Actual source code: glle.c

petsc-master 2019-10-16
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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: #if !defined(PETSC_MISSING_LAPACK_GELSS)
151:     PetscBLASInt rank;
152: #endif
153:     PetscMalloc7(PetscSqr(r),&ImV,3*s,&H,3*ss,&bmat,lwork,&workscalar,5*(3+r),&workreal,r+s,&sing,r+s,&ipiv);

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

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

179:     /* Build stage_error vector
180:            xi = glm.c.^(p+1)/factorial(p+1) - glm.A*glm.c.^p/factorial(p) + glm.U(:,2:end)*e.alpha;
181:     */
182:     for (i=0; i<s; i++) {
183:       scheme->stage_error[i] = CPowF(c[i],p+1);
184:       for (j=0; j<s; j++) scheme->stage_error[i] -= a[i*s+j]*CPowF(c[j],p);
185:       for (j=1; j<r; j++) scheme->stage_error[i] += u[i*r+j]*scheme->alpha[j];
186:     }

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

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

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

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

220:     /* Assemble H
221:     *    % Determine the error estimators phi
222:        H = [[cpow(glm.c,p) + C*e.alpha] [cpow(glm.c,p+1) + C*e.beta] ...
223:                [e.xi - C*(e.gamma + 0*e.epsilon*eye(s-1,1))]]';
224:     % Paper has formula above without the 0, but that term must be left
225:     % out to satisfy the conditions they propose and to make the
226:     % example schemes work
227:     e.H = H;
228:     e.phi = (H \ [1 0 0;1 1 0;0 0 -1])';
229:     e.psi = -e.phi*C;
230:     * */
231:     for (j=0; j<s; j++) {
232:       H[0+j*3] = CPowF(c[j],p);
233:       H[1+j*3] = CPowF(c[j],p+1);
234:       H[2+j*3] = scheme->stage_error[j];
235:       for (k=1; k<r; k++) {
236:         H[0+j*3] += CPowF(c[j],k-1)*scheme->alpha[k];
237:         H[1+j*3] += CPowF(c[j],k-1)*scheme->beta[k];
238:         H[2+j*3] -= CPowF(c[j],k-1)*scheme->gamma[k];
239:       }
240:     }
241:     bmat[0+0*ss] = 1.;  bmat[0+1*ss] = 0.;  bmat[0+2*ss] = 0.;
242:     bmat[1+0*ss] = 1.;  bmat[1+1*ss] = 1.;  bmat[1+2*ss] = 0.;
243:     bmat[2+0*ss] = 0.;  bmat[2+1*ss] = 0.;  bmat[2+2*ss] = -1.;
244:     m     = 3;
245:     PetscBLASIntCast(s,&n);
246:     PetscBLASIntCast(ss,&ldb);
247:     rcond = 1e-12;
248: #if defined(PETSC_MISSING_LAPACK_GELSS)
249:     /* ESSL does not have this routine */
250:     SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"GELSS - Lapack routine is unavailable\nNot able to run GL time stepping.");
251: #else
252: #if defined(PETSC_USE_COMPLEX)
253:     /* ZGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, RWORK, INFO) */
254:     PetscStackCallBLAS("LAPACKgelss",LAPACKgelss_(&m,&n,&m,H,&m,bmat,&ldb,sing,&rcond,&rank,workscalar,&lwork,workreal,&info));
255: #else
256:     /* DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, INFO) */
257:     PetscStackCallBLAS("LAPACKgelss",LAPACKgelss_(&m,&n,&m,H,&m,bmat,&ldb,sing,&rcond,&rank,workscalar,&lwork,&info));
258: #endif
259:     if (info < 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"Bad argument to GELSS");
260:     if (info > 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"SVD failed to converge");
261: #endif

263:     for (j=0; j<3; j++) {
264:       for (k=0; k<s; k++) scheme->phi[k+j*s] = bmat[k+j*ss];
265:     }

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

293:   *inscheme = scheme;
294:   return(0);
295: }

297: static PetscErrorCode TSGLLESchemeDestroy(TSGLLEScheme sc)
298: {

302:   PetscFree5(sc->c,sc->a,sc->b,sc->u,sc->v);
303:   PetscFree6(sc->alpha,sc->beta,sc->gamma,sc->phi,sc->psi,sc->stage_error);
304:   PetscFree(sc);
305:   return(0);
306: }

308: static PetscErrorCode TSGLLEDestroy_Default(TS_GLLE *gl)
309: {
311:   PetscInt       i;

314:   for (i=0; i<gl->nschemes; i++) {
315:     if (gl->schemes[i]) {TSGLLESchemeDestroy(gl->schemes[i]);}
316:   }
317:   PetscFree(gl->schemes);
318:   gl->nschemes = 0;
319:   PetscMemzero(gl->type_name,sizeof(gl->type_name));
320:   return(0);
321: }

323: static PetscErrorCode TSGLLEViewTable_Private(PetscViewer viewer,PetscInt m,PetscInt n,const PetscScalar a[],const char name[])
324: {
326:   PetscBool      iascii;
327:   PetscInt       i,j;

330:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
331:   if (iascii) {
332:     PetscViewerASCIIPrintf(viewer,"%30s = [",name);
333:     for (i=0; i<m; i++) {
334:       if (i) {PetscViewerASCIIPrintf(viewer,"%30s   [","");}
335:       PetscViewerASCIIUseTabs(viewer,PETSC_FALSE);
336:       for (j=0; j<n; j++) {
337:         PetscViewerASCIIPrintf(viewer," %12.8g",PetscRealPart(a[i*n+j]));
338:       }
339:       PetscViewerASCIIPrintf(viewer,"]\n");
340:       PetscViewerASCIIUseTabs(viewer,PETSC_TRUE);
341:     }
342:   }
343:   return(0);
344: }


347: static PetscErrorCode TSGLLESchemeView(TSGLLEScheme sc,PetscBool view_details,PetscViewer viewer)
348: {
350:   PetscBool      iascii;

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

367:       TSGLLEViewTable_Private(viewer,3,sc->s,sc->phi,"Error estimate phi");
368:       TSGLLEViewTable_Private(viewer,3,sc->r,sc->psi,"Error estimate psi");
369:       TSGLLEViewTable_Private(viewer,1,sc->r,sc->alpha,"Modify alpha");
370:       TSGLLEViewTable_Private(viewer,1,sc->r,sc->beta,"Modify beta");
371:       TSGLLEViewTable_Private(viewer,1,sc->r,sc->gamma,"Modify gamma");
372:       TSGLLEViewTable_Private(viewer,1,sc->s,sc->stage_error,"Stage error xi");
373:     }
374:     PetscViewerASCIIPopTab(viewer);
375:   } else SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_SUP,"Viewer type %s not supported",((PetscObject)viewer)->type_name);
376:   return(0);
377: }

379: static PetscErrorCode TSGLLEEstimateHigherMoments_Default(TSGLLEScheme sc,PetscReal h,Vec Ydot[],Vec Xold[],Vec hm[])
380: {
382:   PetscInt       i;

385:   if (sc->r > 64 || sc->s > 64) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Ridiculous number of stages or items passed between stages");
386:   /* build error vectors*/
387:   for (i=0; i<3; i++) {
388:     PetscScalar phih[64];
389:     PetscInt    j;
390:     for (j=0; j<sc->s; j++) phih[j] = sc->phi[i*sc->s+j]*h;
391:     VecZeroEntries(hm[i]);
392:     VecMAXPY(hm[i],sc->s,phih,Ydot);
393:     VecMAXPY(hm[i],sc->r,&sc->psi[i*sc->r],Xold);
394:   }
395:   return(0);
396: }

398: static PetscErrorCode TSGLLECompleteStep_Rescale(TSGLLEScheme sc,PetscReal h,TSGLLEScheme next_sc,PetscReal next_h,Vec Ydot[],Vec Xold[],Vec X[])
399: {
401:   PetscScalar    brow[32],vrow[32];
402:   PetscInt       i,j,r,s;

405:   /* Build the new solution from (X,Ydot) */
406:   r = sc->r;
407:   s = sc->s;
408:   for (i=0; i<r; i++) {
409:     VecZeroEntries(X[i]);
410:     for (j=0; j<s; j++) brow[j] = h*sc->b[i*s+j];
411:     VecMAXPY(X[i],s,brow,Ydot);
412:     for (j=0; j<r; j++) vrow[j] = sc->v[i*r+j];
413:     VecMAXPY(X[i],r,vrow,Xold);
414:   }
415:   return(0);
416: }

418: static PetscErrorCode TSGLLECompleteStep_RescaleAndModify(TSGLLEScheme sc,PetscReal h,TSGLLEScheme next_sc,PetscReal next_h,Vec Ydot[],Vec Xold[],Vec X[])
419: {
421:   PetscScalar    brow[32],vrow[32];
422:   PetscReal      ratio;
423:   PetscInt       i,j,p,r,s;

426:   /* Build the new solution from (X,Ydot) */
427:   p     = sc->p;
428:   r     = sc->r;
429:   s     = sc->s;
430:   ratio = next_h/h;
431:   for (i=0; i<r; i++) {
432:     VecZeroEntries(X[i]);
433:     for (j=0; j<s; j++) {
434:       brow[j] = h*(PetscPowRealInt(ratio,i)*sc->b[i*s+j]
435:                    + (PetscPowRealInt(ratio,i) - PetscPowRealInt(ratio,p+1))*(+ sc->alpha[i]*sc->phi[0*s+j])
436:                    + (PetscPowRealInt(ratio,i) - PetscPowRealInt(ratio,p+2))*(+ sc->beta [i]*sc->phi[1*s+j]
437:                                                                               + sc->gamma[i]*sc->phi[2*s+j]));
438:     }
439:     VecMAXPY(X[i],s,brow,Ydot);
440:     for (j=0; j<r; j++) {
441:       vrow[j] = (PetscPowRealInt(ratio,i)*sc->v[i*r+j]
442:                  + (PetscPowRealInt(ratio,i) - PetscPowRealInt(ratio,p+1))*(+ sc->alpha[i]*sc->psi[0*r+j])
443:                  + (PetscPowRealInt(ratio,i) - PetscPowRealInt(ratio,p+2))*(+ sc->beta [i]*sc->psi[1*r+j]
444:                                                                             + sc->gamma[i]*sc->psi[2*r+j]));
445:     }
446:     VecMAXPY(X[i],r,vrow,Xold);
447:   }
448:   if (r < next_sc->r) {
449:     if (r+1 != next_sc->r) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Cannot accommodate jump in r greater than 1");
450:     VecZeroEntries(X[r]);
451:     for (j=0; j<s; j++) brow[j] = h*PetscPowRealInt(ratio,p+1)*sc->phi[0*s+j];
452:     VecMAXPY(X[r],s,brow,Ydot);
453:     for (j=0; j<r; j++) vrow[j] = PetscPowRealInt(ratio,p+1)*sc->psi[0*r+j];
454:     VecMAXPY(X[r],r,vrow,Xold);
455:   }
456:   return(0);
457: }

459: static PetscErrorCode TSGLLECreate_IRKS(TS ts)
460: {
461:   TS_GLLE        *gl = (TS_GLLE*)ts->data;

465:   gl->Destroy               = TSGLLEDestroy_Default;
466:   gl->EstimateHigherMoments = TSGLLEEstimateHigherMoments_Default;
467:   gl->CompleteStep          = TSGLLECompleteStep_RescaleAndModify;
468:   PetscMalloc1(10,&gl->schemes);
469:   gl->nschemes = 0;

471:   {
472:     /* p=1,q=1, r=s=2, A- and L-stable with error estimates of order 2 and 3
473:     * Listed in Butcher & Podhaisky 2006. On error estimation in general linear methods for stiff ODE.
474:     * irks(0.3,0,[.3,1],[1],1)
475:     * Note: can be made to have classical order (not stage order) 2 by replacing 0.3 with 1-sqrt(1/2)
476:     * but doing so would sacrifice the error estimator.
477:     */
478:     const PetscScalar c[2]    = {3./10., 1.};
479:     const PetscScalar a[2][2] = {{3./10., 0}, {7./10., 3./10.}};
480:     const PetscScalar b[2][2] = {{7./10., 3./10.}, {0,1}};
481:     const PetscScalar u[2][2] = {{1,0},{1,0}};
482:     const PetscScalar v[2][2] = {{1,0},{0,0}};
483:     TSGLLESchemeCreate(1,1,2,2,c,*a,*b,*u,*v,&gl->schemes[gl->nschemes++]);
484:   }

486:   {
487:     /* p=q=2, r=s=3: irks(4/9,0,[1:3]/3,[0.33852],1) */
488:     /* http://www.math.auckland.ac.nz/~hpod/atlas/i2a.html */
489:     const PetscScalar c[3] = {1./3., 2./3., 1}
490:     ,a[3][3] = {{4./9.                ,0                      ,       0},
491:                 {1.03750643704090e+00 ,                  4./9.,       0},
492:                 {7.67024779410304e-01 ,  -3.81140216918943e-01,   4./9.}}
493:     ,b[3][3] = {{0.767024779410304,  -0.381140216918943,   4./9.},
494:                 {0.000000000000000,  0.000000000000000,   1.000000000000000},
495:                 {-2.075048385225385,   0.621728385225383,   1.277197204924873}}
496:     ,u[3][3] = {{1.0000000000000000,  -0.1111111111111109,  -0.0925925925925922},
497:                 {1.0000000000000000,  -0.8152842148186744,  -0.4199095530877056},
498:                 {1.0000000000000000,   0.1696709930641948,   0.0539741070314165}}
499:     ,v[3][3] = {{1.0000000000000000,  0.1696709930641948,   0.0539741070314165},
500:                 {0.000000000000000,   0.000000000000000,   0.000000000000000},
501:                 {0.000000000000000,   0.176122795075129,   0.000000000000000}};
502:     TSGLLESchemeCreate(2,2,3,3,c,*a,*b,*u,*v,&gl->schemes[gl->nschemes++]);
503:   }
504:   {
505:     /* p=q=3, r=s=4: irks(9/40,0,[1:4]/4,[0.3312 1.0050],[0.49541 1;1 0]) */
506:     const PetscScalar c[4] = {0.25,0.5,0.75,1.0}
507:     ,a[4][4] = {{9./40.               ,                      0,                      0,                      0},
508:                 {2.11286958887701e-01 ,    9./40.             ,                      0,                      0},
509:                 {9.46338294287584e-01 ,  -3.42942861246094e-01,   9./40.              ,                      0},
510:                 {0.521490453970721    ,  -0.662474225622980,   0.490476425459734,   9./40.           }}
511:     ,b[4][4] = {{0.521490453970721    ,  -0.662474225622980,   0.490476425459734,   9./40.           },
512:                 {0.000000000000000    ,   0.000000000000000,   0.000000000000000,   1.000000000000000},
513:                 {-0.084677029310348   ,   1.390757514776085,  -1.568157386206001,   2.023192696767826},
514:                 {0.465383797936408    ,   1.478273530625148,  -1.930836081010182,   1.644872111193354}}
515:     ,u[4][4] = {{1.00000000000000000  ,   0.02500000000001035,  -0.02499999999999053,  -0.00442708333332865},
516:                 {1.00000000000000000  ,   0.06371304111232945,  -0.04032173972189845,  -0.01389438413189452},
517:                 {1.00000000000000000  ,  -0.07839543304147778,   0.04738685705116663,   0.02032603595928376},
518:                 {1.00000000000000000  ,   0.42550734619251651,   0.10800718022400080,  -0.01726712647760034}}
519:     ,v[4][4] = {{1.00000000000000000  ,   0.42550734619251651,   0.10800718022400080,  -0.01726712647760034},
520:                 {0.000000000000000    ,   0.000000000000000,   0.000000000000000,   0.000000000000000},
521:                 {0.000000000000000    ,  -1.761115796027561,  -0.521284157173780,   0.258249384305463},
522:                 {0.000000000000000    ,  -1.657693358744728,  -1.052227765232394,   0.521284157173780}};
523:     TSGLLESchemeCreate(3,3,4,4,c,*a,*b,*u,*v,&gl->schemes[gl->nschemes++]);
524:   }
525:   {
526:     /* p=q=4, r=s=5:
527:           irks(3/11,0,[1:5]/5, [0.1715   -0.1238    0.6617],...
528:           [ -0.0812    0.4079    1.0000
529:              1.0000         0         0
530:              0.8270    1.0000         0])
531:     */
532:     const PetscScalar c[5] = {0.2,0.4,0.6,0.8,1.0}
533:     ,a[5][5] = {{2.72727272727352e-01 ,   0.00000000000000e+00,  0.00000000000000e+00 ,  0.00000000000000e+00  ,  0.00000000000000e+00},
534:                 {-1.03980153733431e-01,   2.72727272727405e-01,   0.00000000000000e+00,  0.00000000000000e+00  ,  0.00000000000000e+00},
535:                 {-1.58615400341492e+00,   7.44168951881122e-01,   2.72727272727309e-01,  0.00000000000000e+00  ,  0.00000000000000e+00},
536:                 {-8.73658042865628e-01,   5.37884671894595e-01,  -1.63298538799523e-01,   2.72727272726996e-01 ,  0.00000000000000e+00},
537:                 {2.95489397443992e-01 , -1.18481693910097e+00 , -6.68029812659953e-01 ,  1.00716687860943e+00  , 2.72727272727288e-01}}
538:     ,b[5][5] = {{2.95489397443992e-01 , -1.18481693910097e+00 , -6.68029812659953e-01 ,  1.00716687860943e+00  , 2.72727272727288e-01},
539:                 {0.00000000000000e+00 ,  1.11022302462516e-16 , -2.22044604925031e-16 ,  0.00000000000000e+00  , 1.00000000000000e+00},
540:                 {-4.05882503986005e+00,  -4.00924006567769e+00,  -1.38930610972481e+00,   4.45223930308488e+00 ,  6.32331093108427e-01},
541:                 {8.35690179937017e+00 , -2.26640927349732e+00 ,  6.86647884973826e+00 , -5.22595158025740e+00  , 4.50893068837431e+00},
542:                 {1.27656267027479e+01 ,  2.80882153840821e+00 ,  8.91173096522890e+00 , -1.07936444078906e+01  , 4.82534148988854e+00}}
543:     ,u[5][5] = {{1.00000000000000e+00 , -7.27272727273551e-02 , -3.45454545454419e-02 , -4.12121212119565e-03  ,-2.96969696964014e-04},
544:                 {1.00000000000000e+00 ,  2.31252881006154e-01 , -8.29487834416481e-03 , -9.07191207681020e-03  ,-1.70378403743473e-03},
545:                 {1.00000000000000e+00 ,  1.16925777880663e+00 ,  3.59268562942635e-02 , -4.09013451730615e-02  ,-1.02411119670164e-02},
546:                 {1.00000000000000e+00 ,  1.02634463704356e+00 ,  1.59375044913405e-01 ,  1.89673015035370e-03  ,-4.89987231897569e-03},
547:                 {1.00000000000000e+00 ,  1.27746320298021e+00 ,  2.37186008132728e-01 , -8.28694373940065e-02  ,-5.34396510196430e-02}}
548:     ,v[5][5] = {{1.00000000000000e+00 ,  1.27746320298021e+00 ,  2.37186008132728e-01 , -8.28694373940065e-02  ,-5.34396510196430e-02},
549:                 {0.00000000000000e+00 , -1.77635683940025e-15 , -1.99840144432528e-15 , -9.99200722162641e-16  ,-3.33066907387547e-16},
550:                 {0.00000000000000e+00 ,  4.37280081906924e+00 ,  5.49221645016377e-02 , -8.88913177394943e-02  , 1.12879077989154e-01},
551:                 {0.00000000000000e+00 , -1.22399504837280e+01 , -5.21287338448645e+00 , -8.03952325565291e-01  , 4.60298678047147e-01},
552:                 {0.00000000000000e+00 , -1.85178762883829e+01 , -5.21411849862624e+00 , -1.04283436528809e+00  , 7.49030161063651e-01}};
553:     TSGLLESchemeCreate(4,4,5,5,c,*a,*b,*u,*v,&gl->schemes[gl->nschemes++]);
554:   }
555:   {
556:     /* p=q=5, r=s=6;
557:        irks(1/3,0,[1:6]/6,...
558:           [-0.0489    0.4228   -0.8814    0.9021],...
559:           [-0.3474   -0.6617    0.6294    0.2129
560:             0.0044   -0.4256   -0.1427   -0.8936
561:            -0.8267    0.4821    0.1371   -0.2557
562:            -0.4426   -0.3855   -0.7514    0.3014])
563:     */
564:     const PetscScalar c[6] = {1./6, 2./6, 3./6, 4./6, 5./6, 1.}
565:     ,a[6][6] = {{  3.33333333333940e-01,  0                   ,  0                   ,  0                   ,  0                   ,  0                   },
566:                 { -8.64423857333350e-02,  3.33333333332888e-01,  0                   ,  0                   ,  0                   ,  0                   },
567:                 { -2.16850174258252e+00, -2.23619072028839e+00,  3.33333333335204e-01,  0                   ,  0                   ,  0                   },
568:                 { -4.73160970138997e+00, -3.89265344629268e+00, -2.76318716520933e-01,  3.33333333335759e-01,  0                   ,  0                   },
569:                 { -6.75187540297338e+00, -7.90756533769377e+00,  7.90245051802259e-01, -4.48352364517632e-01,  3.33333333328483e-01,  0                   },
570:                 { -4.26488287921548e+00, -1.19320395589302e+01,  3.38924509887755e+00, -2.23969848002481e+00,  6.62807710124007e-01,  3.33333333335440e-01}}
571:     ,b[6][6] = {{ -4.26488287921548e+00, -1.19320395589302e+01,  3.38924509887755e+00, -2.23969848002481e+00,  6.62807710124007e-01,  3.33333333335440e-01},
572:                 { -8.88178419700125e-16,  4.44089209850063e-16, -1.54737334057131e-15, -8.88178419700125e-16,  0.00000000000000e+00,  1.00000000000001e+00},
573:                 { -2.87780425770651e+01, -1.13520448264971e+01,  2.62002318943161e+01,  2.56943874812797e+01, -3.06702268304488e+01,  6.68067773510103e+00},
574:                 {  5.47971245256474e+01,  6.80366875868284e+01, -6.50952588861999e+01, -8.28643975339097e+01,  8.17416943896414e+01, -1.17819043489036e+01},
575:                 { -2.33332114788869e+02,  6.12942539462634e+01, -4.91850135865944e+01,  1.82716844135480e+02, -1.29788173979395e+02,  3.09968095651099e+01},
576:                 { -1.72049132343751e+02,  8.60194713593999e+00,  7.98154219170200e-01,  1.50371386053218e+02, -1.18515423962066e+02,  2.50898277784663e+01}}
577:     ,u[6][6] = {{  1.00000000000000e+00, -1.66666666666870e-01, -4.16666666664335e-02, -3.85802469124815e-03, -2.25051440302250e-04, -9.64506172339142e-06},
578:                 {  1.00000000000000e+00,  8.64423857327162e-02, -4.11484912671353e-02, -1.11450903217645e-02, -1.47651050487126e-03, -1.34395070766826e-04},
579:                 {  1.00000000000000e+00,  4.57135912953434e+00,  1.06514719719137e+00,  1.33517564218007e-01,  1.11365952968659e-02,  6.12382756769504e-04},
580:                 {  1.00000000000000e+00,  9.23391519753404e+00,  2.22431212392095e+00,  2.91823807741891e-01,  2.52058456411084e-02,  1.22800542949647e-03},
581:                 {  1.00000000000000e+00,  1.48175480533865e+01,  3.73439117461835e+00,  5.14648336541804e-01,  4.76430038853402e-02,  2.56798515502156e-03},
582:                 {  1.00000000000000e+00,  1.50512347758335e+01,  4.10099701165164e+00,  5.66039141003603e-01,  3.91213893800891e-02, -2.99136269067853e-03}}
583:     ,v[6][6] = {{  1.00000000000000e+00,  1.50512347758335e+01,  4.10099701165164e+00,  5.66039141003603e-01,  3.91213893800891e-02, -2.99136269067853e-03},
584:                 {  0.00000000000000e+00, -4.88498130835069e-15, -6.43929354282591e-15, -3.55271367880050e-15, -1.22124532708767e-15, -3.12250225675825e-16},
585:                 {  0.00000000000000e+00,  1.22250171233141e+01, -1.77150760606169e+00,  3.54516769879390e-01,  6.22298845883398e-01,  2.31647447450276e-01},
586:                 {  0.00000000000000e+00, -4.48339457331040e+01, -3.57363126641880e-01,  5.18750173123425e-01,  6.55727990241799e-02,  1.63175368287079e-01},
587:                 {  0.00000000000000e+00,  1.37297394708005e+02, -1.60145272991317e+00, -5.05319555199441e+00,  1.55328940390990e-01,  9.16629423682464e-01},
588:                 {  0.00000000000000e+00,  1.05703241119022e+02, -1.16610260983038e+00, -2.99767252773859e+00, -1.13472315553890e-01,  1.09742849254729e+00}};
589:     TSGLLESchemeCreate(5,5,6,6,c,*a,*b,*u,*v,&gl->schemes[gl->nschemes++]);
590:   }
591:   return(0);
592: }

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

597:    Collective on TS

599:    Input Parameters:
600: +  ts - the TS context
601: -  type - a method

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

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

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

621:    Level: intermediate

623: @*/
624: PetscErrorCode  TSGLLESetType(TS ts,TSGLLEType type)
625: {

631:   PetscTryMethod(ts,"TSGLLESetType_C",(TS,TSGLLEType),(ts,type));
632:   return(0);
633: }

635: /*@C
636:    TSGLLESetAcceptType - sets the acceptance test

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

641:    Logically Collective on TS

643:    Input Parameters:
644: +  ts - the TS context
645: -  type - the type

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

650:    Level: intermediate

652: .seealso: TS, TSGLLE, TSGLLEAcceptRegister(), TSGLLEAdapt, set type
653: @*/
654: PetscErrorCode  TSGLLESetAcceptType(TS ts,TSGLLEAcceptType type)
655: {

661:   PetscTryMethod(ts,"TSGLLESetAcceptType_C",(TS,TSGLLEAcceptType),(ts,type));
662:   return(0);
663: }

665: /*@C
666:    TSGLLEGetAdapt - gets the TSGLLEAdapt object from the TS

668:    Not Collective

670:    Input Parameter:
671: .  ts - the TS context

673:    Output Parameter:
674: .  adapt - the TSGLLEAdapt context

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

680:    Level: advanced

682: .seealso: TSGLLEAdapt, TSGLLEAdaptRegister()
683: @*/
684: PetscErrorCode  TSGLLEGetAdapt(TS ts,TSGLLEAdapt *adapt)
685: {

691:   PetscUseMethod(ts,"TSGLLEGetAdapt_C",(TS,TSGLLEAdapt*),(ts,adapt));
692:   return(0);
693: }

695: static PetscErrorCode TSGLLEAccept_Always(TS ts,PetscReal tleft,PetscReal h,const PetscReal enorms[],PetscBool  *accept)
696: {
698:   *accept = PETSC_TRUE;
699:   return(0);
700: }

702: static PetscErrorCode TSGLLEUpdateWRMS(TS ts)
703: {
704:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
706:   PetscScalar    *x,*w;
707:   PetscInt       n,i;

710:   VecGetArray(gl->X[0],&x);
711:   VecGetArray(gl->W,&w);
712:   VecGetLocalSize(gl->W,&n);
713:   for (i=0; i<n; i++) w[i] = 1./(gl->wrms_atol + gl->wrms_rtol*PetscAbsScalar(x[i]));
714:   VecRestoreArray(gl->X[0],&x);
715:   VecRestoreArray(gl->W,&w);
716:   return(0);
717: }

719: static PetscErrorCode TSGLLEVecNormWRMS(TS ts,Vec X,PetscReal *nrm)
720: {
721:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
723:   PetscScalar    *x,*w;
724:   PetscReal      sum = 0.0,gsum;
725:   PetscInt       n,N,i;

728:   VecGetArray(X,&x);
729:   VecGetArray(gl->W,&w);
730:   VecGetLocalSize(gl->W,&n);
731:   for (i=0; i<n; i++) sum += PetscAbsScalar(PetscSqr(x[i]*w[i]));
732:   VecRestoreArray(X,&x);
733:   VecRestoreArray(gl->W,&w);
734:   MPIU_Allreduce(&sum,&gsum,1,MPIU_REAL,MPIU_SUM,PetscObjectComm((PetscObject)ts));
735:   VecGetSize(gl->W,&N);
736:   *nrm = PetscSqrtReal(gsum/(1.*N));
737:   return(0);
738: }

740: static PetscErrorCode TSGLLESetType_GLLE(TS ts,TSGLLEType type)
741: {
742:   PetscErrorCode ierr,(*r)(TS);
743:   PetscBool      same;
744:   TS_GLLE        *gl = (TS_GLLE*)ts->data;

747:   if (gl->type_name[0]) {
748:     PetscStrcmp(gl->type_name,type,&same);
749:     if (same) return(0);
750:     (*gl->Destroy)(gl);
751:   }

753:   PetscFunctionListFind(TSGLLEList,type,&r);
754:   if (!r) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"Unknown TSGLLE type \"%s\" given",type);
755:   (*r)(ts);
756:   PetscStrcpy(gl->type_name,type);
757:   return(0);
758: }

760: static PetscErrorCode TSGLLESetAcceptType_GLLE(TS ts,TSGLLEAcceptType type)
761: {
762:   PetscErrorCode       ierr;
763:   TSGLLEAcceptFunction r;
764:   TS_GLLE              *gl = (TS_GLLE*)ts->data;

767:   PetscFunctionListFind(TSGLLEAcceptList,type,&r);
768:   if (!r) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"Unknown TSGLLEAccept type \"%s\" given",type);
769:   gl->Accept = r;
770:   PetscStrncpy(gl->accept_name,type,sizeof(gl->accept_name));
771:   return(0);
772: }

774: static PetscErrorCode TSGLLEGetAdapt_GLLE(TS ts,TSGLLEAdapt *adapt)
775: {
777:   TS_GLLE        *gl = (TS_GLLE*)ts->data;

780:   if (!gl->adapt) {
781:     TSGLLEAdaptCreate(PetscObjectComm((PetscObject)ts),&gl->adapt);
782:     PetscObjectIncrementTabLevel((PetscObject)gl->adapt,(PetscObject)ts,1);
783:     PetscLogObjectParent((PetscObject)ts,(PetscObject)gl->adapt);
784:   }
785:   *adapt = gl->adapt;
786:   return(0);
787: }

789: static PetscErrorCode TSGLLEChooseNextScheme(TS ts,PetscReal h,const PetscReal hmnorm[],PetscInt *next_scheme,PetscReal *next_h,PetscBool  *finish)
790: {
792:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
793:   PetscInt       i,n,cur_p,cur,next_sc,candidates[64],orders[64];
794:   PetscReal      errors[64],costs[64],tleft;

797:   cur   = -1;
798:   cur_p = gl->schemes[gl->current_scheme]->p;
799:   tleft = ts->max_time - (ts->ptime + ts->time_step);
800:   for (i=0,n=0; i<gl->nschemes; i++) {
801:     TSGLLEScheme sc = gl->schemes[i];
802:     if (sc->p < gl->min_order || gl->max_order < sc->p) continue;
803:     if (sc->p == cur_p - 1)    errors[n] = PetscAbsScalar(sc->alpha[0])*hmnorm[0];
804:     else if (sc->p == cur_p)   errors[n] = PetscAbsScalar(sc->alpha[0])*hmnorm[1];
805:     else if (sc->p == cur_p+1) errors[n] = PetscAbsScalar(sc->alpha[0])*(hmnorm[2]+hmnorm[3]);
806:     else continue;
807:     candidates[n] = i;
808:     orders[n]     = PetscMin(sc->p,sc->q); /* order of global truncation error */
809:     costs[n]      = sc->s;                 /* estimate the cost as the number of stages */
810:     if (i == gl->current_scheme) cur = n;
811:     n++;
812:   }
813:   if (cur < 0 || gl->nschemes <= cur) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Current scheme not found in scheme list");
814:   TSGLLEAdaptChoose(gl->adapt,n,orders,errors,costs,cur,h,tleft,&next_sc,next_h,finish);
815:   *next_scheme = candidates[next_sc];
816:   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);
817:   return(0);
818: }

820: static PetscErrorCode TSGLLEGetMaxSizes(TS ts,PetscInt *max_r,PetscInt *max_s)
821: {
822:   TS_GLLE *gl = (TS_GLLE*)ts->data;

825:   *max_r = gl->schemes[gl->nschemes-1]->r;
826:   *max_s = gl->schemes[gl->nschemes-1]->s;
827:   return(0);
828: }

830: static PetscErrorCode TSSolve_GLLE(TS ts)
831: {
832:   TS_GLLE             *gl = (TS_GLLE*)ts->data;
833:   PetscInt            i,k,its,lits,max_r,max_s;
834:   PetscBool           final_step,finish;
835:   SNESConvergedReason snesreason;
836:   PetscErrorCode      ierr;

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

841:   TSGLLEGetMaxSizes(ts,&max_r,&max_s);
842:   VecCopy(ts->vec_sol,gl->X[0]);
843:   for (i=1; i<max_r; i++) {
844:     VecZeroEntries(gl->X[i]);
845:   }
846:   TSGLLEUpdateWRMS(ts);

848:   if (0) {
849:     /* Find consistent initial data for DAE */
850:     gl->stage_time = ts->ptime + ts->time_step;
851:     gl->scoeff = 1.;
852:     gl->stage  = 0;

854:     VecCopy(ts->vec_sol,gl->Z);
855:     VecCopy(ts->vec_sol,gl->Y);
856:     SNESSolve(ts->snes,NULL,gl->Y);
857:     SNESGetIterationNumber(ts->snes,&its);
858:     SNESGetLinearSolveIterations(ts->snes,&lits);
859:     SNESGetConvergedReason(ts->snes,&snesreason);

861:     ts->snes_its += its; ts->ksp_its += lits;
862:     if (snesreason < 0 && ts->max_snes_failures > 0 && ++ts->num_snes_failures >= ts->max_snes_failures) {
863:       ts->reason = TS_DIVERGED_NONLINEAR_SOLVE;
864:       PetscInfo2(ts,"Step=%D, nonlinear solve solve failures %D greater than current TS allowed, stopping solve\n",ts->steps,ts->num_snes_failures);
865:       return(0);
866:     }
867:   }

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

871:   for (k=0,final_step=PETSC_FALSE,finish=PETSC_FALSE; k<ts->max_steps && !finish; k++) {
872:     PetscInt          j,r,s,next_scheme = 0,rejections;
873:     PetscReal         h,hmnorm[4],enorm[3],next_h;
874:     PetscBool         accept;
875:     const PetscScalar *c,*a,*u;
876:     Vec               *X,*Ydot,Y;
877:     TSGLLEScheme        scheme = gl->schemes[gl->current_scheme];

879:     r = scheme->r; s = scheme->s;
880:     c = scheme->c;
881:     a = scheme->a; u = scheme->u;
882:     h = ts->time_step;
883:     X = gl->X; Ydot = gl->Ydot; Y = gl->Y;

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

887:     /*
888:       We only call PreStep at the start of each STEP, not each STAGE.  This is because it is
889:       possible to fail (have to restart a step) after multiple stages.
890:     */
891:     TSPreStep(ts);

893:     rejections = 0;
894:     while (1) {
895:       for (i=0; i<s; i++) {
896:         PetscScalar shift;
897:         gl->scoeff     = 1./PetscRealPart(a[i*s+i]);
898:         shift          = gl->scoeff/ts->time_step;
899:         gl->stage      = i;
900:         gl->stage_time = ts->ptime + PetscRealPart(c[i])*h;

902:         /*
903:         * Stage equation: Y = h A Y' + U X
904:         * We assume that A is lower-triangular so that we can solve the stages (Y,Y') sequentially
905:         * Build the affine vector z_i = -[1/(h a_ii)](h sum_j a_ij y'_j + sum_j u_ij x_j)
906:         * Then y'_i = z + 1/(h a_ii) y_i
907:         */
908:         VecZeroEntries(gl->Z);
909:         for (j=0; j<r; j++) {
910:           VecAXPY(gl->Z,-shift*u[i*r+j],X[j]);
911:         }
912:         for (j=0; j<i; j++) {
913:           VecAXPY(gl->Z,-shift*h*a[i*s+j],Ydot[j]);
914:         }
915:         /* Note: Z is used within function evaluation, Ydot = Z + shift*Y */

917:         /* Compute an estimate of Y to start Newton iteration */
918:         if (gl->extrapolate) {
919:           if (i==0) {
920:             /* Linear extrapolation on the first stage */
921:             VecWAXPY(Y,c[i]*h,X[1],X[0]);
922:           } else {
923:             /* Linear extrapolation from the last stage */
924:             VecAXPY(Y,(c[i]-c[i-1])*h,Ydot[i-1]);
925:           }
926:         } else if (i==0) {        /* Directly use solution from the last step, otherwise reuse the last stage (do nothing) */
927:           VecCopy(X[0],Y);
928:         }

930:         /* Solve this stage (Ydot[i] is computed during function evaluation) */
931:         SNESSolve(ts->snes,NULL,Y);
932:         SNESGetIterationNumber(ts->snes,&its);
933:         SNESGetLinearSolveIterations(ts->snes,&lits);
934:         SNESGetConvergedReason(ts->snes,&snesreason);
935:         ts->snes_its += its; ts->ksp_its += lits;
936:         if (snesreason < 0 && ts->max_snes_failures > 0 && ++ts->num_snes_failures >= ts->max_snes_failures) {
937:           ts->reason = TS_DIVERGED_NONLINEAR_SOLVE;
938:           PetscInfo2(ts,"Step=%D, nonlinear solve solve failures %D greater than current TS allowed, stopping solve\n",ts->steps,ts->num_snes_failures);
939:           return(0);
940:         }
941:       }

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

945:       (*gl->EstimateHigherMoments)(scheme,h,Ydot,gl->X,gl->himom);
946:       /* hmnorm[i] = h^{p+i}x^{(p+i)} with i=0,1,2; hmnorm[3] = h^{p+2}(dx'/dx) x^{(p+1)} */
947:       for (i=0; i<3; i++) {
948:         TSGLLEVecNormWRMS(ts,gl->himom[i],&hmnorm[i+1]);
949:       }
950:       enorm[0] = PetscRealPart(scheme->alpha[0])*hmnorm[1];
951:       enorm[1] = PetscRealPart(scheme->beta[0]) *hmnorm[2];
952:       enorm[2] = PetscRealPart(scheme->gamma[0])*hmnorm[3];
953:       (*gl->Accept)(ts,ts->max_time-gl->stage_time,h,enorm,&accept);
954:       if (accept) goto accepted;
955:       rejections++;
956:       PetscInfo3(ts,"Step %D (t=%g) not accepted, rejections=%D\n",k,gl->stage_time,rejections);
957:       if (rejections > gl->max_step_rejections) break;
958:       /*
959:         There are lots of reasons why a step might be rejected, including solvers not converging and other factors that
960:         TSGLLEChooseNextScheme does not support.  Additionally, the error estimates may be very screwed up, so I'm not
961:         convinced that it's safe to just compute a new error estimate using the same interface as the current adaptor
962:         (the adaptor interface probably has to change).  Here we make an arbitrary and naive choice.  This assumes that
963:         steps were written in Nordsieck form.  The "correct" method would be to re-complete the previous time step with
964:         the correct "next" step size.  It is unclear to me whether the present ad-hoc method of rescaling X is stable.
965:       */
966:       h *= 0.5;
967:       for (i=1; i<scheme->r; i++) {
968:         VecScale(X[i],PetscPowRealInt(0.5,i));
969:       }
970:     }
971:     SETERRQ3(PETSC_COMM_SELF,PETSC_ERR_CONV_FAILED,"Time step %D (t=%g) not accepted after %D failures\n",k,gl->stage_time,rejections);

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

978:     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]);
979:     if (!final_step) {
980:       TSGLLEChooseNextScheme(ts,h,hmnorm,&next_scheme,&next_h,&final_step);
981:     } else {
982:       /* Dummy values to complete the current step in a consistent manner */
983:       next_scheme = gl->current_scheme;
984:       next_h      = h;
985:       finish      = PETSC_TRUE;
986:     }

988:     X        = gl->Xold;
989:     gl->Xold = gl->X;
990:     gl->X    = X;
991:     (*gl->CompleteStep)(scheme,h,gl->schemes[next_scheme],next_h,Ydot,gl->Xold,gl->X);

993:     TSGLLEUpdateWRMS(ts);

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

1000:     TSPostEvaluate(ts);
1001:     TSPostStep(ts);
1002:     TSMonitor(ts,ts->steps,ts->ptime,ts->vec_sol);

1004:     gl->current_scheme = next_scheme;
1005:     ts->time_step      = next_h;
1006:   }
1007:   return(0);
1008: }

1010: /*------------------------------------------------------------*/

1012: static PetscErrorCode TSReset_GLLE(TS ts)
1013: {
1014:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
1015:   PetscInt       max_r,max_s;

1019:   if (gl->setupcalled) {
1020:     TSGLLEGetMaxSizes(ts,&max_r,&max_s);
1021:     VecDestroyVecs(max_r,&gl->Xold);
1022:     VecDestroyVecs(max_r,&gl->X);
1023:     VecDestroyVecs(max_s,&gl->Ydot);
1024:     VecDestroyVecs(3,&gl->himom);
1025:     VecDestroy(&gl->W);
1026:     VecDestroy(&gl->Y);
1027:     VecDestroy(&gl->Z);
1028:   }
1029:   gl->setupcalled = PETSC_FALSE;
1030:   return(0);
1031: }

1033: static PetscErrorCode TSDestroy_GLLE(TS ts)
1034: {
1035:   TS_GLLE        *gl = (TS_GLLE*)ts->data;

1039:   TSReset_GLLE(ts);
1040:   if (ts->dm) {
1041:     DMCoarsenHookRemove(ts->dm,DMCoarsenHook_TSGLLE,DMRestrictHook_TSGLLE,ts);
1042:     DMSubDomainHookRemove(ts->dm,DMSubDomainHook_TSGLLE,DMSubDomainRestrictHook_TSGLLE,ts);
1043:   }
1044:   if (gl->adapt) {TSGLLEAdaptDestroy(&gl->adapt);}
1045:   if (gl->Destroy) {(*gl->Destroy)(gl);}
1046:   PetscFree(ts->data);
1047:   PetscObjectComposeFunction((PetscObject)ts,"TSGLLESetType_C",NULL);
1048:   PetscObjectComposeFunction((PetscObject)ts,"TSGLLESetAcceptType_C",NULL);
1049:   PetscObjectComposeFunction((PetscObject)ts,"TSGLLEGetAdapt_C",NULL);
1050:   return(0);
1051: }

1053: /*
1054:     This defines the nonlinear equation that is to be solved with SNES
1055:     g(x) = f(t,x,z+shift*x) = 0
1056: */
1057: static PetscErrorCode SNESTSFormFunction_GLLE(SNES snes,Vec x,Vec f,TS ts)
1058: {
1059:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
1061:   Vec            Z,Ydot;
1062:   DM             dm,dmsave;

1065:   SNESGetDM(snes,&dm);
1066:   TSGLLEGetVecs(ts,dm,&Z,&Ydot);
1067:   VecWAXPY(Ydot,gl->scoeff/ts->time_step,x,Z);
1068:   dmsave = ts->dm;
1069:   ts->dm = dm;
1070:   TSComputeIFunction(ts,gl->stage_time,x,Ydot,f,PETSC_FALSE);
1071:   ts->dm = dmsave;
1072:   TSGLLERestoreVecs(ts,dm,&Z,&Ydot);
1073:   return(0);
1074: }

1076: static PetscErrorCode SNESTSFormJacobian_GLLE(SNES snes,Vec x,Mat A,Mat B,TS ts)
1077: {
1078:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
1080:   Vec            Z,Ydot;
1081:   DM             dm,dmsave;

1084:   SNESGetDM(snes,&dm);
1085:   TSGLLEGetVecs(ts,dm,&Z,&Ydot);
1086:   dmsave = ts->dm;
1087:   ts->dm = dm;
1088:   /* gl->Xdot will have already been computed in SNESTSFormFunction_GLLE */
1089:   TSComputeIJacobian(ts,gl->stage_time,x,gl->Ydot[gl->stage],gl->scoeff/ts->time_step,A,B,PETSC_FALSE);
1090:   ts->dm = dmsave;
1091:   TSGLLERestoreVecs(ts,dm,&Z,&Ydot);
1092:   return(0);
1093: }


1096: static PetscErrorCode TSSetUp_GLLE(TS ts)
1097: {
1098:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
1099:   PetscInt       max_r,max_s;
1101:   DM             dm;

1104:   gl->setupcalled = PETSC_TRUE;
1105:   TSGLLEGetMaxSizes(ts,&max_r,&max_s);
1106:   VecDuplicateVecs(ts->vec_sol,max_r,&gl->X);
1107:   VecDuplicateVecs(ts->vec_sol,max_r,&gl->Xold);
1108:   VecDuplicateVecs(ts->vec_sol,max_s,&gl->Ydot);
1109:   VecDuplicateVecs(ts->vec_sol,3,&gl->himom);
1110:   VecDuplicate(ts->vec_sol,&gl->W);
1111:   VecDuplicate(ts->vec_sol,&gl->Y);
1112:   VecDuplicate(ts->vec_sol,&gl->Z);

1114:   /* Default acceptance tests and adaptivity */
1115:   if (!gl->Accept) {TSGLLESetAcceptType(ts,TSGLLEACCEPT_ALWAYS);}
1116:   if (!gl->adapt)  {TSGLLEGetAdapt(ts,&gl->adapt);}

1118:   if (gl->current_scheme < 0) {
1119:     PetscInt i;
1120:     for (i=0;; i++) {
1121:       if (gl->schemes[i]->p == gl->start_order) break;
1122:       if (i+1 == gl->nschemes) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"No schemes available with requested start order %d",i);
1123:     }
1124:     gl->current_scheme = i;
1125:   }
1126:   TSGetDM(ts,&dm);
1127:   DMCoarsenHookAdd(dm,DMCoarsenHook_TSGLLE,DMRestrictHook_TSGLLE,ts);
1128:   DMSubDomainHookAdd(dm,DMSubDomainHook_TSGLLE,DMSubDomainRestrictHook_TSGLLE,ts);
1129:   return(0);
1130: }
1131: /*------------------------------------------------------------*/

1133: static PetscErrorCode TSSetFromOptions_GLLE(PetscOptionItems *PetscOptionsObject,TS ts)
1134: {
1135:   TS_GLLE        *gl        = (TS_GLLE*)ts->data;
1136:   char           tname[256] = TSGLLE_IRKS,completef[256] = "rescale-and-modify";

1140:   PetscOptionsHead(PetscOptionsObject,"General Linear ODE solver options");
1141:   {
1142:     PetscBool flg;
1143:     PetscOptionsFList("-ts_gl_type","Type of GL method","TSGLLESetType",TSGLLEList,gl->type_name[0] ? gl->type_name : tname,tname,sizeof(tname),&flg);
1144:     if (flg || !gl->type_name[0]) {
1145:       TSGLLESetType(ts,tname);
1146:     }
1147:     PetscOptionsInt("-ts_gl_max_step_rejections","Maximum number of times to attempt a step","None",gl->max_step_rejections,&gl->max_step_rejections,NULL);
1148:     PetscOptionsInt("-ts_gl_max_order","Maximum order to try","TSGLLESetMaxOrder",gl->max_order,&gl->max_order,NULL);
1149:     PetscOptionsInt("-ts_gl_min_order","Minimum order to try","TSGLLESetMinOrder",gl->min_order,&gl->min_order,NULL);
1150:     PetscOptionsInt("-ts_gl_start_order","Initial order to try","TSGLLESetMinOrder",gl->start_order,&gl->start_order,NULL);
1151:     PetscOptionsEnum("-ts_gl_error_direction","Which direction to look when estimating error","TSGLLESetErrorDirection",TSGLLEErrorDirections,(PetscEnum)gl->error_direction,(PetscEnum*)&gl->error_direction,NULL);
1152:     PetscOptionsBool("-ts_gl_extrapolate","Extrapolate stage solution from previous solution (sometimes unstable)","TSGLLESetExtrapolate",gl->extrapolate,&gl->extrapolate,NULL);
1153:     PetscOptionsReal("-ts_gl_atol","Absolute tolerance","TSGLLESetTolerances",gl->wrms_atol,&gl->wrms_atol,NULL);
1154:     PetscOptionsReal("-ts_gl_rtol","Relative tolerance","TSGLLESetTolerances",gl->wrms_rtol,&gl->wrms_rtol,NULL);
1155:     PetscOptionsString("-ts_gl_complete","Method to use for completing the step","none",completef,completef,sizeof(completef),&flg);
1156:     if (flg) {
1157:       PetscBool match1,match2;
1158:       PetscStrcmp(completef,"rescale",&match1);
1159:       PetscStrcmp(completef,"rescale-and-modify",&match2);
1160:       if (match1)      gl->CompleteStep = TSGLLECompleteStep_Rescale;
1161:       else if (match2) gl->CompleteStep = TSGLLECompleteStep_RescaleAndModify;
1162:       else SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"%s",completef);
1163:     }
1164:     {
1165:       char type[256] = TSGLLEACCEPT_ALWAYS;
1166:       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);
1167:       if (flg || !gl->accept_name[0]) {
1168:         TSGLLESetAcceptType(ts,type);
1169:       }
1170:     }
1171:     {
1172:       TSGLLEAdapt adapt;
1173:       TSGLLEGetAdapt(ts,&adapt);
1174:       TSGLLEAdaptSetFromOptions(PetscOptionsObject,adapt);
1175:     }
1176:   }
1177:   PetscOptionsTail();
1178:   return(0);
1179: }

1181: static PetscErrorCode TSView_GLLE(TS ts,PetscViewer viewer)
1182: {
1183:   TS_GLLE        *gl = (TS_GLLE*)ts->data;
1184:   PetscInt       i;
1185:   PetscBool      iascii,details;

1189:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
1190:   if (iascii) {
1191:     PetscViewerASCIIPrintf(viewer,"  min order %D, max order %D, current order %D\n",gl->min_order,gl->max_order,gl->schemes[gl->current_scheme]->p);
1192:     PetscViewerASCIIPrintf(viewer,"  Error estimation: %s\n",TSGLLEErrorDirections[gl->error_direction]);
1193:     PetscViewerASCIIPrintf(viewer,"  Extrapolation: %s\n",gl->extrapolate ? "yes" : "no");
1194:     PetscViewerASCIIPrintf(viewer,"  Acceptance test: %s\n",gl->accept_name[0] ? gl->accept_name : "(not yet set)");
1195:     PetscViewerASCIIPushTab(viewer);
1196:     TSGLLEAdaptView(gl->adapt,viewer);
1197:     PetscViewerASCIIPopTab(viewer);
1198:     PetscViewerASCIIPrintf(viewer,"  type: %s\n",gl->type_name[0] ? gl->type_name : "(not yet set)");
1199:     PetscViewerASCIIPrintf(viewer,"Schemes within family (%d):\n",gl->nschemes);
1200:     details = PETSC_FALSE;
1201:     PetscOptionsGetBool(((PetscObject)ts)->options,((PetscObject)ts)->prefix,"-ts_gl_view_detailed",&details,NULL);
1202:     PetscViewerASCIIPushTab(viewer);
1203:     for (i=0; i<gl->nschemes; i++) {
1204:       TSGLLESchemeView(gl->schemes[i],details,viewer);
1205:     }
1206:     if (gl->View) {
1207:       (*gl->View)(gl,viewer);
1208:     }
1209:     PetscViewerASCIIPopTab(viewer);
1210:   }
1211:   return(0);
1212: }

1214: /*@C
1215:    TSGLLERegister -  adds a TSGLLE implementation

1217:    Not Collective

1219:    Input Parameters:
1220: +  name_scheme - name of user-defined general linear scheme
1221: -  routine_create - routine to create method context

1223:    Notes:
1224:    TSGLLERegister() may be called multiple times to add several user-defined families.

1226:    Sample usage:
1227: .vb
1228:    TSGLLERegister("my_scheme",MySchemeCreate);
1229: .ve

1231:    Then, your scheme can be chosen with the procedural interface via
1232: $     TSGLLESetType(ts,"my_scheme")
1233:    or at runtime via the option
1234: $     -ts_gl_type my_scheme

1236:    Level: advanced

1238: .seealso: TSGLLERegisterAll()
1239: @*/
1240: PetscErrorCode  TSGLLERegister(const char sname[],PetscErrorCode (*function)(TS))
1241: {

1245:   TSGLLEInitializePackage();
1246:   PetscFunctionListAdd(&TSGLLEList,sname,function);
1247:   return(0);
1248: }

1250: /*@C
1251:    TSGLLEAcceptRegister -  adds a TSGLLE acceptance scheme

1253:    Not Collective

1255:    Input Parameters:
1256: +  name_scheme - name of user-defined acceptance scheme
1257: -  routine_create - routine to create method context

1259:    Notes:
1260:    TSGLLEAcceptRegister() may be called multiple times to add several user-defined families.

1262:    Sample usage:
1263: .vb
1264:    TSGLLEAcceptRegister("my_scheme",MySchemeCreate);
1265: .ve

1267:    Then, your scheme can be chosen with the procedural interface via
1268: $     TSGLLESetAcceptType(ts,"my_scheme")
1269:    or at runtime via the option
1270: $     -ts_gl_accept_type my_scheme

1272:    Level: advanced

1274: .seealso: TSGLLERegisterAll()
1275: @*/
1276: PetscErrorCode  TSGLLEAcceptRegister(const char sname[],TSGLLEAcceptFunction function)
1277: {

1281:   PetscFunctionListAdd(&TSGLLEAcceptList,sname,function);
1282:   return(0);
1283: }

1285: /*@C
1286:   TSGLLERegisterAll - Registers all of the general linear methods in TSGLLE

1288:   Not Collective

1290:   Level: advanced

1292: .seealso:  TSGLLERegisterDestroy()
1293: @*/
1294: PetscErrorCode  TSGLLERegisterAll(void)
1295: {

1299:   if (TSGLLERegisterAllCalled) return(0);
1300:   TSGLLERegisterAllCalled = PETSC_TRUE;

1302:   TSGLLERegister(TSGLLE_IRKS,              TSGLLECreate_IRKS);
1303:   TSGLLEAcceptRegister(TSGLLEACCEPT_ALWAYS,TSGLLEAccept_Always);
1304:   return(0);
1305: }

1307: /*@C
1308:   TSGLLEInitializePackage - This function initializes everything in the TSGLLE package. It is called
1309:   from TSInitializePackage().

1311:   Level: developer

1313: .seealso: PetscInitialize()
1314: @*/
1315: PetscErrorCode  TSGLLEInitializePackage(void)
1316: {

1320:   if (TSGLLEPackageInitialized) return(0);
1321:   TSGLLEPackageInitialized = PETSC_TRUE;
1322:   TSGLLERegisterAll();
1323:   PetscRegisterFinalize(TSGLLEFinalizePackage);
1324:   return(0);
1325: }

1327: /*@C
1328:   TSGLLEFinalizePackage - This function destroys everything in the TSGLLE package. It is
1329:   called from PetscFinalize().

1331:   Level: developer

1333: .seealso: PetscFinalize()
1334: @*/
1335: PetscErrorCode  TSGLLEFinalizePackage(void)
1336: {

1340:   PetscFunctionListDestroy(&TSGLLEList);
1341:   PetscFunctionListDestroy(&TSGLLEAcceptList);
1342:   TSGLLEPackageInitialized = PETSC_FALSE;
1343:   TSGLLERegisterAllCalled  = PETSC_FALSE;
1344:   return(0);
1345: }

1347: /* ------------------------------------------------------------ */
1348: /*MC
1349:       TSGLLE - DAE solver using implicit General Linear methods

1351:   These methods contain Runge-Kutta and multistep schemes as special cases.  These special cases have some fundamental
1352:   limitations.  For example, diagonally implicit Runge-Kutta cannot have stage order greater than 1 which limits their
1353:   applicability to very stiff systems.  Meanwhile, multistep methods cannot be A-stable for order greater than 2 and BDF
1354:   are not 0-stable for order greater than 6.  GL methods can be A- and L-stable with arbitrarily high stage order and
1355:   reliable error estimates for both 1 and 2 orders higher to facilitate adaptive step sizes and adaptive order schemes.
1356:   All this is possible while preserving a singly diagonally implicit structure.

1358:   Options database keys:
1359: +  -ts_gl_type <type> - the class of general linear method (irks)
1360: .  -ts_gl_rtol <tol>  - relative error
1361: .  -ts_gl_atol <tol>  - absolute error
1362: .  -ts_gl_min_order <p> - minimum order method to consider (default=1)
1363: .  -ts_gl_max_order <p> - maximum order method to consider (default=3)
1364: .  -ts_gl_start_order <p> - order of starting method (default=1)
1365: .  -ts_gl_complete <method> - method to use for completing the step (rescale-and-modify or rescale)
1366: -  -ts_adapt_type <method> - adaptive controller to use (none step both)

1368:   Notes:
1369:   This integrator can be applied to DAE.

1371:   Diagonally implicit general linear (DIGL) methods are a generalization of diagonally implicit Runge-Kutta (DIRK).
1372:   They are represented by the tableau

1374: .vb
1375:   A  |  U
1376:   -------
1377:   B  |  V
1378: .ve

1380:   combined with a vector c of abscissa.  "Diagonally implicit" means that A is lower triangular.
1381:   A step of the general method reads

1383: .vb
1384:   [ Y ] = [A  U] [  Y'   ]
1385:   [X^k] = [B  V] [X^{k-1}]
1386: .ve

1388:   where Y is the multivector of stage values, Y' is the multivector of stage derivatives, X^k is the Nordsieck vector of
1389:   the solution at step k.  The Nordsieck vector consists of the first r moments of the solution, given by

1391: .vb
1392:   X = [x_0,x_1,...,x_{r-1}] = [x, h x', h^2 x'', ..., h^{r-1} x^{(r-1)} ]
1393: .ve

1395:   If A is lower triangular, we can solve the stages (Y,Y') sequentially

1397: .vb
1398:   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}
1399: .ve

1401:   and then construct the pieces to carry to the next step

1403: .vb
1404:   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}
1405: .ve

1407:   Note that when the equations are cast in implicit form, we are using the stage equation to define y'_i
1408:   in terms of y_i and known stuff (y_j for j<i and x_j for all j).


1411:   Error estimation

1413:   At present, the most attractive GL methods for stiff problems are singly diagonally implicit schemes which posses
1414:   Inherent Runge-Kutta Stability (IRKS).  These methods have r=s, the number of items passed between steps is equal to
1415:   the number of stages.  The order and stage-order are one less than the number of stages.  We use the error estimates
1416:   in the 2007 paper which provide the following estimates

1418: .vb
1419:   h^{p+1} X^{(p+1)}          = phi_0^T Y' + [0 psi_0^T] Xold
1420:   h^{p+2} X^{(p+2)}          = phi_1^T Y' + [0 psi_1^T] Xold
1421:   h^{p+2} (dx'/dx) X^{(p+1)} = phi_2^T Y' + [0 psi_2^T] Xold
1422: .ve

1424:   These estimates are accurate to O(h^{p+3}).

1426:   Changing the step size

1428:   We use the generalized "rescale and modify" scheme, see equation (4.5) of the 2007 paper.

1430:   Level: beginner

1432:   References:
1433: +  1. - John Butcher and Z. Jackieweicz and W. Wright, On error propagation in general linear methods for
1434:   ordinary differential equations, Journal of Complexity, Vol 23, 2007.
1435: -  2. - John Butcher, Numerical methods for ordinary differential equations, second edition, Wiley, 2009.

1437: .seealso:  TSCreate(), TS, TSSetType()

1439: M*/
1440: PETSC_EXTERN PetscErrorCode TSCreate_GLLE(TS ts)
1441: {
1442:   TS_GLLE        *gl;

1446:   TSGLLEInitializePackage();

1448:   PetscNewLog(ts,&gl);
1449:   ts->data = (void*)gl;

1451:   ts->ops->reset          = TSReset_GLLE;
1452:   ts->ops->destroy        = TSDestroy_GLLE;
1453:   ts->ops->view           = TSView_GLLE;
1454:   ts->ops->setup          = TSSetUp_GLLE;
1455:   ts->ops->solve          = TSSolve_GLLE;
1456:   ts->ops->setfromoptions = TSSetFromOptions_GLLE;
1457:   ts->ops->snesfunction   = SNESTSFormFunction_GLLE;
1458:   ts->ops->snesjacobian   = SNESTSFormJacobian_GLLE;

1460:   ts->usessnes = PETSC_TRUE;


1463:   gl->max_step_rejections = 1;
1464:   gl->min_order           = 1;
1465:   gl->max_order           = 3;
1466:   gl->start_order         = 1;
1467:   gl->current_scheme      = -1;
1468:   gl->extrapolate         = PETSC_FALSE;

1470:   gl->wrms_atol = 1e-8;
1471:   gl->wrms_rtol = 1e-5;

1473:   PetscObjectComposeFunction((PetscObject)ts,"TSGLLESetType_C",      &TSGLLESetType_GLLE);
1474:   PetscObjectComposeFunction((PetscObject)ts,"TSGLLESetAcceptType_C",&TSGLLESetAcceptType_GLLE);
1475:   PetscObjectComposeFunction((PetscObject)ts,"TSGLLEGetAdapt_C",     &TSGLLEGetAdapt_GLLE);
1476:   return(0);
1477: }