Actual source code: glle.c

petsc-master 2020-04-03
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  2:  #include <../src/ts/impls/implicit/glle/glle.h>
  3:  #include <petscdm.h>
  4:  #include <petscblaslapack.h>

  6: static const char        *TSGLLEErrorDirections[] = {"FORWARD","BACKWARD","TSGLLEErrorDirection","TSGLLEERROR_",0};
  7: static PetscFunctionList TSGLLEList;
  8: static PetscFunctionList TSGLLEAcceptList;
  9: static PetscBool         TSGLLEPackageInitialized;
 10: static PetscBool         TSGLLERegisterAllCalled;

 12: /* This function is pure */
 13: static PetscScalar Factorial(PetscInt n)
 14: {
 15:   PetscInt i;
 16:   if (n < 12) {                 /* Can compute with 32-bit integers */
 17:     PetscInt f = 1;
 18:     for (i=2; i<=n; i++) f *= i;
 19:     return (PetscScalar)f;
 20:   } else {
 21:     PetscScalar f = 1.;
 22:     for (i=2; i<=n; i++) f *= (PetscScalar)i;
 23:     return f;
 24:   }
 25: }

 27: /* This function is pure */
 28: static PetscScalar CPowF(PetscScalar c,PetscInt p)
 29: {
 30:   return PetscPowRealInt(PetscRealPart(c),p)/Factorial(p);
 31: }

 33: static PetscErrorCode TSGLLEGetVecs(TS ts,DM dm,Vec *Z,Vec *Ydotstage)
 34: {
 35:   TS_GLLE        *gl = (TS_GLLE*)ts->data;

 39:   if (Z) {
 40:     if (dm && dm != ts->dm) {
 41:       DMGetNamedGlobalVector(dm,"TSGLLE_Z",Z);
 42:     } else *Z = gl->Z;
 43:   }
 44:   if (Ydotstage) {
 45:     if (dm && dm != ts->dm) {
 46:       DMGetNamedGlobalVector(dm,"TSGLLE_Ydot",Ydotstage);
 47:     } else *Ydotstage = gl->Ydot[gl->stage];
 48:   }
 49:   return(0);
 50: }


 53: static PetscErrorCode TSGLLERestoreVecs(TS ts,DM dm,Vec *Z,Vec *Ydotstage)
 54: {

 58:   if (Z) {
 59:     if (dm && dm != ts->dm) {
 60:       DMRestoreNamedGlobalVector(dm,"TSGLLE_Z",Z);
 61:     }
 62:   }
 63:   if (Ydotstage) {

 65:     if (dm && dm != ts->dm) {
 66:       DMRestoreNamedGlobalVector(dm,"TSGLLE_Ydot",Ydotstage);
 67:     }
 68:   }
 69:   return(0);
 70: }

 72: static PetscErrorCode DMCoarsenHook_TSGLLE(DM fine,DM coarse,void *ctx)
 73: {
 75:   return(0);
 76: }

 78: static PetscErrorCode DMRestrictHook_TSGLLE(DM fine,Mat restrct,Vec rscale,Mat inject,DM coarse,void *ctx)
 79: {
 80:   TS             ts = (TS)ctx;
 82:   Vec            Ydot,Ydot_c;

 85:   TSGLLEGetVecs(ts,fine,NULL,&Ydot);
 86:   TSGLLEGetVecs(ts,coarse,NULL,&Ydot_c);
 87:   MatRestrict(restrct,Ydot,Ydot_c);
 88:   VecPointwiseMult(Ydot_c,rscale,Ydot_c);
 89:   TSGLLERestoreVecs(ts,fine,NULL,&Ydot);
 90:   TSGLLERestoreVecs(ts,coarse,NULL,&Ydot_c);
 91:   return(0);
 92: }

 94: static PetscErrorCode DMSubDomainHook_TSGLLE(DM dm,DM subdm,void *ctx)
 95: {
 97:   return(0);
 98: }

100: static PetscErrorCode DMSubDomainRestrictHook_TSGLLE(DM dm,VecScatter gscat, VecScatter lscat,DM subdm,void *ctx)
101: {
102:   TS             ts = (TS)ctx;
104:   Vec            Ydot,Ydot_s;

107:   TSGLLEGetVecs(ts,dm,NULL,&Ydot);
108:   TSGLLEGetVecs(ts,subdm,NULL,&Ydot_s);

110:   VecScatterBegin(gscat,Ydot,Ydot_s,INSERT_VALUES,SCATTER_FORWARD);
111:   VecScatterEnd(gscat,Ydot,Ydot_s,INSERT_VALUES,SCATTER_FORWARD);

113:   TSGLLERestoreVecs(ts,dm,NULL,&Ydot);
114:   TSGLLERestoreVecs(ts,subdm,NULL,&Ydot_s);
115:   return(0);
116: }

118: static PetscErrorCode TSGLLESchemeCreate(PetscInt p,PetscInt q,PetscInt r,PetscInt s,const PetscScalar *c,
119:                                        const PetscScalar *a,const PetscScalar *b,const PetscScalar *u,const PetscScalar *v,TSGLLEScheme *inscheme)
120: {
121:   TSGLLEScheme     scheme;
122:   PetscInt       j;

126:   if (p < 1) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Scheme order must be positive");
127:   if (r < 1) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"At least one item must be carried between steps");
128:   if (s < 1) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"At least one stage is required");
130:   *inscheme = 0;
131:   PetscNew(&scheme);
132:   scheme->p = p;
133:   scheme->q = q;
134:   scheme->r = r;
135:   scheme->s = s;

137:   PetscMalloc5(s,&scheme->c,s*s,&scheme->a,r*s,&scheme->b,r*s,&scheme->u,r*r,&scheme->v);
138:   PetscArraycpy(scheme->c,c,s);
139:   for (j=0; j<s*s; j++) scheme->a[j] = (PetscAbsScalar(a[j]) < 1e-12) ? 0 : a[j];
140:   for (j=0; j<r*s; j++) scheme->b[j] = (PetscAbsScalar(b[j]) < 1e-12) ? 0 : b[j];
141:   for (j=0; j<s*r; j++) scheme->u[j] = (PetscAbsScalar(u[j]) < 1e-12) ? 0 : u[j];
142:   for (j=0; j<r*r; j++) scheme->v[j] = (PetscAbsScalar(v[j]) < 1e-12) ? 0 : v[j];

144:   PetscMalloc6(r,&scheme->alpha,r,&scheme->beta,r,&scheme->gamma,3*s,&scheme->phi,3*r,&scheme->psi,r,&scheme->stage_error);
145:   {
146:     PetscInt     i,j,k,ss=s+2;
147:     PetscBLASInt m,n,one=1,*ipiv,lwork=4*((s+3)*3+3),info,ldb;
148:     PetscReal    rcond,*sing,*workreal;
149:     PetscScalar  *ImV,*H,*bmat,*workscalar,*c=scheme->c,*a=scheme->a,*b=scheme->b,*u=scheme->u,*v=scheme->v;
150:     PetscBLASInt rank;
151:     PetscMalloc7(PetscSqr(r),&ImV,3*s,&H,3*ss,&bmat,lwork,&workscalar,5*(3+r),&workreal,r+s,&sing,r+s,&ipiv);

153:     /* column-major input */
154:     for (i=0; i<r-1; i++) {
155:       for (j=0; j<r-1; j++) ImV[i+j*r] = 1.0*(i==j) - v[(i+1)*r+j+1];
156:     }
157:     /* Build right hand side for alpha (tp - glm.B(2:end,:)*(glm.c.^(p)./factorial(p))) */
158:     for (i=1; i<r; i++) {
159:       scheme->alpha[i] = 1./Factorial(p+1-i);
160:       for (j=0; j<s; j++) scheme->alpha[i] -= b[i*s+j]*CPowF(c[j],p);
161:     }
162:     PetscBLASIntCast(r-1,&m);
163:     PetscBLASIntCast(r,&n);
164:     PetscStackCallBLAS("LAPACKgesv",LAPACKgesv_(&m,&one,ImV,&n,ipiv,scheme->alpha+1,&n,&info));
165:     if (info < 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"Bad argument to GESV");
166:     if (info > 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_MAT_LU_ZRPVT,"Bad LU factorization");

168:     /* Build right hand side for beta (tp1 - glm.B(2:end,:)*(glm.c.^(p+1)./factorial(p+1)) - e.alpha) */
169:     for (i=1; i<r; i++) {
170:       scheme->beta[i] = 1./Factorial(p+2-i) - scheme->alpha[i];
171:       for (j=0; j<s; j++) scheme->beta[i] -= b[i*s+j]*CPowF(c[j],p+1);
172:     }
173:     PetscStackCallBLAS("LAPACKgetrs",LAPACKgetrs_("No transpose",&m,&one,ImV,&n,ipiv,scheme->beta+1,&n,&info));
174:     if (info < 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"Bad argument to GETRS");
175:     if (info > 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"Should not happen");

177:     /* Build stage_error vector
178:            xi = glm.c.^(p+1)/factorial(p+1) - glm.A*glm.c.^p/factorial(p) + glm.U(:,2:end)*e.alpha;
179:     */
180:     for (i=0; i<s; i++) {
181:       scheme->stage_error[i] = CPowF(c[i],p+1);
182:       for (j=0; j<s; j++) scheme->stage_error[i] -= a[i*s+j]*CPowF(c[j],p);
183:       for (j=1; j<r; j++) scheme->stage_error[i] += u[i*r+j]*scheme->alpha[j];
184:     }

186:     /* alpha[0] (epsilon in B,J,W 2007)
187:            epsilon = 1/factorial(p+1) - B(1,:)*c.^p/factorial(p) + V(1,2:end)*e.alpha;
188:     */
189:     scheme->alpha[0] = 1./Factorial(p+1);
190:     for (j=0; j<s; j++) scheme->alpha[0] -= b[0*s+j]*CPowF(c[j],p);
191:     for (j=1; j<r; j++) scheme->alpha[0] += v[0*r+j]*scheme->alpha[j];

193:     /* right hand side for gamma (glm.B(2:end,:)*e.xi - e.epsilon*eye(s-1,1)) */
194:     for (i=1; i<r; i++) {
195:       scheme->gamma[i] = (i==1 ? -1. : 0)*scheme->alpha[0];
196:       for (j=0; j<s; j++) scheme->gamma[i] += b[i*s+j]*scheme->stage_error[j];
197:     }
198:     PetscStackCallBLAS("LAPACKgetrs",LAPACKgetrs_("No transpose",&m,&one,ImV,&n,ipiv,scheme->gamma+1,&n,&info));
199:     if (info < 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"Bad argument to GETRS");
200:     if (info > 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"Should not happen");

202:     /* beta[0] (rho in B,J,W 2007)
203:         e.rho = 1/factorial(p+2) - glm.B(1,:)*glm.c.^(p+1)/factorial(p+1) ...
204:             + glm.V(1,2:end)*e.beta;% - e.epsilon;
205:     % Note: The paper (B,J,W 2007) includes the last term in their definition
206:     * */
207:     scheme->beta[0] = 1./Factorial(p+2);
208:     for (j=0; j<s; j++) scheme->beta[0] -= b[0*s+j]*CPowF(c[j],p+1);
209:     for (j=1; j<r; j++) scheme->beta[0] += v[0*r+j]*scheme->beta[j];

211:     /* gamma[0] (sigma in B,J,W 2007)
212:     *   e.sigma = glm.B(1,:)*e.xi + glm.V(1,2:end)*e.gamma;
213:     * */
214:     scheme->gamma[0] = 0.0;
215:     for (j=0; j<s; j++) scheme->gamma[0] += b[0*s+j]*scheme->stage_error[j];
216:     for (j=1; j<r; j++) scheme->gamma[0] += v[0*s+j]*scheme->gamma[j];

218:     /* Assemble H
219:     *    % Determine the error estimators phi
220:        H = [[cpow(glm.c,p) + C*e.alpha] [cpow(glm.c,p+1) + C*e.beta] ...
221:                [e.xi - C*(e.gamma + 0*e.epsilon*eye(s-1,1))]]';
222:     % Paper has formula above without the 0, but that term must be left
223:     % out to satisfy the conditions they propose and to make the
224:     % example schemes work
225:     e.H = H;
226:     e.phi = (H \ [1 0 0;1 1 0;0 0 -1])';
227:     e.psi = -e.phi*C;
228:     * */
229:     for (j=0; j<s; j++) {
230:       H[0+j*3] = CPowF(c[j],p);
231:       H[1+j*3] = CPowF(c[j],p+1);
232:       H[2+j*3] = scheme->stage_error[j];
233:       for (k=1; k<r; k++) {
234:         H[0+j*3] += CPowF(c[j],k-1)*scheme->alpha[k];
235:         H[1+j*3] += CPowF(c[j],k-1)*scheme->beta[k];
236:         H[2+j*3] -= CPowF(c[j],k-1)*scheme->gamma[k];
237:       }
238:     }
239:     bmat[0+0*ss] = 1.;  bmat[0+1*ss] = 0.;  bmat[0+2*ss] = 0.;
240:     bmat[1+0*ss] = 1.;  bmat[1+1*ss] = 1.;  bmat[1+2*ss] = 0.;
241:     bmat[2+0*ss] = 0.;  bmat[2+1*ss] = 0.;  bmat[2+2*ss] = -1.;
242:     m     = 3;
243:     PetscBLASIntCast(s,&n);
244:     PetscBLASIntCast(ss,&ldb);
245:     rcond = 1e-12;
246: #if defined(PETSC_USE_COMPLEX)
247:     /* ZGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, RWORK, INFO) */
248:     PetscStackCallBLAS("LAPACKgelss",LAPACKgelss_(&m,&n,&m,H,&m,bmat,&ldb,sing,&rcond,&rank,workscalar,&lwork,workreal,&info));
249: #else
250:     /* DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, INFO) */
251:     PetscStackCallBLAS("LAPACKgelss",LAPACKgelss_(&m,&n,&m,H,&m,bmat,&ldb,sing,&rcond,&rank,workscalar,&lwork,&info));
252: #endif
253:     if (info < 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"Bad argument to GELSS");
254:     if (info > 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"SVD failed to converge");

256:     for (j=0; j<3; j++) {
257:       for (k=0; k<s; k++) scheme->phi[k+j*s] = bmat[k+j*ss];
258:     }

260:     /* the other part of the error estimator, psi in B,J,W 2007 */
261:     scheme->psi[0*r+0] = 0.;
262:     scheme->psi[1*r+0] = 0.;
263:     scheme->psi[2*r+0] = 0.;
264:     for (j=1; j<r; j++) {
265:       scheme->psi[0*r+j] = 0.;
266:       scheme->psi[1*r+j] = 0.;
267:       scheme->psi[2*r+j] = 0.;
268:       for (k=0; k<s; k++) {
269:         scheme->psi[0*r+j] -= CPowF(c[k],j-1)*scheme->phi[0*s+k];
270:         scheme->psi[1*r+j] -= CPowF(c[k],j-1)*scheme->phi[1*s+k];
271:         scheme->psi[2*r+j] -= CPowF(c[k],j-1)*scheme->phi[2*s+k];
272:       }
273:     }
274:     PetscFree7(ImV,H,bmat,workscalar,workreal,sing,ipiv);
275:   }
276:   /* Check which properties are satisfied */
277:   scheme->stiffly_accurate = PETSC_TRUE;
278:   if (scheme->c[s-1] != 1.) scheme->stiffly_accurate = PETSC_FALSE;
279:   for (j=0; j<s; j++) if (a[(s-1)*s+j] != b[j]) scheme->stiffly_accurate = PETSC_FALSE;
280:   for (j=0; j<r; j++) if (u[(s-1)*r+j] != v[j]) scheme->stiffly_accurate = PETSC_FALSE;
281:   scheme->fsal = scheme->stiffly_accurate; /* FSAL is stronger */
282:   for (j=0; j<s-1; j++) if (r>1 && b[1*s+j] != 0.) scheme->fsal = PETSC_FALSE;
283:   if (b[1*s+r-1] != 1.) scheme->fsal = PETSC_FALSE;
284:   for (j=0; j<r; j++) if (r>1 && v[1*r+j] != 0.) scheme->fsal = PETSC_FALSE;

286:   *inscheme = scheme;
287:   return(0);
288: }

290: static PetscErrorCode TSGLLESchemeDestroy(TSGLLEScheme sc)
291: {

295:   PetscFree5(sc->c,sc->a,sc->b,sc->u,sc->v);
296:   PetscFree6(sc->alpha,sc->beta,sc->gamma,sc->phi,sc->psi,sc->stage_error);
297:   PetscFree(sc);
298:   return(0);
299: }

301: static PetscErrorCode TSGLLEDestroy_Default(TS_GLLE *gl)
302: {
304:   PetscInt       i;

307:   for (i=0; i<gl->nschemes; i++) {
308:     if (gl->schemes[i]) {TSGLLESchemeDestroy(gl->schemes[i]);}
309:   }
310:   PetscFree(gl->schemes);
311:   gl->nschemes = 0;
312:   PetscMemzero(gl->type_name,sizeof(gl->type_name));
313:   return(0);
314: }

316: static PetscErrorCode TSGLLEViewTable_Private(PetscViewer viewer,PetscInt m,PetscInt n,const PetscScalar a[],const char name[])
317: {
319:   PetscBool      iascii;
320:   PetscInt       i,j;

323:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
324:   if (iascii) {
325:     PetscViewerASCIIPrintf(viewer,"%30s = [",name);
326:     for (i=0; i<m; i++) {
327:       if (i) {PetscViewerASCIIPrintf(viewer,"%30s   [","");}
328:       PetscViewerASCIIUseTabs(viewer,PETSC_FALSE);
329:       for (j=0; j<n; j++) {
330:         PetscViewerASCIIPrintf(viewer," %12.8g",PetscRealPart(a[i*n+j]));
331:       }
332:       PetscViewerASCIIPrintf(viewer,"]\n");
333:       PetscViewerASCIIUseTabs(viewer,PETSC_TRUE);
334:     }
335:   }
336:   return(0);
337: }


340: static PetscErrorCode TSGLLESchemeView(TSGLLEScheme sc,PetscBool view_details,PetscViewer viewer)
341: {
343:   PetscBool      iascii;

346:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
347:   if (iascii) {
348:     PetscViewerASCIIPrintf(viewer,"GL scheme p,q,r,s = %d,%d,%d,%d\n",sc->p,sc->q,sc->r,sc->s);
349:     PetscViewerASCIIPushTab(viewer);
350:     PetscViewerASCIIPrintf(viewer,"Stiffly accurate: %s,  FSAL: %s\n",sc->stiffly_accurate ? "yes" : "no",sc->fsal ? "yes" : "no");
351:     PetscViewerASCIIPrintf(viewer,"Leading error constants: %10.3e  %10.3e  %10.3e\n",
352:                                   PetscRealPart(sc->alpha[0]),PetscRealPart(sc->beta[0]),PetscRealPart(sc->gamma[0]));
353:     TSGLLEViewTable_Private(viewer,1,sc->s,sc->c,"Abscissas c");
354:     if (view_details) {
355:       TSGLLEViewTable_Private(viewer,sc->s,sc->s,sc->a,"A");
356:       TSGLLEViewTable_Private(viewer,sc->r,sc->s,sc->b,"B");
357:       TSGLLEViewTable_Private(viewer,sc->s,sc->r,sc->u,"U");
358:       TSGLLEViewTable_Private(viewer,sc->r,sc->r,sc->v,"V");

360:       TSGLLEViewTable_Private(viewer,3,sc->s,sc->phi,"Error estimate phi");
361:       TSGLLEViewTable_Private(viewer,3,sc->r,sc->psi,"Error estimate psi");
362:       TSGLLEViewTable_Private(viewer,1,sc->r,sc->alpha,"Modify alpha");
363:       TSGLLEViewTable_Private(viewer,1,sc->r,sc->beta,"Modify beta");
364:       TSGLLEViewTable_Private(viewer,1,sc->r,sc->gamma,"Modify gamma");
365:       TSGLLEViewTable_Private(viewer,1,sc->s,sc->stage_error,"Stage error xi");
366:     }
367:     PetscViewerASCIIPopTab(viewer);
368:   } else SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_SUP,"Viewer type %s not supported",((PetscObject)viewer)->type_name);
369:   return(0);
370: }

372: static PetscErrorCode TSGLLEEstimateHigherMoments_Default(TSGLLEScheme sc,PetscReal h,Vec Ydot[],Vec Xold[],Vec hm[])
373: {
375:   PetscInt       i;

378:   if (sc->r > 64 || sc->s > 64) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Ridiculous number of stages or items passed between stages");
379:   /* build error vectors*/
380:   for (i=0; i<3; i++) {
381:     PetscScalar phih[64];
382:     PetscInt    j;
383:     for (j=0; j<sc->s; j++) phih[j] = sc->phi[i*sc->s+j]*h;
384:     VecZeroEntries(hm[i]);
385:     VecMAXPY(hm[i],sc->s,phih,Ydot);
386:     VecMAXPY(hm[i],sc->r,&sc->psi[i*sc->r],Xold);
387:   }
388:   return(0);
389: }

391: static PetscErrorCode TSGLLECompleteStep_Rescale(TSGLLEScheme sc,PetscReal h,TSGLLEScheme next_sc,PetscReal next_h,Vec Ydot[],Vec Xold[],Vec X[])
392: {
394:   PetscScalar    brow[32],vrow[32];
395:   PetscInt       i,j,r,s;

398:   /* Build the new solution from (X,Ydot) */
399:   r = sc->r;
400:   s = sc->s;
401:   for (i=0; i<r; i++) {
402:     VecZeroEntries(X[i]);
403:     for (j=0; j<s; j++) brow[j] = h*sc->b[i*s+j];
404:     VecMAXPY(X[i],s,brow,Ydot);
405:     for (j=0; j<r; j++) vrow[j] = sc->v[i*r+j];
406:     VecMAXPY(X[i],r,vrow,Xold);
407:   }
408:   return(0);
409: }

411: static PetscErrorCode TSGLLECompleteStep_RescaleAndModify(TSGLLEScheme sc,PetscReal h,TSGLLEScheme next_sc,PetscReal next_h,Vec Ydot[],Vec Xold[],Vec X[])
412: {
414:   PetscScalar    brow[32],vrow[32];
415:   PetscReal      ratio;
416:   PetscInt       i,j,p,r,s;

419:   /* Build the new solution from (X,Ydot) */
420:   p     = sc->p;
421:   r     = sc->r;
422:   s     = sc->s;
423:   ratio = next_h/h;
424:   for (i=0; i<r; i++) {
425:     VecZeroEntries(X[i]);
426:     for (j=0; j<s; j++) {
427:       brow[j] = h*(PetscPowRealInt(ratio,i)*sc->b[i*s+j]
428:                    + (PetscPowRealInt(ratio,i) - PetscPowRealInt(ratio,p+1))*(+ sc->alpha[i]*sc->phi[0*s+j])
429:                    + (PetscPowRealInt(ratio,i) - PetscPowRealInt(ratio,p+2))*(+ sc->beta [i]*sc->phi[1*s+j]
430:                                                                               + sc->gamma[i]*sc->phi[2*s+j]));
431:     }
432:     VecMAXPY(X[i],s,brow,Ydot);
433:     for (j=0; j<r; j++) {
434:       vrow[j] = (PetscPowRealInt(ratio,i)*sc->v[i*r+j]
435:                  + (PetscPowRealInt(ratio,i) - PetscPowRealInt(ratio,p+1))*(+ sc->alpha[i]*sc->psi[0*r+j])
436:                  + (PetscPowRealInt(ratio,i) - PetscPowRealInt(ratio,p+2))*(+ sc->beta [i]*sc->psi[1*r+j]
437:                                                                             + sc->gamma[i]*sc->psi[2*r+j]));
438:     }
439:     VecMAXPY(X[i],r,vrow,Xold);
440:   }
441:   if (r < next_sc->r) {
442:     if (r+1 != next_sc->r) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Cannot accommodate jump in r greater than 1");
443:     VecZeroEntries(X[r]);
444:     for (j=0; j<s; j++) brow[j] = h*PetscPowRealInt(ratio,p+1)*sc->phi[0*s+j];
445:     VecMAXPY(X[r],s,brow,Ydot);
446:     for (j=0; j<r; j++) vrow[j] = PetscPowRealInt(ratio,p+1)*sc->psi[0*r+j];
447:     VecMAXPY(X[r],r,vrow,Xold);
448:   }
449:   return(0);
450: }

452: static PetscErrorCode TSGLLECreate_IRKS(TS ts)
453: {
454:   TS_GLLE        *gl = (TS_GLLE*)ts->data;

458:   gl->Destroy               = TSGLLEDestroy_Default;
459:   gl->EstimateHigherMoments = TSGLLEEstimateHigherMoments_Default;
460:   gl->CompleteStep          = TSGLLECompleteStep_RescaleAndModify;
461:   PetscMalloc1(10,&gl->schemes);
462:   gl->nschemes = 0;

464:   {
465:     /* p=1,q=1, r=s=2, A- and L-stable with error estimates of order 2 and 3
466:     * Listed in Butcher & Podhaisky 2006. On error estimation in general linear methods for stiff ODE.
467:     * irks(0.3,0,[.3,1],[1],1)
468:     * Note: can be made to have classical order (not stage order) 2 by replacing 0.3 with 1-sqrt(1/2)
469:     * but doing so would sacrifice the error estimator.
470:     */
471:     const PetscScalar c[2]    = {3./10., 1.};
472:     const PetscScalar a[2][2] = {{3./10., 0}, {7./10., 3./10.}};
473:     const PetscScalar b[2][2] = {{7./10., 3./10.}, {0,1}};
474:     const PetscScalar u[2][2] = {{1,0},{1,0}};
475:     const PetscScalar v[2][2] = {{1,0},{0,0}};
476:     TSGLLESchemeCreate(1,1,2,2,c,*a,*b,*u,*v,&gl->schemes[gl->nschemes++]);
477:   }

479:   {
480:     /* p=q=2, r=s=3: irks(4/9,0,[1:3]/3,[0.33852],1) */
481:     /* http://www.math.auckland.ac.nz/~hpod/atlas/i2a.html */
482:     const PetscScalar c[3] = {1./3., 2./3., 1}
483:     ,a[3][3] = {{4./9.                ,0                      ,       0},
484:                 {1.03750643704090e+00 ,                  4./9.,       0},
485:                 {7.67024779410304e-01 ,  -3.81140216918943e-01,   4./9.}}
486:     ,b[3][3] = {{0.767024779410304,  -0.381140216918943,   4./9.},
487:                 {0.000000000000000,  0.000000000000000,   1.000000000000000},
488:                 {-2.075048385225385,   0.621728385225383,   1.277197204924873}}
489:     ,u[3][3] = {{1.0000000000000000,  -0.1111111111111109,  -0.0925925925925922},
490:                 {1.0000000000000000,  -0.8152842148186744,  -0.4199095530877056},
491:                 {1.0000000000000000,   0.1696709930641948,   0.0539741070314165}}
492:     ,v[3][3] = {{1.0000000000000000,  0.1696709930641948,   0.0539741070314165},
493:                 {0.000000000000000,   0.000000000000000,   0.000000000000000},
494:                 {0.000000000000000,   0.176122795075129,   0.000000000000000}};
495:     TSGLLESchemeCreate(2,2,3,3,c,*a,*b,*u,*v,&gl->schemes[gl->nschemes++]);
496:   }
497:   {
498:     /* p=q=3, r=s=4: irks(9/40,0,[1:4]/4,[0.3312 1.0050],[0.49541 1;1 0]) */
499:     const PetscScalar c[4] = {0.25,0.5,0.75,1.0}
500:     ,a[4][4] = {{9./40.               ,                      0,                      0,                      0},
501:                 {2.11286958887701e-01 ,    9./40.             ,                      0,                      0},
502:                 {9.46338294287584e-01 ,  -3.42942861246094e-01,   9./40.              ,                      0},
503:                 {0.521490453970721    ,  -0.662474225622980,   0.490476425459734,   9./40.           }}
504:     ,b[4][4] = {{0.521490453970721    ,  -0.662474225622980,   0.490476425459734,   9./40.           },
505:                 {0.000000000000000    ,   0.000000000000000,   0.000000000000000,   1.000000000000000},
506:                 {-0.084677029310348   ,   1.390757514776085,  -1.568157386206001,   2.023192696767826},
507:                 {0.465383797936408    ,   1.478273530625148,  -1.930836081010182,   1.644872111193354}}
508:     ,u[4][4] = {{1.00000000000000000  ,   0.02500000000001035,  -0.02499999999999053,  -0.00442708333332865},
509:                 {1.00000000000000000  ,   0.06371304111232945,  -0.04032173972189845,  -0.01389438413189452},
510:                 {1.00000000000000000  ,  -0.07839543304147778,   0.04738685705116663,   0.02032603595928376},
511:                 {1.00000000000000000  ,   0.42550734619251651,   0.10800718022400080,  -0.01726712647760034}}
512:     ,v[4][4] = {{1.00000000000000000  ,   0.42550734619251651,   0.10800718022400080,  -0.01726712647760034},
513:                 {0.000000000000000    ,   0.000000000000000,   0.000000000000000,   0.000000000000000},
514:                 {0.000000000000000    ,  -1.761115796027561,  -0.521284157173780,   0.258249384305463},
515:                 {0.000000000000000    ,  -1.657693358744728,  -1.052227765232394,   0.521284157173780}};
516:     TSGLLESchemeCreate(3,3,4,4,c,*a,*b,*u,*v,&gl->schemes[gl->nschemes++]);
517:   }
518:   {
519:     /* p=q=4, r=s=5:
520:           irks(3/11,0,[1:5]/5, [0.1715   -0.1238    0.6617],...
521:           [ -0.0812    0.4079    1.0000
522:              1.0000         0         0
523:              0.8270    1.0000         0])
524:     */
525:     const PetscScalar c[5] = {0.2,0.4,0.6,0.8,1.0}
526:     ,a[5][5] = {{2.72727272727352e-01 ,   0.00000000000000e+00,  0.00000000000000e+00 ,  0.00000000000000e+00  ,  0.00000000000000e+00},
527:                 {-1.03980153733431e-01,   2.72727272727405e-01,   0.00000000000000e+00,  0.00000000000000e+00  ,  0.00000000000000e+00},
528:                 {-1.58615400341492e+00,   7.44168951881122e-01,   2.72727272727309e-01,  0.00000000000000e+00  ,  0.00000000000000e+00},
529:                 {-8.73658042865628e-01,   5.37884671894595e-01,  -1.63298538799523e-01,   2.72727272726996e-01 ,  0.00000000000000e+00},
530:                 {2.95489397443992e-01 , -1.18481693910097e+00 , -6.68029812659953e-01 ,  1.00716687860943e+00  , 2.72727272727288e-01}}
531:     ,b[5][5] = {{2.95489397443992e-01 , -1.18481693910097e+00 , -6.68029812659953e-01 ,  1.00716687860943e+00  , 2.72727272727288e-01},
532:                 {0.00000000000000e+00 ,  1.11022302462516e-16 , -2.22044604925031e-16 ,  0.00000000000000e+00  , 1.00000000000000e+00},
533:                 {-4.05882503986005e+00,  -4.00924006567769e+00,  -1.38930610972481e+00,   4.45223930308488e+00 ,  6.32331093108427e-01},
534:                 {8.35690179937017e+00 , -2.26640927349732e+00 ,  6.86647884973826e+00 , -5.22595158025740e+00  , 4.50893068837431e+00},
535:                 {1.27656267027479e+01 ,  2.80882153840821e+00 ,  8.91173096522890e+00 , -1.07936444078906e+01  , 4.82534148988854e+00}}
536:     ,u[5][5] = {{1.00000000000000e+00 , -7.27272727273551e-02 , -3.45454545454419e-02 , -4.12121212119565e-03  ,-2.96969696964014e-04},
537:                 {1.00000000000000e+00 ,  2.31252881006154e-01 , -8.29487834416481e-03 , -9.07191207681020e-03  ,-1.70378403743473e-03},
538:                 {1.00000000000000e+00 ,  1.16925777880663e+00 ,  3.59268562942635e-02 , -4.09013451730615e-02  ,-1.02411119670164e-02},
539:                 {1.00000000000000e+00 ,  1.02634463704356e+00 ,  1.59375044913405e-01 ,  1.89673015035370e-03  ,-4.89987231897569e-03},
540:                 {1.00000000000000e+00 ,  1.27746320298021e+00 ,  2.37186008132728e-01 , -8.28694373940065e-02  ,-5.34396510196430e-02}}
541:     ,v[5][5] = {{1.00000000000000e+00 ,  1.27746320298021e+00 ,  2.37186008132728e-01 , -8.28694373940065e-02  ,-5.34396510196430e-02},
542:                 {0.00000000000000e+00 , -1.77635683940025e-15 , -1.99840144432528e-15 , -9.99200722162641e-16  ,-3.33066907387547e-16},
543:                 {0.00000000000000e+00 ,  4.37280081906924e+00 ,  5.49221645016377e-02 , -8.88913177394943e-02  , 1.12879077989154e-01},
544:                 {0.00000000000000e+00 , -1.22399504837280e+01 , -5.21287338448645e+00 , -8.03952325565291e-01  , 4.60298678047147e-01},
545:                 {0.00000000000000e+00 , -1.85178762883829e+01 , -5.21411849862624e+00 , -1.04283436528809e+00  , 7.49030161063651e-01}};
546:     TSGLLESchemeCreate(4,4,5,5,c,*a,*b,*u,*v,&gl->schemes[gl->nschemes++]);
547:   }
548:   {
549:     /* p=q=5, r=s=6;
550:        irks(1/3,0,[1:6]/6,...
551:           [-0.0489    0.4228   -0.8814    0.9021],...
552:           [-0.3474   -0.6617    0.6294    0.2129
553:             0.0044   -0.4256   -0.1427   -0.8936
554:            -0.8267    0.4821    0.1371   -0.2557
555:            -0.4426   -0.3855   -0.7514    0.3014])
556:     */
557:     const PetscScalar c[6] = {1./6, 2./6, 3./6, 4./6, 5./6, 1.}
558:     ,a[6][6] = {{  3.33333333333940e-01,  0                   ,  0                   ,  0                   ,  0                   ,  0                   },
559:                 { -8.64423857333350e-02,  3.33333333332888e-01,  0                   ,  0                   ,  0                   ,  0                   },
560:                 { -2.16850174258252e+00, -2.23619072028839e+00,  3.33333333335204e-01,  0                   ,  0                   ,  0                   },
561:                 { -4.73160970138997e+00, -3.89265344629268e+00, -2.76318716520933e-01,  3.33333333335759e-01,  0                   ,  0                   },
562:                 { -6.75187540297338e+00, -7.90756533769377e+00,  7.90245051802259e-01, -4.48352364517632e-01,  3.33333333328483e-01,  0                   },
563:                 { -4.26488287921548e+00, -1.19320395589302e+01,  3.38924509887755e+00, -2.23969848002481e+00,  6.62807710124007e-01,  3.33333333335440e-01}}
564:     ,b[6][6] = {{ -4.26488287921548e+00, -1.19320395589302e+01,  3.38924509887755e+00, -2.23969848002481e+00,  6.62807710124007e-01,  3.33333333335440e-01},
565:                 { -8.88178419700125e-16,  4.44089209850063e-16, -1.54737334057131e-15, -8.88178419700125e-16,  0.00000000000000e+00,  1.00000000000001e+00},
566:                 { -2.87780425770651e+01, -1.13520448264971e+01,  2.62002318943161e+01,  2.56943874812797e+01, -3.06702268304488e+01,  6.68067773510103e+00},
567:                 {  5.47971245256474e+01,  6.80366875868284e+01, -6.50952588861999e+01, -8.28643975339097e+01,  8.17416943896414e+01, -1.17819043489036e+01},
568:                 { -2.33332114788869e+02,  6.12942539462634e+01, -4.91850135865944e+01,  1.82716844135480e+02, -1.29788173979395e+02,  3.09968095651099e+01},
569:                 { -1.72049132343751e+02,  8.60194713593999e+00,  7.98154219170200e-01,  1.50371386053218e+02, -1.18515423962066e+02,  2.50898277784663e+01}}
570:     ,u[6][6] = {{  1.00000000000000e+00, -1.66666666666870e-01, -4.16666666664335e-02, -3.85802469124815e-03, -2.25051440302250e-04, -9.64506172339142e-06},
571:                 {  1.00000000000000e+00,  8.64423857327162e-02, -4.11484912671353e-02, -1.11450903217645e-02, -1.47651050487126e-03, -1.34395070766826e-04},
572:                 {  1.00000000000000e+00,  4.57135912953434e+00,  1.06514719719137e+00,  1.33517564218007e-01,  1.11365952968659e-02,  6.12382756769504e-04},
573:                 {  1.00000000000000e+00,  9.23391519753404e+00,  2.22431212392095e+00,  2.91823807741891e-01,  2.52058456411084e-02,  1.22800542949647e-03},
574:                 {  1.00000000000000e+00,  1.48175480533865e+01,  3.73439117461835e+00,  5.14648336541804e-01,  4.76430038853402e-02,  2.56798515502156e-03},
575:                 {  1.00000000000000e+00,  1.50512347758335e+01,  4.10099701165164e+00,  5.66039141003603e-01,  3.91213893800891e-02, -2.99136269067853e-03}}
576:     ,v[6][6] = {{  1.00000000000000e+00,  1.50512347758335e+01,  4.10099701165164e+00,  5.66039141003603e-01,  3.91213893800891e-02, -2.99136269067853e-03},
577:                 {  0.00000000000000e+00, -4.88498130835069e-15, -6.43929354282591e-15, -3.55271367880050e-15, -1.22124532708767e-15, -3.12250225675825e-16},
578:                 {  0.00000000000000e+00,  1.22250171233141e+01, -1.77150760606169e+00,  3.54516769879390e-01,  6.22298845883398e-01,  2.31647447450276e-01},
579:                 {  0.00000000000000e+00, -4.48339457331040e+01, -3.57363126641880e-01,  5.18750173123425e-01,  6.55727990241799e-02,  1.63175368287079e-01},
580:                 {  0.00000000000000e+00,  1.37297394708005e+02, -1.60145272991317e+00, -5.05319555199441e+00,  1.55328940390990e-01,  9.16629423682464e-01},
581:                 {  0.00000000000000e+00,  1.05703241119022e+02, -1.16610260983038e+00, -2.99767252773859e+00, -1.13472315553890e-01,  1.09742849254729e+00}};
582:     TSGLLESchemeCreate(5,5,6,6,c,*a,*b,*u,*v,&gl->schemes[gl->nschemes++]);
583:   }
584:   return(0);
585: }

587: /*@C
588:    TSGLLESetType - sets the class of general linear method to use for time-stepping

590:    Collective on TS

592:    Input Parameters:
593: +  ts - the TS context
594: -  type - a method

596:    Options Database Key:
597: .  -ts_gl_type <type> - sets the method, use -help for a list of available method (e.g. irks)

599:    Notes:
600:    See "petsc/include/petscts.h" for available methods (for instance)
601: .    TSGLLE_IRKS - Diagonally implicit methods with inherent Runge-Kutta stability (for stiff problems)

603:    Normally, it is best to use the TSSetFromOptions() command and
604:    then set the TSGLLE type from the options database rather than by using
605:    this routine.  Using the options database provides the user with
606:    maximum flexibility in evaluating the many different solvers.
607:    The TSGLLESetType() routine is provided for those situations where it
608:    is necessary to set the timestepping solver independently of the
609:    command line or options database.  This might be the case, for example,
610:    when the choice of solver changes during the execution of the
611:    program, and the user's application is taking responsibility for
612:    choosing the appropriate method.

614:    Level: intermediate

616: @*/
617: PetscErrorCode  TSGLLESetType(TS ts,TSGLLEType type)
618: {

624:   PetscTryMethod(ts,"TSGLLESetType_C",(TS,TSGLLEType),(ts,type));
625:   return(0);
626: }

628: /*@C
629:    TSGLLESetAcceptType - sets the acceptance test

631:    Time integrators that need to control error must have the option to reject a time step based on local error
632:    estimates.  This function allows different schemes to be set.

634:    Logically Collective on TS

636:    Input Parameters:
637: +  ts - the TS context
638: -  type - the type

640:    Options Database Key:
641: .  -ts_gl_accept_type <type> - sets the method used to determine whether to accept or reject a step

643:    Level: intermediate

645: .seealso: TS, TSGLLE, TSGLLEAcceptRegister(), TSGLLEAdapt, set type
646: @*/
647: PetscErrorCode  TSGLLESetAcceptType(TS ts,TSGLLEAcceptType type)
648: {

654:   PetscTryMethod(ts,"TSGLLESetAcceptType_C",(TS,TSGLLEAcceptType),(ts,type));
655:   return(0);
656: }

658: /*@C
659:    TSGLLEGetAdapt - gets the TSGLLEAdapt object from the TS

661:    Not Collective

663:    Input Parameter:
664: .  ts - the TS context

666:    Output Parameter:
667: .  adapt - the TSGLLEAdapt context

669:    Notes:
670:    This allows the user set options on the TSGLLEAdapt object.  Usually it is better to do this using the options
671:    database, so this function is rarely needed.

673:    Level: advanced

675: .seealso: TSGLLEAdapt, TSGLLEAdaptRegister()
676: @*/
677: PetscErrorCode  TSGLLEGetAdapt(TS ts,TSGLLEAdapt *adapt)
678: {

684:   PetscUseMethod(ts,"TSGLLEGetAdapt_C",(TS,TSGLLEAdapt*),(ts,adapt));
685:   return(0);
686: }

688: static PetscErrorCode TSGLLEAccept_Always(TS ts,PetscReal tleft,PetscReal h,const PetscReal enorms[],PetscBool  *accept)
689: {
691:   *accept = PETSC_TRUE;
692:   return(0);
693: }

695: static PetscErrorCode TSGLLEUpdateWRMS(TS ts)
696: {
697:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
699:   PetscScalar    *x,*w;
700:   PetscInt       n,i;

703:   VecGetArray(gl->X[0],&x);
704:   VecGetArray(gl->W,&w);
705:   VecGetLocalSize(gl->W,&n);
706:   for (i=0; i<n; i++) w[i] = 1./(gl->wrms_atol + gl->wrms_rtol*PetscAbsScalar(x[i]));
707:   VecRestoreArray(gl->X[0],&x);
708:   VecRestoreArray(gl->W,&w);
709:   return(0);
710: }

712: static PetscErrorCode TSGLLEVecNormWRMS(TS ts,Vec X,PetscReal *nrm)
713: {
714:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
716:   PetscScalar    *x,*w;
717:   PetscReal      sum = 0.0,gsum;
718:   PetscInt       n,N,i;

721:   VecGetArray(X,&x);
722:   VecGetArray(gl->W,&w);
723:   VecGetLocalSize(gl->W,&n);
724:   for (i=0; i<n; i++) sum += PetscAbsScalar(PetscSqr(x[i]*w[i]));
725:   VecRestoreArray(X,&x);
726:   VecRestoreArray(gl->W,&w);
727:   MPIU_Allreduce(&sum,&gsum,1,MPIU_REAL,MPIU_SUM,PetscObjectComm((PetscObject)ts));
728:   VecGetSize(gl->W,&N);
729:   *nrm = PetscSqrtReal(gsum/(1.*N));
730:   return(0);
731: }

733: static PetscErrorCode TSGLLESetType_GLLE(TS ts,TSGLLEType type)
734: {
735:   PetscErrorCode ierr,(*r)(TS);
736:   PetscBool      same;
737:   TS_GLLE        *gl = (TS_GLLE*)ts->data;

740:   if (gl->type_name[0]) {
741:     PetscStrcmp(gl->type_name,type,&same);
742:     if (same) return(0);
743:     (*gl->Destroy)(gl);
744:   }

746:   PetscFunctionListFind(TSGLLEList,type,&r);
747:   if (!r) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"Unknown TSGLLE type \"%s\" given",type);
748:   (*r)(ts);
749:   PetscStrcpy(gl->type_name,type);
750:   return(0);
751: }

753: static PetscErrorCode TSGLLESetAcceptType_GLLE(TS ts,TSGLLEAcceptType type)
754: {
755:   PetscErrorCode       ierr;
756:   TSGLLEAcceptFunction r;
757:   TS_GLLE              *gl = (TS_GLLE*)ts->data;

760:   PetscFunctionListFind(TSGLLEAcceptList,type,&r);
761:   if (!r) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"Unknown TSGLLEAccept type \"%s\" given",type);
762:   gl->Accept = r;
763:   PetscStrncpy(gl->accept_name,type,sizeof(gl->accept_name));
764:   return(0);
765: }

767: static PetscErrorCode TSGLLEGetAdapt_GLLE(TS ts,TSGLLEAdapt *adapt)
768: {
770:   TS_GLLE        *gl = (TS_GLLE*)ts->data;

773:   if (!gl->adapt) {
774:     TSGLLEAdaptCreate(PetscObjectComm((PetscObject)ts),&gl->adapt);
775:     PetscObjectIncrementTabLevel((PetscObject)gl->adapt,(PetscObject)ts,1);
776:     PetscLogObjectParent((PetscObject)ts,(PetscObject)gl->adapt);
777:   }
778:   *adapt = gl->adapt;
779:   return(0);
780: }

782: static PetscErrorCode TSGLLEChooseNextScheme(TS ts,PetscReal h,const PetscReal hmnorm[],PetscInt *next_scheme,PetscReal *next_h,PetscBool  *finish)
783: {
785:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
786:   PetscInt       i,n,cur_p,cur,next_sc,candidates[64],orders[64];
787:   PetscReal      errors[64],costs[64],tleft;

790:   cur   = -1;
791:   cur_p = gl->schemes[gl->current_scheme]->p;
792:   tleft = ts->max_time - (ts->ptime + ts->time_step);
793:   for (i=0,n=0; i<gl->nschemes; i++) {
794:     TSGLLEScheme sc = gl->schemes[i];
795:     if (sc->p < gl->min_order || gl->max_order < sc->p) continue;
796:     if (sc->p == cur_p - 1)    errors[n] = PetscAbsScalar(sc->alpha[0])*hmnorm[0];
797:     else if (sc->p == cur_p)   errors[n] = PetscAbsScalar(sc->alpha[0])*hmnorm[1];
798:     else if (sc->p == cur_p+1) errors[n] = PetscAbsScalar(sc->alpha[0])*(hmnorm[2]+hmnorm[3]);
799:     else continue;
800:     candidates[n] = i;
801:     orders[n]     = PetscMin(sc->p,sc->q); /* order of global truncation error */
802:     costs[n]      = sc->s;                 /* estimate the cost as the number of stages */
803:     if (i == gl->current_scheme) cur = n;
804:     n++;
805:   }
806:   if (cur < 0 || gl->nschemes <= cur) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Current scheme not found in scheme list");
807:   TSGLLEAdaptChoose(gl->adapt,n,orders,errors,costs,cur,h,tleft,&next_sc,next_h,finish);
808:   *next_scheme = candidates[next_sc];
809:   PetscInfo7(ts,"Adapt chose scheme %d (%d,%d,%d,%d) with step size %6.2e, finish=%d\n",*next_scheme,gl->schemes[*next_scheme]->p,gl->schemes[*next_scheme]->q,gl->schemes[*next_scheme]->r,gl->schemes[*next_scheme]->s,*next_h,*finish);
810:   return(0);
811: }

813: static PetscErrorCode TSGLLEGetMaxSizes(TS ts,PetscInt *max_r,PetscInt *max_s)
814: {
815:   TS_GLLE *gl = (TS_GLLE*)ts->data;

818:   *max_r = gl->schemes[gl->nschemes-1]->r;
819:   *max_s = gl->schemes[gl->nschemes-1]->s;
820:   return(0);
821: }

823: static PetscErrorCode TSSolve_GLLE(TS ts)
824: {
825:   TS_GLLE             *gl = (TS_GLLE*)ts->data;
826:   PetscInt            i,k,its,lits,max_r,max_s;
827:   PetscBool           final_step,finish;
828:   SNESConvergedReason snesreason;
829:   PetscErrorCode      ierr;

832:   TSMonitor(ts,ts->steps,ts->ptime,ts->vec_sol);

834:   TSGLLEGetMaxSizes(ts,&max_r,&max_s);
835:   VecCopy(ts->vec_sol,gl->X[0]);
836:   for (i=1; i<max_r; i++) {
837:     VecZeroEntries(gl->X[i]);
838:   }
839:   TSGLLEUpdateWRMS(ts);

841:   if (0) {
842:     /* Find consistent initial data for DAE */
843:     gl->stage_time = ts->ptime + ts->time_step;
844:     gl->scoeff = 1.;
845:     gl->stage  = 0;

847:     VecCopy(ts->vec_sol,gl->Z);
848:     VecCopy(ts->vec_sol,gl->Y);
849:     SNESSolve(ts->snes,NULL,gl->Y);
850:     SNESGetIterationNumber(ts->snes,&its);
851:     SNESGetLinearSolveIterations(ts->snes,&lits);
852:     SNESGetConvergedReason(ts->snes,&snesreason);

854:     ts->snes_its += its; ts->ksp_its += lits;
855:     if (snesreason < 0 && ts->max_snes_failures > 0 && ++ts->num_snes_failures >= ts->max_snes_failures) {
856:       ts->reason = TS_DIVERGED_NONLINEAR_SOLVE;
857:       PetscInfo2(ts,"Step=%D, nonlinear solve solve failures %D greater than current TS allowed, stopping solve\n",ts->steps,ts->num_snes_failures);
858:       return(0);
859:     }
860:   }

862:   if (gl->current_scheme < 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ORDER,"A starting scheme has not been provided");

864:   for (k=0,final_step=PETSC_FALSE,finish=PETSC_FALSE; k<ts->max_steps && !finish; k++) {
865:     PetscInt          j,r,s,next_scheme = 0,rejections;
866:     PetscReal         h,hmnorm[4],enorm[3],next_h;
867:     PetscBool         accept;
868:     const PetscScalar *c,*a,*u;
869:     Vec               *X,*Ydot,Y;
870:     TSGLLEScheme        scheme = gl->schemes[gl->current_scheme];

872:     r = scheme->r; s = scheme->s;
873:     c = scheme->c;
874:     a = scheme->a; u = scheme->u;
875:     h = ts->time_step;
876:     X = gl->X; Ydot = gl->Ydot; Y = gl->Y;

878:     if (ts->ptime > ts->max_time) break;

880:     /*
881:       We only call PreStep at the start of each STEP, not each STAGE.  This is because it is
882:       possible to fail (have to restart a step) after multiple stages.
883:     */
884:     TSPreStep(ts);

886:     rejections = 0;
887:     while (1) {
888:       for (i=0; i<s; i++) {
889:         PetscScalar shift;
890:         gl->scoeff     = 1./PetscRealPart(a[i*s+i]);
891:         shift          = gl->scoeff/ts->time_step;
892:         gl->stage      = i;
893:         gl->stage_time = ts->ptime + PetscRealPart(c[i])*h;

895:         /*
896:         * Stage equation: Y = h A Y' + U X
897:         * We assume that A is lower-triangular so that we can solve the stages (Y,Y') sequentially
898:         * Build the affine vector z_i = -[1/(h a_ii)](h sum_j a_ij y'_j + sum_j u_ij x_j)
899:         * Then y'_i = z + 1/(h a_ii) y_i
900:         */
901:         VecZeroEntries(gl->Z);
902:         for (j=0; j<r; j++) {
903:           VecAXPY(gl->Z,-shift*u[i*r+j],X[j]);
904:         }
905:         for (j=0; j<i; j++) {
906:           VecAXPY(gl->Z,-shift*h*a[i*s+j],Ydot[j]);
907:         }
908:         /* Note: Z is used within function evaluation, Ydot = Z + shift*Y */

910:         /* Compute an estimate of Y to start Newton iteration */
911:         if (gl->extrapolate) {
912:           if (i==0) {
913:             /* Linear extrapolation on the first stage */
914:             VecWAXPY(Y,c[i]*h,X[1],X[0]);
915:           } else {
916:             /* Linear extrapolation from the last stage */
917:             VecAXPY(Y,(c[i]-c[i-1])*h,Ydot[i-1]);
918:           }
919:         } else if (i==0) {        /* Directly use solution from the last step, otherwise reuse the last stage (do nothing) */
920:           VecCopy(X[0],Y);
921:         }

923:         /* Solve this stage (Ydot[i] is computed during function evaluation) */
924:         SNESSolve(ts->snes,NULL,Y);
925:         SNESGetIterationNumber(ts->snes,&its);
926:         SNESGetLinearSolveIterations(ts->snes,&lits);
927:         SNESGetConvergedReason(ts->snes,&snesreason);
928:         ts->snes_its += its; ts->ksp_its += lits;
929:         if (snesreason < 0 && ts->max_snes_failures > 0 && ++ts->num_snes_failures >= ts->max_snes_failures) {
930:           ts->reason = TS_DIVERGED_NONLINEAR_SOLVE;
931:           PetscInfo2(ts,"Step=%D, nonlinear solve solve failures %D greater than current TS allowed, stopping solve\n",ts->steps,ts->num_snes_failures);
932:           return(0);
933:         }
934:       }

936:       gl->stage_time = ts->ptime + ts->time_step;

938:       (*gl->EstimateHigherMoments)(scheme,h,Ydot,gl->X,gl->himom);
939:       /* hmnorm[i] = h^{p+i}x^{(p+i)} with i=0,1,2; hmnorm[3] = h^{p+2}(dx'/dx) x^{(p+1)} */
940:       for (i=0; i<3; i++) {
941:         TSGLLEVecNormWRMS(ts,gl->himom[i],&hmnorm[i+1]);
942:       }
943:       enorm[0] = PetscRealPart(scheme->alpha[0])*hmnorm[1];
944:       enorm[1] = PetscRealPart(scheme->beta[0]) *hmnorm[2];
945:       enorm[2] = PetscRealPart(scheme->gamma[0])*hmnorm[3];
946:       (*gl->Accept)(ts,ts->max_time-gl->stage_time,h,enorm,&accept);
947:       if (accept) goto accepted;
948:       rejections++;
949:       PetscInfo3(ts,"Step %D (t=%g) not accepted, rejections=%D\n",k,gl->stage_time,rejections);
950:       if (rejections > gl->max_step_rejections) break;
951:       /*
952:         There are lots of reasons why a step might be rejected, including solvers not converging and other factors that
953:         TSGLLEChooseNextScheme does not support.  Additionally, the error estimates may be very screwed up, so I'm not
954:         convinced that it's safe to just compute a new error estimate using the same interface as the current adaptor
955:         (the adaptor interface probably has to change).  Here we make an arbitrary and naive choice.  This assumes that
956:         steps were written in Nordsieck form.  The "correct" method would be to re-complete the previous time step with
957:         the correct "next" step size.  It is unclear to me whether the present ad-hoc method of rescaling X is stable.
958:       */
959:       h *= 0.5;
960:       for (i=1; i<scheme->r; i++) {
961:         VecScale(X[i],PetscPowRealInt(0.5,i));
962:       }
963:     }
964:     SETERRQ3(PETSC_COMM_SELF,PETSC_ERR_CONV_FAILED,"Time step %D (t=%g) not accepted after %D failures\n",k,gl->stage_time,rejections);

966: accepted:
967:     /* This term is not error, but it *would* be the leading term for a lower order method */
968:     TSGLLEVecNormWRMS(ts,gl->X[scheme->r-1],&hmnorm[0]);
969:     /* Correct scaling so that these are equivalent to norms of the Nordsieck vectors */

971:     PetscInfo4(ts,"Last moment norm %10.2e, estimated error norms %10.2e %10.2e %10.2e\n",hmnorm[0],enorm[0],enorm[1],enorm[2]);
972:     if (!final_step) {
973:       TSGLLEChooseNextScheme(ts,h,hmnorm,&next_scheme,&next_h,&final_step);
974:     } else {
975:       /* Dummy values to complete the current step in a consistent manner */
976:       next_scheme = gl->current_scheme;
977:       next_h      = h;
978:       finish      = PETSC_TRUE;
979:     }

981:     X        = gl->Xold;
982:     gl->Xold = gl->X;
983:     gl->X    = X;
984:     (*gl->CompleteStep)(scheme,h,gl->schemes[next_scheme],next_h,Ydot,gl->Xold,gl->X);

986:     TSGLLEUpdateWRMS(ts);

988:     /* Post the solution for the user, we could avoid this copy with a small bit of cleverness */
989:     VecCopy(gl->X[0],ts->vec_sol);
990:     ts->ptime += h;
991:     ts->steps++;

993:     TSPostEvaluate(ts);
994:     TSPostStep(ts);
995:     TSMonitor(ts,ts->steps,ts->ptime,ts->vec_sol);

997:     gl->current_scheme = next_scheme;
998:     ts->time_step      = next_h;
999:   }
1000:   return(0);
1001: }

1003: /*------------------------------------------------------------*/

1005: static PetscErrorCode TSReset_GLLE(TS ts)
1006: {
1007:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
1008:   PetscInt       max_r,max_s;

1012:   if (gl->setupcalled) {
1013:     TSGLLEGetMaxSizes(ts,&max_r,&max_s);
1014:     VecDestroyVecs(max_r,&gl->Xold);
1015:     VecDestroyVecs(max_r,&gl->X);
1016:     VecDestroyVecs(max_s,&gl->Ydot);
1017:     VecDestroyVecs(3,&gl->himom);
1018:     VecDestroy(&gl->W);
1019:     VecDestroy(&gl->Y);
1020:     VecDestroy(&gl->Z);
1021:   }
1022:   gl->setupcalled = PETSC_FALSE;
1023:   return(0);
1024: }

1026: static PetscErrorCode TSDestroy_GLLE(TS ts)
1027: {
1028:   TS_GLLE        *gl = (TS_GLLE*)ts->data;

1032:   TSReset_GLLE(ts);
1033:   if (ts->dm) {
1034:     DMCoarsenHookRemove(ts->dm,DMCoarsenHook_TSGLLE,DMRestrictHook_TSGLLE,ts);
1035:     DMSubDomainHookRemove(ts->dm,DMSubDomainHook_TSGLLE,DMSubDomainRestrictHook_TSGLLE,ts);
1036:   }
1037:   if (gl->adapt) {TSGLLEAdaptDestroy(&gl->adapt);}
1038:   if (gl->Destroy) {(*gl->Destroy)(gl);}
1039:   PetscFree(ts->data);
1040:   PetscObjectComposeFunction((PetscObject)ts,"TSGLLESetType_C",NULL);
1041:   PetscObjectComposeFunction((PetscObject)ts,"TSGLLESetAcceptType_C",NULL);
1042:   PetscObjectComposeFunction((PetscObject)ts,"TSGLLEGetAdapt_C",NULL);
1043:   return(0);
1044: }

1046: /*
1047:     This defines the nonlinear equation that is to be solved with SNES
1048:     g(x) = f(t,x,z+shift*x) = 0
1049: */
1050: static PetscErrorCode SNESTSFormFunction_GLLE(SNES snes,Vec x,Vec f,TS ts)
1051: {
1052:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
1054:   Vec            Z,Ydot;
1055:   DM             dm,dmsave;

1058:   SNESGetDM(snes,&dm);
1059:   TSGLLEGetVecs(ts,dm,&Z,&Ydot);
1060:   VecWAXPY(Ydot,gl->scoeff/ts->time_step,x,Z);
1061:   dmsave = ts->dm;
1062:   ts->dm = dm;
1063:   TSComputeIFunction(ts,gl->stage_time,x,Ydot,f,PETSC_FALSE);
1064:   ts->dm = dmsave;
1065:   TSGLLERestoreVecs(ts,dm,&Z,&Ydot);
1066:   return(0);
1067: }

1069: static PetscErrorCode SNESTSFormJacobian_GLLE(SNES snes,Vec x,Mat A,Mat B,TS ts)
1070: {
1071:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
1073:   Vec            Z,Ydot;
1074:   DM             dm,dmsave;

1077:   SNESGetDM(snes,&dm);
1078:   TSGLLEGetVecs(ts,dm,&Z,&Ydot);
1079:   dmsave = ts->dm;
1080:   ts->dm = dm;
1081:   /* gl->Xdot will have already been computed in SNESTSFormFunction_GLLE */
1082:   TSComputeIJacobian(ts,gl->stage_time,x,gl->Ydot[gl->stage],gl->scoeff/ts->time_step,A,B,PETSC_FALSE);
1083:   ts->dm = dmsave;
1084:   TSGLLERestoreVecs(ts,dm,&Z,&Ydot);
1085:   return(0);
1086: }


1089: static PetscErrorCode TSSetUp_GLLE(TS ts)
1090: {
1091:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
1092:   PetscInt       max_r,max_s;
1094:   DM             dm;

1097:   gl->setupcalled = PETSC_TRUE;
1098:   TSGLLEGetMaxSizes(ts,&max_r,&max_s);
1099:   VecDuplicateVecs(ts->vec_sol,max_r,&gl->X);
1100:   VecDuplicateVecs(ts->vec_sol,max_r,&gl->Xold);
1101:   VecDuplicateVecs(ts->vec_sol,max_s,&gl->Ydot);
1102:   VecDuplicateVecs(ts->vec_sol,3,&gl->himom);
1103:   VecDuplicate(ts->vec_sol,&gl->W);
1104:   VecDuplicate(ts->vec_sol,&gl->Y);
1105:   VecDuplicate(ts->vec_sol,&gl->Z);

1107:   /* Default acceptance tests and adaptivity */
1108:   if (!gl->Accept) {TSGLLESetAcceptType(ts,TSGLLEACCEPT_ALWAYS);}
1109:   if (!gl->adapt)  {TSGLLEGetAdapt(ts,&gl->adapt);}

1111:   if (gl->current_scheme < 0) {
1112:     PetscInt i;
1113:     for (i=0;; i++) {
1114:       if (gl->schemes[i]->p == gl->start_order) break;
1115:       if (i+1 == gl->nschemes) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"No schemes available with requested start order %d",i);
1116:     }
1117:     gl->current_scheme = i;
1118:   }
1119:   TSGetDM(ts,&dm);
1120:   DMCoarsenHookAdd(dm,DMCoarsenHook_TSGLLE,DMRestrictHook_TSGLLE,ts);
1121:   DMSubDomainHookAdd(dm,DMSubDomainHook_TSGLLE,DMSubDomainRestrictHook_TSGLLE,ts);
1122:   return(0);
1123: }
1124: /*------------------------------------------------------------*/

1126: static PetscErrorCode TSSetFromOptions_GLLE(PetscOptionItems *PetscOptionsObject,TS ts)
1127: {
1128:   TS_GLLE        *gl        = (TS_GLLE*)ts->data;
1129:   char           tname[256] = TSGLLE_IRKS,completef[256] = "rescale-and-modify";

1133:   PetscOptionsHead(PetscOptionsObject,"General Linear ODE solver options");
1134:   {
1135:     PetscBool flg;
1136:     PetscOptionsFList("-ts_gl_type","Type of GL method","TSGLLESetType",TSGLLEList,gl->type_name[0] ? gl->type_name : tname,tname,sizeof(tname),&flg);
1137:     if (flg || !gl->type_name[0]) {
1138:       TSGLLESetType(ts,tname);
1139:     }
1140:     PetscOptionsInt("-ts_gl_max_step_rejections","Maximum number of times to attempt a step","None",gl->max_step_rejections,&gl->max_step_rejections,NULL);
1141:     PetscOptionsInt("-ts_gl_max_order","Maximum order to try","TSGLLESetMaxOrder",gl->max_order,&gl->max_order,NULL);
1142:     PetscOptionsInt("-ts_gl_min_order","Minimum order to try","TSGLLESetMinOrder",gl->min_order,&gl->min_order,NULL);
1143:     PetscOptionsInt("-ts_gl_start_order","Initial order to try","TSGLLESetMinOrder",gl->start_order,&gl->start_order,NULL);
1144:     PetscOptionsEnum("-ts_gl_error_direction","Which direction to look when estimating error","TSGLLESetErrorDirection",TSGLLEErrorDirections,(PetscEnum)gl->error_direction,(PetscEnum*)&gl->error_direction,NULL);
1145:     PetscOptionsBool("-ts_gl_extrapolate","Extrapolate stage solution from previous solution (sometimes unstable)","TSGLLESetExtrapolate",gl->extrapolate,&gl->extrapolate,NULL);
1146:     PetscOptionsReal("-ts_gl_atol","Absolute tolerance","TSGLLESetTolerances",gl->wrms_atol,&gl->wrms_atol,NULL);
1147:     PetscOptionsReal("-ts_gl_rtol","Relative tolerance","TSGLLESetTolerances",gl->wrms_rtol,&gl->wrms_rtol,NULL);
1148:     PetscOptionsString("-ts_gl_complete","Method to use for completing the step","none",completef,completef,sizeof(completef),&flg);
1149:     if (flg) {
1150:       PetscBool match1,match2;
1151:       PetscStrcmp(completef,"rescale",&match1);
1152:       PetscStrcmp(completef,"rescale-and-modify",&match2);
1153:       if (match1)      gl->CompleteStep = TSGLLECompleteStep_Rescale;
1154:       else if (match2) gl->CompleteStep = TSGLLECompleteStep_RescaleAndModify;
1155:       else SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"%s",completef);
1156:     }
1157:     {
1158:       char type[256] = TSGLLEACCEPT_ALWAYS;
1159:       PetscOptionsFList("-ts_gl_accept_type","Method to use for determining whether to accept a step","TSGLLESetAcceptType",TSGLLEAcceptList,gl->accept_name[0] ? gl->accept_name : type,type,sizeof(type),&flg);
1160:       if (flg || !gl->accept_name[0]) {
1161:         TSGLLESetAcceptType(ts,type);
1162:       }
1163:     }
1164:     {
1165:       TSGLLEAdapt adapt;
1166:       TSGLLEGetAdapt(ts,&adapt);
1167:       TSGLLEAdaptSetFromOptions(PetscOptionsObject,adapt);
1168:     }
1169:   }
1170:   PetscOptionsTail();
1171:   return(0);
1172: }

1174: static PetscErrorCode TSView_GLLE(TS ts,PetscViewer viewer)
1175: {
1176:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
1177:   PetscInt       i;
1178:   PetscBool      iascii,details;

1182:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
1183:   if (iascii) {
1184:     PetscViewerASCIIPrintf(viewer,"  min order %D, max order %D, current order %D\n",gl->min_order,gl->max_order,gl->schemes[gl->current_scheme]->p);
1185:     PetscViewerASCIIPrintf(viewer,"  Error estimation: %s\n",TSGLLEErrorDirections[gl->error_direction]);
1186:     PetscViewerASCIIPrintf(viewer,"  Extrapolation: %s\n",gl->extrapolate ? "yes" : "no");
1187:     PetscViewerASCIIPrintf(viewer,"  Acceptance test: %s\n",gl->accept_name[0] ? gl->accept_name : "(not yet set)");
1188:     PetscViewerASCIIPushTab(viewer);
1189:     TSGLLEAdaptView(gl->adapt,viewer);
1190:     PetscViewerASCIIPopTab(viewer);
1191:     PetscViewerASCIIPrintf(viewer,"  type: %s\n",gl->type_name[0] ? gl->type_name : "(not yet set)");
1192:     PetscViewerASCIIPrintf(viewer,"Schemes within family (%d):\n",gl->nschemes);
1193:     details = PETSC_FALSE;
1194:     PetscOptionsGetBool(((PetscObject)ts)->options,((PetscObject)ts)->prefix,"-ts_gl_view_detailed",&details,NULL);
1195:     PetscViewerASCIIPushTab(viewer);
1196:     for (i=0; i<gl->nschemes; i++) {
1197:       TSGLLESchemeView(gl->schemes[i],details,viewer);
1198:     }
1199:     if (gl->View) {
1200:       (*gl->View)(gl,viewer);
1201:     }
1202:     PetscViewerASCIIPopTab(viewer);
1203:   }
1204:   return(0);
1205: }

1207: /*@C
1208:    TSGLLERegister -  adds a TSGLLE implementation

1210:    Not Collective

1212:    Input Parameters:
1213: +  name_scheme - name of user-defined general linear scheme
1214: -  routine_create - routine to create method context

1216:    Notes:
1217:    TSGLLERegister() may be called multiple times to add several user-defined families.

1219:    Sample usage:
1220: .vb
1221:    TSGLLERegister("my_scheme",MySchemeCreate);
1222: .ve

1224:    Then, your scheme can be chosen with the procedural interface via
1225: $     TSGLLESetType(ts,"my_scheme")
1226:    or at runtime via the option
1227: $     -ts_gl_type my_scheme

1229:    Level: advanced

1231: .seealso: TSGLLERegisterAll()
1232: @*/
1233: PetscErrorCode  TSGLLERegister(const char sname[],PetscErrorCode (*function)(TS))
1234: {

1238:   TSGLLEInitializePackage();
1239:   PetscFunctionListAdd(&TSGLLEList,sname,function);
1240:   return(0);
1241: }

1243: /*@C
1244:    TSGLLEAcceptRegister -  adds a TSGLLE acceptance scheme

1246:    Not Collective

1248:    Input Parameters:
1249: +  name_scheme - name of user-defined acceptance scheme
1250: -  routine_create - routine to create method context

1252:    Notes:
1253:    TSGLLEAcceptRegister() may be called multiple times to add several user-defined families.

1255:    Sample usage:
1256: .vb
1257:    TSGLLEAcceptRegister("my_scheme",MySchemeCreate);
1258: .ve

1260:    Then, your scheme can be chosen with the procedural interface via
1261: $     TSGLLESetAcceptType(ts,"my_scheme")
1262:    or at runtime via the option
1263: $     -ts_gl_accept_type my_scheme

1265:    Level: advanced

1267: .seealso: TSGLLERegisterAll()
1268: @*/
1269: PetscErrorCode  TSGLLEAcceptRegister(const char sname[],TSGLLEAcceptFunction function)
1270: {

1274:   PetscFunctionListAdd(&TSGLLEAcceptList,sname,function);
1275:   return(0);
1276: }

1278: /*@C
1279:   TSGLLERegisterAll - Registers all of the general linear methods in TSGLLE

1281:   Not Collective

1283:   Level: advanced

1285: .seealso:  TSGLLERegisterDestroy()
1286: @*/
1287: PetscErrorCode  TSGLLERegisterAll(void)
1288: {

1292:   if (TSGLLERegisterAllCalled) return(0);
1293:   TSGLLERegisterAllCalled = PETSC_TRUE;

1295:   TSGLLERegister(TSGLLE_IRKS,              TSGLLECreate_IRKS);
1296:   TSGLLEAcceptRegister(TSGLLEACCEPT_ALWAYS,TSGLLEAccept_Always);
1297:   return(0);
1298: }

1300: /*@C
1301:   TSGLLEInitializePackage - This function initializes everything in the TSGLLE package. It is called
1302:   from TSInitializePackage().

1304:   Level: developer

1306: .seealso: PetscInitialize()
1307: @*/
1308: PetscErrorCode  TSGLLEInitializePackage(void)
1309: {

1313:   if (TSGLLEPackageInitialized) return(0);
1314:   TSGLLEPackageInitialized = PETSC_TRUE;
1315:   TSGLLERegisterAll();
1316:   PetscRegisterFinalize(TSGLLEFinalizePackage);
1317:   return(0);
1318: }

1320: /*@C
1321:   TSGLLEFinalizePackage - This function destroys everything in the TSGLLE package. It is
1322:   called from PetscFinalize().

1324:   Level: developer

1326: .seealso: PetscFinalize()
1327: @*/
1328: PetscErrorCode  TSGLLEFinalizePackage(void)
1329: {

1333:   PetscFunctionListDestroy(&TSGLLEList);
1334:   PetscFunctionListDestroy(&TSGLLEAcceptList);
1335:   TSGLLEPackageInitialized = PETSC_FALSE;
1336:   TSGLLERegisterAllCalled  = PETSC_FALSE;
1337:   return(0);
1338: }

1340: /* ------------------------------------------------------------ */
1341: /*MC
1342:       TSGLLE - DAE solver using implicit General Linear methods

1344:   These methods contain Runge-Kutta and multistep schemes as special cases.  These special cases have some fundamental
1345:   limitations.  For example, diagonally implicit Runge-Kutta cannot have stage order greater than 1 which limits their
1346:   applicability to very stiff systems.  Meanwhile, multistep methods cannot be A-stable for order greater than 2 and BDF
1347:   are not 0-stable for order greater than 6.  GL methods can be A- and L-stable with arbitrarily high stage order and
1348:   reliable error estimates for both 1 and 2 orders higher to facilitate adaptive step sizes and adaptive order schemes.
1349:   All this is possible while preserving a singly diagonally implicit structure.

1351:   Options database keys:
1352: +  -ts_gl_type <type> - the class of general linear method (irks)
1353: .  -ts_gl_rtol <tol>  - relative error
1354: .  -ts_gl_atol <tol>  - absolute error
1355: .  -ts_gl_min_order <p> - minimum order method to consider (default=1)
1356: .  -ts_gl_max_order <p> - maximum order method to consider (default=3)
1357: .  -ts_gl_start_order <p> - order of starting method (default=1)
1358: .  -ts_gl_complete <method> - method to use for completing the step (rescale-and-modify or rescale)
1359: -  -ts_adapt_type <method> - adaptive controller to use (none step both)

1361:   Notes:
1362:   This integrator can be applied to DAE.

1364:   Diagonally implicit general linear (DIGL) methods are a generalization of diagonally implicit Runge-Kutta (DIRK).
1365:   They are represented by the tableau

1367: .vb
1368:   A  |  U
1369:   -------
1370:   B  |  V
1371: .ve

1373:   combined with a vector c of abscissa.  "Diagonally implicit" means that A is lower triangular.
1374:   A step of the general method reads

1376: .vb
1377:   [ Y ] = [A  U] [  Y'   ]
1378:   [X^k] = [B  V] [X^{k-1}]
1379: .ve

1381:   where Y is the multivector of stage values, Y' is the multivector of stage derivatives, X^k is the Nordsieck vector of
1382:   the solution at step k.  The Nordsieck vector consists of the first r moments of the solution, given by

1384: .vb
1385:   X = [x_0,x_1,...,x_{r-1}] = [x, h x', h^2 x'', ..., h^{r-1} x^{(r-1)} ]
1386: .ve

1388:   If A is lower triangular, we can solve the stages (Y,Y') sequentially

1390: .vb
1391:   y_i = h sum_{j=0}^{s-1} (a_ij y'_j) + sum_{j=0}^{r-1} u_ij x_j,    i=0,...,{s-1}
1392: .ve

1394:   and then construct the pieces to carry to the next step

1396: .vb
1397:   xx_i = h sum_{j=0}^{s-1} b_ij y'_j  + sum_{j=0}^{r-1} v_ij x_j,    i=0,...,{r-1}
1398: .ve

1400:   Note that when the equations are cast in implicit form, we are using the stage equation to define y'_i
1401:   in terms of y_i and known stuff (y_j for j<i and x_j for all j).


1404:   Error estimation

1406:   At present, the most attractive GL methods for stiff problems are singly diagonally implicit schemes which posses
1407:   Inherent Runge-Kutta Stability (IRKS).  These methods have r=s, the number of items passed between steps is equal to
1408:   the number of stages.  The order and stage-order are one less than the number of stages.  We use the error estimates
1409:   in the 2007 paper which provide the following estimates

1411: .vb
1412:   h^{p+1} X^{(p+1)}          = phi_0^T Y' + [0 psi_0^T] Xold
1413:   h^{p+2} X^{(p+2)}          = phi_1^T Y' + [0 psi_1^T] Xold
1414:   h^{p+2} (dx'/dx) X^{(p+1)} = phi_2^T Y' + [0 psi_2^T] Xold
1415: .ve

1417:   These estimates are accurate to O(h^{p+3}).

1419:   Changing the step size

1421:   We use the generalized "rescale and modify" scheme, see equation (4.5) of the 2007 paper.

1423:   Level: beginner

1425:   References:
1426: +  1. - John Butcher and Z. Jackieweicz and W. Wright, On error propagation in general linear methods for
1427:   ordinary differential equations, Journal of Complexity, Vol 23, 2007.
1428: -  2. - John Butcher, Numerical methods for ordinary differential equations, second edition, Wiley, 2009.

1430: .seealso:  TSCreate(), TS, TSSetType()

1432: M*/
1433: PETSC_EXTERN PetscErrorCode TSCreate_GLLE(TS ts)
1434: {
1435:   TS_GLLE        *gl;

1439:   TSGLLEInitializePackage();

1441:   PetscNewLog(ts,&gl);
1442:   ts->data = (void*)gl;

1444:   ts->ops->reset          = TSReset_GLLE;
1445:   ts->ops->destroy        = TSDestroy_GLLE;
1446:   ts->ops->view           = TSView_GLLE;
1447:   ts->ops->setup          = TSSetUp_GLLE;
1448:   ts->ops->solve          = TSSolve_GLLE;
1449:   ts->ops->setfromoptions = TSSetFromOptions_GLLE;
1450:   ts->ops->snesfunction   = SNESTSFormFunction_GLLE;
1451:   ts->ops->snesjacobian   = SNESTSFormJacobian_GLLE;

1453:   ts->usessnes = PETSC_TRUE;


1456:   gl->max_step_rejections = 1;
1457:   gl->min_order           = 1;
1458:   gl->max_order           = 3;
1459:   gl->start_order         = 1;
1460:   gl->current_scheme      = -1;
1461:   gl->extrapolate         = PETSC_FALSE;

1463:   gl->wrms_atol = 1e-8;
1464:   gl->wrms_rtol = 1e-5;

1466:   PetscObjectComposeFunction((PetscObject)ts,"TSGLLESetType_C",      &TSGLLESetType_GLLE);
1467:   PetscObjectComposeFunction((PetscObject)ts,"TSGLLESetAcceptType_C",&TSGLLESetAcceptType_GLLE);
1468:   PetscObjectComposeFunction((PetscObject)ts,"TSGLLEGetAdapt_C",     &TSGLLEGetAdapt_GLLE);
1469:   return(0);
1470: }