Actual source code: basicsymplectic.c
1: /*
2: Code for Timestepping with basic symplectic integrators for separable Hamiltonian systems
3: */
4: #include <petsc/private/tsimpl.h>
5: #include <petscdm.h>
7: static TSBasicSymplecticType TSBasicSymplecticDefault = TSBASICSYMPLECTICSIEULER;
8: static PetscBool TSBasicSymplecticRegisterAllCalled;
9: static PetscBool TSBasicSymplecticPackageInitialized;
11: typedef struct _BasicSymplecticScheme *BasicSymplecticScheme;
12: typedef struct _BasicSymplecticSchemeLink *BasicSymplecticSchemeLink;
14: struct _BasicSymplecticScheme {
15: char *name;
16: PetscInt order;
17: PetscInt s; /* number of stages */
18: PetscReal *c, *d;
19: };
20: struct _BasicSymplecticSchemeLink {
21: struct _BasicSymplecticScheme sch;
22: BasicSymplecticSchemeLink next;
23: };
24: static BasicSymplecticSchemeLink BasicSymplecticSchemeList;
25: typedef struct {
26: TS subts_p, subts_q; /* sub TS contexts that holds the RHSFunction pointers */
27: IS is_p, is_q; /* IS sets for position and momentum respectively */
28: Vec update; /* a nest work vector for generalized coordinates */
29: BasicSymplecticScheme scheme;
30: } TS_BasicSymplectic;
32: /*MC
33: TSBASICSYMPLECTICSIEULER - first order semi-implicit Euler method
35: Level: intermediate
37: .seealso: [](chapter_ts), `TSBASICSYMPLECTIC`
38: M*/
40: /*MC
41: TSBASICSYMPLECTICVELVERLET - second order Velocity Verlet method (leapfrog method with starting process and determining velocity and position at the same time)
43: Level: intermediate
45: .seealso: [](chapter_ts), `TSBASICSYMPLECTIC`
46: M*/
48: /*@C
49: TSBasicSymplecticRegisterAll - Registers all of the basic symplectic integration methods in `TSBASICSYMPLECTIC`
51: Not Collective, but should be called by all processes which will need the schemes to be registered
53: Level: advanced
55: .seealso: [](chapter_ts), `TSBASICSYMPLECTIC`, `TSBasicSymplecticRegisterDestroy()`
56: @*/
57: PetscErrorCode TSBasicSymplecticRegisterAll(void)
58: {
59: PetscFunctionBegin;
60: if (TSBasicSymplecticRegisterAllCalled) PetscFunctionReturn(PETSC_SUCCESS);
61: TSBasicSymplecticRegisterAllCalled = PETSC_TRUE;
62: {
63: PetscReal c[1] = {1.0}, d[1] = {1.0};
64: PetscCall(TSBasicSymplecticRegister(TSBASICSYMPLECTICSIEULER, 1, 1, c, d));
65: }
66: {
67: PetscReal c[2] = {0, 1.0}, d[2] = {0.5, 0.5};
68: PetscCall(TSBasicSymplecticRegister(TSBASICSYMPLECTICVELVERLET, 2, 2, c, d));
69: }
70: {
71: PetscReal c[3] = {1, -2.0 / 3.0, 2.0 / 3.0}, d[3] = {-1.0 / 24.0, 3.0 / 4.0, 7.0 / 24.0};
72: PetscCall(TSBasicSymplecticRegister(TSBASICSYMPLECTIC3, 3, 3, c, d));
73: }
74: {
75: #define CUBE../../../../..OFTWO 1.2599210498948731647672106
76: PetscReal c[4] = {1.0 / 2.0 / (2.0 - CUBE../../../../..OFTWO), (1.0 - CUBE../../../../..OFTWO) / 2.0 / (2.0 - CUBE../../../../..OFTWO), (1.0 - CUBE../../../../..OFTWO) / 2.0 / (2.0 - CUBE../../../../..OFTWO), 1.0 / 2.0 / (2.0 - CUBE../../../../..OFTWO)}, d[4] = {1.0 / (2.0 - CUBE../../../../..OFTWO), -CUBE../../../../..OFTWO / (2.0 - CUBE../../../../..OFTWO), 1.0 / (2.0 - CUBE../../../../..OFTWO), 0};
77: PetscCall(TSBasicSymplecticRegister(TSBASICSYMPLECTIC4, 4, 4, c, d));
78: }
79: PetscFunctionReturn(PETSC_SUCCESS);
80: }
82: /*@C
83: TSBasicSymplecticRegisterDestroy - Frees the list of schemes that were registered by `TSBasicSymplecticRegister()`.
85: Not Collective
87: Level: advanced
89: .seealso: [](chapter_ts), `TSBasicSymplecticRegister()`, `TSBasicSymplecticRegisterAll()`, `TSBASICSYMPLECTIC`
90: @*/
91: PetscErrorCode TSBasicSymplecticRegisterDestroy(void)
92: {
93: BasicSymplecticSchemeLink link;
95: PetscFunctionBegin;
96: while ((link = BasicSymplecticSchemeList)) {
97: BasicSymplecticScheme scheme = &link->sch;
98: BasicSymplecticSchemeList = link->next;
99: PetscCall(PetscFree2(scheme->c, scheme->d));
100: PetscCall(PetscFree(scheme->name));
101: PetscCall(PetscFree(link));
102: }
103: TSBasicSymplecticRegisterAllCalled = PETSC_FALSE;
104: PetscFunctionReturn(PETSC_SUCCESS);
105: }
107: /*@C
108: TSBasicSymplecticInitializePackage - This function initializes everything in the `TSBASICSYMPLECTIC` package. It is called
109: from `TSInitializePackage()`.
111: Level: developer
113: .seealso: [](chapter_ts), `PetscInitialize()`, `TSBASICSYMPLECTIC`
114: @*/
115: PetscErrorCode TSBasicSymplecticInitializePackage(void)
116: {
117: PetscFunctionBegin;
118: if (TSBasicSymplecticPackageInitialized) PetscFunctionReturn(PETSC_SUCCESS);
119: TSBasicSymplecticPackageInitialized = PETSC_TRUE;
120: PetscCall(TSBasicSymplecticRegisterAll());
121: PetscCall(PetscRegisterFinalize(TSBasicSymplecticFinalizePackage));
122: PetscFunctionReturn(PETSC_SUCCESS);
123: }
125: /*@C
126: TSBasicSymplecticFinalizePackage - This function destroys everything in the `TSBASICSYMPLECTIC` package. It is
127: called from `PetscFinalize()`.
129: Level: developer
131: .seealso: [](chapter_ts), `PetscFinalize()`, `TSBASICSYMPLECTIC`
132: @*/
133: PetscErrorCode TSBasicSymplecticFinalizePackage(void)
134: {
135: PetscFunctionBegin;
136: TSBasicSymplecticPackageInitialized = PETSC_FALSE;
137: PetscCall(TSBasicSymplecticRegisterDestroy());
138: PetscFunctionReturn(PETSC_SUCCESS);
139: }
141: /*@C
142: TSBasicSymplecticRegister - register a basic symplectic integration scheme by providing the coefficients.
144: Not Collective, but the same schemes should be registered on all processes on which they will be used
146: Input Parameters:
147: + name - identifier for method
148: . order - approximation order of method
149: . s - number of stages, this is the dimension of the matrices below
150: . c - coefficients for updating generalized position (dimension s)
151: - d - coefficients for updating generalized momentum (dimension s)
153: Level: advanced
155: Notes:
156: Several symplectic methods are provided, this function is only needed to create new methods.
158: .seealso: [](chapter_ts), `TSBASICSYMPLECTIC`
159: @*/
160: PetscErrorCode TSBasicSymplecticRegister(TSRosWType name, PetscInt order, PetscInt s, PetscReal c[], PetscReal d[])
161: {
162: BasicSymplecticSchemeLink link;
163: BasicSymplecticScheme scheme;
165: PetscFunctionBegin;
170: PetscCall(TSBasicSymplecticInitializePackage());
171: PetscCall(PetscNew(&link));
172: scheme = &link->sch;
173: PetscCall(PetscStrallocpy(name, &scheme->name));
174: scheme->order = order;
175: scheme->s = s;
176: PetscCall(PetscMalloc2(s, &scheme->c, s, &scheme->d));
177: PetscCall(PetscArraycpy(scheme->c, c, s));
178: PetscCall(PetscArraycpy(scheme->d, d, s));
179: link->next = BasicSymplecticSchemeList;
180: BasicSymplecticSchemeList = link;
181: PetscFunctionReturn(PETSC_SUCCESS);
182: }
184: /*
185: The simplified form of the equations are:
187: $ p_{i+1} = p_i + c_i*g(q_i)*h
188: $ q_{i+1} = q_i + d_i*f(p_{i+1},t_{i+1})*h
190: Several symplectic integrators are given below. An illustrative way to use them is to consider a particle with position q and velocity p.
192: To apply a timestep with values c_{1,2},d_{1,2} to the particle, carry out the following steps:
194: - Update the velocity of the particle by adding to it its acceleration multiplied by c_1
195: - Update the position of the particle by adding to it its (updated) velocity multiplied by d_1
196: - Update the velocity of the particle by adding to it its acceleration (at the updated position) multiplied by c_2
197: - Update the position of the particle by adding to it its (double-updated) velocity multiplied by d_2
199: */
200: static PetscErrorCode TSStep_BasicSymplectic(TS ts)
201: {
202: TS_BasicSymplectic *bsymp = (TS_BasicSymplectic *)ts->data;
203: BasicSymplecticScheme scheme = bsymp->scheme;
204: Vec solution = ts->vec_sol, update = bsymp->update, q, p, q_update, p_update;
205: IS is_q = bsymp->is_q, is_p = bsymp->is_p;
206: TS subts_q = bsymp->subts_q, subts_p = bsymp->subts_p;
207: PetscBool stageok;
208: PetscReal next_time_step = ts->time_step;
209: PetscInt iter;
211: PetscFunctionBegin;
212: PetscCall(VecGetSubVector(solution, is_q, &q));
213: PetscCall(VecGetSubVector(solution, is_p, &p));
214: PetscCall(VecGetSubVector(update, is_q, &q_update));
215: PetscCall(VecGetSubVector(update, is_p, &p_update));
217: for (iter = 0; iter < scheme->s; iter++) {
218: PetscCall(TSPreStage(ts, ts->ptime));
219: /* update velocity p */
220: if (scheme->c[iter]) {
221: PetscCall(TSComputeRHSFunction(subts_p, ts->ptime, q, p_update));
222: PetscCall(VecAXPY(p, scheme->c[iter] * ts->time_step, p_update));
223: }
224: /* update position q */
225: if (scheme->d[iter]) {
226: PetscCall(TSComputeRHSFunction(subts_q, ts->ptime, p, q_update));
227: PetscCall(VecAXPY(q, scheme->d[iter] * ts->time_step, q_update));
228: ts->ptime = ts->ptime + scheme->d[iter] * ts->time_step;
229: }
230: PetscCall(TSPostStage(ts, ts->ptime, 0, &solution));
231: PetscCall(TSAdaptCheckStage(ts->adapt, ts, ts->ptime, solution, &stageok));
232: if (!stageok) {
233: ts->reason = TS_DIVERGED_STEP_REJECTED;
234: PetscFunctionReturn(PETSC_SUCCESS);
235: }
236: PetscCall(TSFunctionDomainError(ts, ts->ptime + ts->time_step, update, &stageok));
237: if (!stageok) {
238: ts->reason = TS_DIVERGED_STEP_REJECTED;
239: PetscFunctionReturn(PETSC_SUCCESS);
240: }
241: }
243: ts->time_step = next_time_step;
244: PetscCall(VecRestoreSubVector(solution, is_q, &q));
245: PetscCall(VecRestoreSubVector(solution, is_p, &p));
246: PetscCall(VecRestoreSubVector(update, is_q, &q_update));
247: PetscCall(VecRestoreSubVector(update, is_p, &p_update));
248: PetscFunctionReturn(PETSC_SUCCESS);
249: }
251: static PetscErrorCode DMCoarsenHook_BasicSymplectic(DM fine, DM coarse, void *ctx)
252: {
253: PetscFunctionBegin;
254: PetscFunctionReturn(PETSC_SUCCESS);
255: }
257: static PetscErrorCode DMRestrictHook_BasicSymplectic(DM fine, Mat restrct, Vec rscale, Mat inject, DM coarse, void *ctx)
258: {
259: PetscFunctionBegin;
260: PetscFunctionReturn(PETSC_SUCCESS);
261: }
263: static PetscErrorCode DMSubDomainHook_BasicSymplectic(DM dm, DM subdm, void *ctx)
264: {
265: PetscFunctionBegin;
266: PetscFunctionReturn(PETSC_SUCCESS);
267: }
269: static PetscErrorCode DMSubDomainRestrictHook_BasicSymplectic(DM dm, VecScatter gscat, VecScatter lscat, DM subdm, void *ctx)
270: {
271: PetscFunctionBegin;
272: PetscFunctionReturn(PETSC_SUCCESS);
273: }
275: static PetscErrorCode TSSetUp_BasicSymplectic(TS ts)
276: {
277: TS_BasicSymplectic *bsymp = (TS_BasicSymplectic *)ts->data;
278: DM dm;
280: PetscFunctionBegin;
281: PetscCall(TSRHSSplitGetIS(ts, "position", &bsymp->is_q));
282: PetscCall(TSRHSSplitGetIS(ts, "momentum", &bsymp->is_p));
283: PetscCheck(bsymp->is_q && bsymp->is_p, PetscObjectComm((PetscObject)ts), PETSC_ERR_USER, "Must set up RHSSplits with TSRHSSplitSetIS() using split names position and momentum respectively in order to use -ts_type basicsymplectic");
284: PetscCall(TSRHSSplitGetSubTS(ts, "position", &bsymp->subts_q));
285: PetscCall(TSRHSSplitGetSubTS(ts, "momentum", &bsymp->subts_p));
286: PetscCheck(bsymp->subts_q && bsymp->subts_p, PetscObjectComm((PetscObject)ts), PETSC_ERR_USER, "Must set up the RHSFunctions for position and momentum using TSRHSSplitSetRHSFunction() or calling TSSetRHSFunction() for each sub-TS");
288: PetscCall(VecDuplicate(ts->vec_sol, &bsymp->update));
290: PetscCall(TSGetAdapt(ts, &ts->adapt));
291: PetscCall(TSAdaptCandidatesClear(ts->adapt)); /* make sure to use fixed time stepping */
292: PetscCall(TSGetDM(ts, &dm));
293: if (dm) {
294: PetscCall(DMCoarsenHookAdd(dm, DMCoarsenHook_BasicSymplectic, DMRestrictHook_BasicSymplectic, ts));
295: PetscCall(DMSubDomainHookAdd(dm, DMSubDomainHook_BasicSymplectic, DMSubDomainRestrictHook_BasicSymplectic, ts));
296: }
297: PetscFunctionReturn(PETSC_SUCCESS);
298: }
300: static PetscErrorCode TSReset_BasicSymplectic(TS ts)
301: {
302: TS_BasicSymplectic *bsymp = (TS_BasicSymplectic *)ts->data;
304: PetscFunctionBegin;
305: PetscCall(VecDestroy(&bsymp->update));
306: PetscFunctionReturn(PETSC_SUCCESS);
307: }
309: static PetscErrorCode TSDestroy_BasicSymplectic(TS ts)
310: {
311: PetscFunctionBegin;
312: PetscCall(TSReset_BasicSymplectic(ts));
313: PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSBasicSymplecticSetType_C", NULL));
314: PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSBasicSymplecticGetType_C", NULL));
315: PetscCall(PetscFree(ts->data));
316: PetscFunctionReturn(PETSC_SUCCESS);
317: }
319: static PetscErrorCode TSSetFromOptions_BasicSymplectic(TS ts, PetscOptionItems *PetscOptionsObject)
320: {
321: TS_BasicSymplectic *bsymp = (TS_BasicSymplectic *)ts->data;
323: PetscFunctionBegin;
324: PetscOptionsHeadBegin(PetscOptionsObject, "Basic symplectic integrator options");
325: {
326: BasicSymplecticSchemeLink link;
327: PetscInt count, choice;
328: PetscBool flg;
329: const char **namelist;
331: for (link = BasicSymplecticSchemeList, count = 0; link; link = link->next, count++)
332: ;
333: PetscCall(PetscMalloc1(count, (char ***)&namelist));
334: for (link = BasicSymplecticSchemeList, count = 0; link; link = link->next, count++) namelist[count] = link->sch.name;
335: PetscCall(PetscOptionsEList("-ts_basicsymplectic_type", "Family of basic symplectic integration method", "TSBasicSymplecticSetType", (const char *const *)namelist, count, bsymp->scheme->name, &choice, &flg));
336: if (flg) PetscCall(TSBasicSymplecticSetType(ts, namelist[choice]));
337: PetscCall(PetscFree(namelist));
338: }
339: PetscOptionsHeadEnd();
340: PetscFunctionReturn(PETSC_SUCCESS);
341: }
343: static PetscErrorCode TSView_BasicSymplectic(TS ts, PetscViewer viewer)
344: {
345: PetscFunctionBegin;
346: PetscFunctionReturn(PETSC_SUCCESS);
347: }
349: static PetscErrorCode TSInterpolate_BasicSymplectic(TS ts, PetscReal t, Vec X)
350: {
351: TS_BasicSymplectic *bsymp = (TS_BasicSymplectic *)ts->data;
352: Vec update = bsymp->update;
353: PetscReal alpha = (ts->ptime - t) / ts->time_step;
355: PetscFunctionBegin;
356: PetscCall(VecWAXPY(X, -ts->time_step, update, ts->vec_sol));
357: PetscCall(VecAXPBY(X, 1.0 - alpha, alpha, ts->vec_sol));
358: PetscFunctionReturn(PETSC_SUCCESS);
359: }
361: static PetscErrorCode TSComputeLinearStability_BasicSymplectic(TS ts, PetscReal xr, PetscReal xi, PetscReal *yr, PetscReal *yi)
362: {
363: PetscFunctionBegin;
364: *yr = 1.0 + xr;
365: *yi = xi;
366: PetscFunctionReturn(PETSC_SUCCESS);
367: }
369: /*@C
370: TSBasicSymplecticSetType - Set the type of the basic symplectic method
372: Logically Collective
374: Input Parameters:
375: + ts - timestepping context
376: - bsymptype - type of the symplectic scheme
378: Options Database Key:
379: . -ts_basicsymplectic_type <scheme> - select the scheme
381: Level: intermediate
383: Note:
384: The symplectic solver always expects a two-way splitting with the split names being "position" and "momentum". Each split is associated with an `IS` object and a sub-`TS`
385: that is intended to store the user-provided RHS function.
387: .seealso: [](chapter_ts), `TSBASICSYMPLECTIC`, `TSBasicSymplecticType`, `TSBasicSymplecticSetType()`
388: @*/
389: PetscErrorCode TSBasicSymplecticSetType(TS ts, TSBasicSymplecticType bsymptype)
390: {
391: PetscFunctionBegin;
393: PetscTryMethod(ts, "TSBasicSymplecticSetType_C", (TS, TSBasicSymplecticType), (ts, bsymptype));
394: PetscFunctionReturn(PETSC_SUCCESS);
395: }
397: /*@C
398: TSBasicSymplecticGetType - Get the type of the basic symplectic method
400: Logically Collective
402: Input Parameters:
403: + ts - timestepping context
404: - bsymptype - type of the basic symplectic scheme
406: Level: intermediate
408: .seealso: [](chapter_ts), `TSBASICSYMPLECTIC`, `TSBasicSymplecticType`, `TSBasicSymplecticSetType()`
409: @*/
410: PetscErrorCode TSBasicSymplecticGetType(TS ts, TSBasicSymplecticType *bsymptype)
411: {
412: PetscFunctionBegin;
414: PetscUseMethod(ts, "TSBasicSymplecticGetType_C", (TS, TSBasicSymplecticType *), (ts, bsymptype));
415: PetscFunctionReturn(PETSC_SUCCESS);
416: }
418: static PetscErrorCode TSBasicSymplecticSetType_BasicSymplectic(TS ts, TSBasicSymplecticType bsymptype)
419: {
420: TS_BasicSymplectic *bsymp = (TS_BasicSymplectic *)ts->data;
421: BasicSymplecticSchemeLink link;
422: PetscBool match;
424: PetscFunctionBegin;
425: if (bsymp->scheme) {
426: PetscCall(PetscStrcmp(bsymp->scheme->name, bsymptype, &match));
427: if (match) PetscFunctionReturn(PETSC_SUCCESS);
428: }
429: for (link = BasicSymplecticSchemeList; link; link = link->next) {
430: PetscCall(PetscStrcmp(link->sch.name, bsymptype, &match));
431: if (match) {
432: bsymp->scheme = &link->sch;
433: PetscFunctionReturn(PETSC_SUCCESS);
434: }
435: }
436: SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_ARG_UNKNOWN_TYPE, "Could not find '%s'", bsymptype);
437: }
439: static PetscErrorCode TSBasicSymplecticGetType_BasicSymplectic(TS ts, TSBasicSymplecticType *bsymptype)
440: {
441: TS_BasicSymplectic *bsymp = (TS_BasicSymplectic *)ts->data;
443: PetscFunctionBegin;
444: *bsymptype = bsymp->scheme->name;
445: PetscFunctionReturn(PETSC_SUCCESS);
446: }
448: /*MC
449: TSBASICSYMPLECTIC - ODE solver using basic symplectic integration schemes
451: These methods are intended for separable Hamiltonian systems
452: .vb
453: qdot = dH(q,p,t)/dp
454: pdot = -dH(q,p,t)/dq
455: .ve
457: where the Hamiltonian can be split into the sum of kinetic energy and potential energy
458: .vb
459: H(q,p,t) = T(p,t) + V(q,t).
460: .ve
462: As a result, the system can be genearlly represented by
463: .vb
464: qdot = f(p,t) = dT(p,t)/dp
465: pdot = g(q,t) = -dV(q,t)/dq
466: .ve
468: and solved iteratively with
469: .vb
470: q_new = q_old + d_i*h*f(p_old,t_old)
471: t_new = t_old + d_i*h
472: p_new = p_old + c_i*h*g(p_new,t_new)
473: i=0,1,...,n.
474: .ve
476: The solution vector should contain both q and p, which correspond to (generalized) position and momentum respectively. Note that the momentum component
477: could simply be velocity in some representations. The symplectic solver always expects a two-way splitting with the split names being "position" and "momentum".
478: Each split is associated with an `IS` object and a sub-`TS` that is intended to store the user-provided RHS function.
480: Level: beginner
482: Reference:
483: . * - wikipedia (https://en.wikipedia.org/wiki/Symplectic_integrator)
485: .seealso: [](chapter_ts), `TSCreate()`, `TSSetType()`, `TSRHSSplitSetIS()`, `TSRHSSplitSetRHSFunction()`, `TSType`
486: M*/
487: PETSC_EXTERN PetscErrorCode TSCreate_BasicSymplectic(TS ts)
488: {
489: TS_BasicSymplectic *bsymp;
491: PetscFunctionBegin;
492: PetscCall(TSBasicSymplecticInitializePackage());
493: PetscCall(PetscNew(&bsymp));
494: ts->data = (void *)bsymp;
496: ts->ops->setup = TSSetUp_BasicSymplectic;
497: ts->ops->step = TSStep_BasicSymplectic;
498: ts->ops->reset = TSReset_BasicSymplectic;
499: ts->ops->destroy = TSDestroy_BasicSymplectic;
500: ts->ops->setfromoptions = TSSetFromOptions_BasicSymplectic;
501: ts->ops->view = TSView_BasicSymplectic;
502: ts->ops->interpolate = TSInterpolate_BasicSymplectic;
503: ts->ops->linearstability = TSComputeLinearStability_BasicSymplectic;
505: PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSBasicSymplecticSetType_C", TSBasicSymplecticSetType_BasicSymplectic));
506: PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSBasicSymplecticGetType_C", TSBasicSymplecticGetType_BasicSymplectic));
508: PetscCall(TSBasicSymplecticSetType(ts, TSBasicSymplecticDefault));
509: PetscFunctionReturn(PETSC_SUCCESS);
510: }