   irk->nstages = 3;
    866:   PetscCall(TSIRKSetType(ts, TSIRKDefault));
    867:   PetscFunctionReturn(PETSC_SUCCESS);
    868: }