Actual source code: rosw.c
1: /*
2: Code for timestepping with Rosenbrock W methods
4: Notes:
5: The general system is written as
7: F(t,U,Udot) = G(t,U)
9: where F represents the stiff part of the physics and G represents the non-stiff part.
10: This method is designed to be linearly implicit on F and can use an approximate and lagged Jacobian.
12: */
13: #include <petsc/private/tsimpl.h>
14: #include <petscdm.h>
16: #include <petsc/private/kernels/blockinvert.h>
18: static TSRosWType TSRosWDefault = TSROSWRA34PW2;
19: static PetscBool TSRosWRegisterAllCalled;
20: static PetscBool TSRosWPackageInitialized;
22: typedef struct _RosWTableau *RosWTableau;
23: struct _RosWTableau {
24: char *name;
25: PetscInt order; /* Classical approximation order of the method */
26: PetscInt s; /* Number of stages */
27: PetscInt pinterp; /* Interpolation order */
28: PetscReal *A; /* Propagation table, strictly lower triangular */
29: PetscReal *Gamma; /* Stage table, lower triangular with nonzero diagonal */
30: PetscBool *GammaZeroDiag; /* Diagonal entries that are zero in stage table Gamma, vector indicating explicit statages */
31: PetscReal *GammaExplicitCorr; /* Coefficients for correction terms needed for explicit stages in transformed variables*/
32: PetscReal *b; /* Step completion table */
33: PetscReal *bembed; /* Step completion table for embedded method of order one less */
34: PetscReal *ASum; /* Row sum of A */
35: PetscReal *GammaSum; /* Row sum of Gamma, only needed for non-autonomous systems */
36: PetscReal *At; /* Propagation table in transformed variables */
37: PetscReal *bt; /* Step completion table in transformed variables */
38: PetscReal *bembedt; /* Step completion table of order one less in transformed variables */
39: PetscReal *GammaInv; /* Inverse of Gamma, used for transformed variables */
40: PetscReal ccfl; /* Placeholder for CFL coefficient relative to forward Euler */
41: PetscReal *binterpt; /* Dense output formula */
42: };
43: typedef struct _RosWTableauLink *RosWTableauLink;
44: struct _RosWTableauLink {
45: struct _RosWTableau tab;
46: RosWTableauLink next;
47: };
48: static RosWTableauLink RosWTableauList;
50: typedef struct {
51: RosWTableau tableau;
52: Vec *Y; /* States computed during the step, used to complete the step */
53: Vec Ydot; /* Work vector holding Ydot during residual evaluation */
54: Vec Ystage; /* Work vector for the state value at each stage */
55: Vec Zdot; /* Ydot = Zdot + shift*Y */
56: Vec Zstage; /* Y = Zstage + Y */
57: Vec vec_sol_prev; /* Solution from the previous step (used for interpolation and rollback)*/
58: PetscScalar *work; /* Scalar work space of length number of stages, used to prepare VecMAXPY() */
59: PetscReal scoeff; /* shift = scoeff/dt */
60: PetscReal stage_time;
61: PetscReal stage_explicit; /* Flag indicates that the current stage is explicit */
62: PetscBool recompute_jacobian; /* Recompute the Jacobian at each stage, default is to freeze the Jacobian at the start of each step */
63: TSStepStatus status;
64: } TS_RosW;
66: /*MC
67: TSROSWTHETA1 - One stage first order L-stable Rosenbrock-W scheme (aka theta method).
69: Only an approximate Jacobian is needed.
71: Level: intermediate
73: .seealso: TSROSW
74: M*/
76: /*MC
77: TSROSWTHETA2 - One stage second order A-stable Rosenbrock-W scheme (aka theta method).
79: Only an approximate Jacobian is needed.
81: Level: intermediate
83: .seealso: TSROSW
84: M*/
86: /*MC
87: TSROSW2M - Two stage second order L-stable Rosenbrock-W scheme.
89: Only an approximate Jacobian is needed. By default, it is only recomputed once per step. This method is a reflection of TSROSW2P.
91: Level: intermediate
93: .seealso: TSROSW
94: M*/
96: /*MC
97: TSROSW2P - Two stage second order L-stable Rosenbrock-W scheme.
99: Only an approximate Jacobian is needed. By default, it is only recomputed once per step. This method is a reflection of TSROSW2M.
101: Level: intermediate
103: .seealso: TSROSW
104: M*/
106: /*MC
107: TSROSWRA3PW - Three stage third order Rosenbrock-W scheme for PDAE of index 1.
109: Only an approximate Jacobian is needed. By default, it is only recomputed once per step.
111: This is strongly A-stable with R(infty) = 0.73. The embedded method of order 2 is strongly A-stable with R(infty) = 0.73.
113: References:
114: . * - Rang and Angermann, New Rosenbrock W methods of order 3 for partial differential algebraic equations of index 1, 2005.
116: Level: intermediate
118: .seealso: TSROSW
119: M*/
121: /*MC
122: TSROSWRA34PW2 - Four stage third order L-stable Rosenbrock-W scheme for PDAE of index 1.
124: Only an approximate Jacobian is needed. By default, it is only recomputed once per step.
126: This is strongly A-stable with R(infty) = 0. The embedded method of order 2 is strongly A-stable with R(infty) = 0.48.
128: References:
129: . * - Rang and Angermann, New Rosenbrock W methods of order 3 for partial differential algebraic equations of index 1, 2005.
131: Level: intermediate
133: .seealso: TSROSW
134: M*/
136: /*MC
137: TSROSWRODAS3 - Four stage third order L-stable Rosenbrock scheme
139: By default, the Jacobian is only recomputed once per step.
141: Both the third order and embedded second order methods are stiffly accurate and L-stable.
143: References:
144: . * - Sandu et al, Benchmarking stiff ODE solvers for atmospheric chemistry problems II, Rosenbrock solvers, 1997.
146: Level: intermediate
148: .seealso: TSROSW, TSROSWSANDU3
149: M*/
151: /*MC
152: TSROSWSANDU3 - Three stage third order L-stable Rosenbrock scheme
154: By default, the Jacobian is only recomputed once per step.
156: The third order method is L-stable, but not stiffly accurate.
157: The second order embedded method is strongly A-stable with R(infty) = 0.5.
158: The internal stages are L-stable.
159: This method is called ROS3 in the paper.
161: References:
162: . * - Sandu et al, Benchmarking stiff ODE solvers for atmospheric chemistry problems II, Rosenbrock solvers, 1997.
164: Level: intermediate
166: .seealso: TSROSW, TSROSWRODAS3
167: M*/
169: /*MC
170: TSROSWASSP3P3S1C - A-stable Rosenbrock-W method with SSP explicit part, third order, three stages
172: By default, the Jacobian is only recomputed once per step.
174: A-stable SPP explicit order 3, 3 stages, CFL 1 (eff = 1/3)
176: References:
177: . * - Emil Constantinescu
179: Level: intermediate
181: .seealso: TSROSW, TSROSWLASSP3P4S2C, TSROSWLLSSP3P4S2C, SSP
182: M*/
184: /*MC
185: TSROSWLASSP3P4S2C - L-stable Rosenbrock-W method with SSP explicit part, third order, four stages
187: By default, the Jacobian is only recomputed once per step.
189: L-stable (A-stable embedded) SPP explicit order 3, 4 stages, CFL 2 (eff = 1/2)
191: References:
192: . * - Emil Constantinescu
194: Level: intermediate
196: .seealso: TSROSW, TSROSWASSP3P3S1C, TSROSWLLSSP3P4S2C, TSSSP
197: M*/
199: /*MC
200: TSROSWLLSSP3P4S2C - L-stable Rosenbrock-W method with SSP explicit part, third order, four stages
202: By default, the Jacobian is only recomputed once per step.
204: L-stable (L-stable embedded) SPP explicit order 3, 4 stages, CFL 2 (eff = 1/2)
206: References:
207: . * - Emil Constantinescu
209: Level: intermediate
211: .seealso: TSROSW, TSROSWASSP3P3S1C, TSROSWLASSP3P4S2C, TSSSP
212: M*/
214: /*MC
215: TSROSWGRK4T - four stage, fourth order Rosenbrock (not W) method from Kaps and Rentrop
217: By default, the Jacobian is only recomputed once per step.
219: A(89.3 degrees)-stable, |R(infty)| = 0.454.
221: This method does not provide a dense output formula.
223: References:
224: + * - Kaps and Rentrop, Generalized Runge Kutta methods of order four with stepsize control for stiff ordinary differential equations, 1979.
225: - * - Hairer and Wanner, Solving Ordinary Differential Equations II, Section 4 Table 7.2.
227: Hairer's code ros4.f
229: Level: intermediate
231: .seealso: TSROSW, TSROSWSHAMP4, TSROSWVELDD4, TSROSW4L
232: M*/
234: /*MC
235: TSROSWSHAMP4 - four stage, fourth order Rosenbrock (not W) method from Shampine
237: By default, the Jacobian is only recomputed once per step.
239: A-stable, |R(infty)| = 1/3.
241: This method does not provide a dense output formula.
243: References:
244: + * - Shampine, Implementation of Rosenbrock methods, 1982.
245: - * - Hairer and Wanner, Solving Ordinary Differential Equations II, Section 4 Table 7.2.
247: Hairer's code ros4.f
249: Level: intermediate
251: .seealso: TSROSW, TSROSWGRK4T, TSROSWVELDD4, TSROSW4L
252: M*/
254: /*MC
255: TSROSWVELDD4 - four stage, fourth order Rosenbrock (not W) method from van Veldhuizen
257: By default, the Jacobian is only recomputed once per step.
259: A(89.5 degrees)-stable, |R(infty)| = 0.24.
261: This method does not provide a dense output formula.
263: References:
264: + * - van Veldhuizen, D stability and Kaps Rentrop methods, 1984.
265: - * - Hairer and Wanner, Solving Ordinary Differential Equations II, Section 4 Table 7.2.
267: Hairer's code ros4.f
269: Level: intermediate
271: .seealso: TSROSW, TSROSWGRK4T, TSROSWSHAMP4, TSROSW4L
272: M*/
274: /*MC
275: TSROSW4L - four stage, fourth order Rosenbrock (not W) method
277: By default, the Jacobian is only recomputed once per step.
279: A-stable and L-stable
281: This method does not provide a dense output formula.
283: References:
284: . * - Hairer and Wanner, Solving Ordinary Differential Equations II, Section 4 Table 7.2.
286: Hairer's code ros4.f
288: Level: intermediate
290: .seealso: TSROSW, TSROSWGRK4T, TSROSWSHAMP4, TSROSW4L
291: M*/
293: /*@C
294: TSRosWRegisterAll - Registers all of the Rosenbrock-W methods in TSRosW
296: Not Collective, but should be called by all processes which will need the schemes to be registered
298: Level: advanced
300: .seealso: TSRosWRegisterDestroy()
301: @*/
302: PetscErrorCode TSRosWRegisterAll(void)
303: {
304: if (TSRosWRegisterAllCalled) return 0;
305: TSRosWRegisterAllCalled = PETSC_TRUE;
307: {
308: const PetscReal A = 0;
309: const PetscReal Gamma = 1;
310: const PetscReal b = 1;
311: const PetscReal binterpt=1;
313: TSRosWRegister(TSROSWTHETA1,1,1,&A,&Gamma,&b,NULL,1,&binterpt);
314: }
316: {
317: const PetscReal A = 0;
318: const PetscReal Gamma = 0.5;
319: const PetscReal b = 1;
320: const PetscReal binterpt=1;
322: TSRosWRegister(TSROSWTHETA2,2,1,&A,&Gamma,&b,NULL,1,&binterpt);
323: }
325: {
326: /*const PetscReal g = 1. + 1./PetscSqrtReal(2.0); Direct evaluation: 1.707106781186547524401. Used for setting up arrays of values known at compile time below. */
327: const PetscReal
328: A[2][2] = {{0,0}, {1.,0}},
329: Gamma[2][2] = {{1.707106781186547524401,0}, {-2.*1.707106781186547524401,1.707106781186547524401}},
330: b[2] = {0.5,0.5},
331: b1[2] = {1.0,0.0};
332: PetscReal binterpt[2][2];
333: binterpt[0][0] = 1.707106781186547524401 - 1.0;
334: binterpt[1][0] = 2.0 - 1.707106781186547524401;
335: binterpt[0][1] = 1.707106781186547524401 - 1.5;
336: binterpt[1][1] = 1.5 - 1.707106781186547524401;
338: TSRosWRegister(TSROSW2P,2,2,&A[0][0],&Gamma[0][0],b,b1,2,&binterpt[0][0]);
339: }
340: {
341: /*const PetscReal g = 1. - 1./PetscSqrtReal(2.0); Direct evaluation: 0.2928932188134524755992. Used for setting up arrays of values known at compile time below. */
342: const PetscReal
343: A[2][2] = {{0,0}, {1.,0}},
344: Gamma[2][2] = {{0.2928932188134524755992,0}, {-2.*0.2928932188134524755992,0.2928932188134524755992}},
345: b[2] = {0.5,0.5},
346: b1[2] = {1.0,0.0};
347: PetscReal binterpt[2][2];
348: binterpt[0][0] = 0.2928932188134524755992 - 1.0;
349: binterpt[1][0] = 2.0 - 0.2928932188134524755992;
350: binterpt[0][1] = 0.2928932188134524755992 - 1.5;
351: binterpt[1][1] = 1.5 - 0.2928932188134524755992;
353: TSRosWRegister(TSROSW2M,2,2,&A[0][0],&Gamma[0][0],b,b1,2,&binterpt[0][0]);
354: }
355: {
356: /*const PetscReal g = 7.8867513459481287e-01; Directly written in-place below */
357: PetscReal binterpt[3][2];
358: const PetscReal
359: A[3][3] = {{0,0,0},
360: {1.5773502691896257e+00,0,0},
361: {0.5,0,0}},
362: Gamma[3][3] = {{7.8867513459481287e-01,0,0},
363: {-1.5773502691896257e+00,7.8867513459481287e-01,0},
364: {-6.7075317547305480e-01,-1.7075317547305482e-01,7.8867513459481287e-01}},
365: b[3] = {1.0566243270259355e-01,4.9038105676657971e-02,8.4529946162074843e-01},
366: b2[3] = {-1.7863279495408180e-01,1./3.,8.4529946162074843e-01};
368: binterpt[0][0] = -0.8094010767585034;
369: binterpt[1][0] = -0.5;
370: binterpt[2][0] = 2.3094010767585034;
371: binterpt[0][1] = 0.9641016151377548;
372: binterpt[1][1] = 0.5;
373: binterpt[2][1] = -1.4641016151377548;
375: TSRosWRegister(TSROSWRA3PW,3,3,&A[0][0],&Gamma[0][0],b,b2,2,&binterpt[0][0]);
376: }
377: {
378: PetscReal binterpt[4][3];
379: /*const PetscReal g = 4.3586652150845900e-01; Directly written in-place below */
380: const PetscReal
381: A[4][4] = {{0,0,0,0},
382: {8.7173304301691801e-01,0,0,0},
383: {8.4457060015369423e-01,-1.1299064236484185e-01,0,0},
384: {0,0,1.,0}},
385: Gamma[4][4] = {{4.3586652150845900e-01,0,0,0},
386: {-8.7173304301691801e-01,4.3586652150845900e-01,0,0},
387: {-9.0338057013044082e-01,5.4180672388095326e-02,4.3586652150845900e-01,0},
388: {2.4212380706095346e-01,-1.2232505839045147e+00,5.4526025533510214e-01,4.3586652150845900e-01}},
389: b[4] = {2.4212380706095346e-01,-1.2232505839045147e+00,1.5452602553351020e+00,4.3586652150845900e-01},
390: b2[4] = {3.7810903145819369e-01,-9.6042292212423178e-02,5.0000000000000000e-01,2.1793326075422950e-01};
392: binterpt[0][0]=1.0564298455794094;
393: binterpt[1][0]=2.296429974281067;
394: binterpt[2][0]=-1.307599564525376;
395: binterpt[3][0]=-1.045260255335102;
396: binterpt[0][1]=-1.3864882699759573;
397: binterpt[1][1]=-8.262611700275677;
398: binterpt[2][1]=7.250979895056055;
399: binterpt[3][1]=2.398120075195581;
400: binterpt[0][2]=0.5721822314575016;
401: binterpt[1][2]=4.742931142090097;
402: binterpt[2][2]=-4.398120075195578;
403: binterpt[3][2]=-0.9169932983520199;
405: TSRosWRegister(TSROSWRA34PW2,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
406: }
407: {
408: /* const PetscReal g = 0.5; Directly written in-place below */
409: const PetscReal
410: A[4][4] = {{0,0,0,0},
411: {0,0,0,0},
412: {1.,0,0,0},
413: {0.75,-0.25,0.5,0}},
414: Gamma[4][4] = {{0.5,0,0,0},
415: {1.,0.5,0,0},
416: {-0.25,-0.25,0.5,0},
417: {1./12,1./12,-2./3,0.5}},
418: b[4] = {5./6,-1./6,-1./6,0.5},
419: b2[4] = {0.75,-0.25,0.5,0};
421: TSRosWRegister(TSROSWRODAS3,3,4,&A[0][0],&Gamma[0][0],b,b2,0,NULL);
422: }
423: {
424: /*const PetscReal g = 0.43586652150845899941601945119356; Directly written in-place below */
425: const PetscReal
426: A[3][3] = {{0,0,0},
427: {0.43586652150845899941601945119356,0,0},
428: {0.43586652150845899941601945119356,0,0}},
429: Gamma[3][3] = {{0.43586652150845899941601945119356,0,0},
430: {-0.19294655696029095575009695436041,0.43586652150845899941601945119356,0},
431: {0,1.74927148125794685173529749738960,0.43586652150845899941601945119356}},
432: b[3] = {-0.75457412385404315829818998646589,1.94100407061964420292840123379419,-0.18642994676560104463021124732829},
433: b2[3] = {-1.53358745784149585370766523913002,2.81745131148625772213931745457622,-0.28386385364476186843165221544619};
435: PetscReal binterpt[3][2];
436: binterpt[0][0] = 3.793692883777660870425141387941;
437: binterpt[1][0] = -2.918692883777660870425141387941;
438: binterpt[2][0] = 0.125;
439: binterpt[0][1] = -0.725741064379812106687651020584;
440: binterpt[1][1] = 0.559074397713145440020984353917;
441: binterpt[2][1] = 0.16666666666666666666666666666667;
443: TSRosWRegister(TSROSWSANDU3,3,3,&A[0][0],&Gamma[0][0],b,b2,2,&binterpt[0][0]);
444: }
445: {
446: /*const PetscReal s3 = PetscSqrtReal(3.),g = (3.0+s3)/6.0;
447: * Direct evaluation: s3 = 1.732050807568877293527;
448: * g = 0.7886751345948128822546;
449: * Values are directly inserted below to ensure availability at compile time (compiler warnings otherwise...) */
450: const PetscReal
451: A[3][3] = {{0,0,0},
452: {1,0,0},
453: {0.25,0.25,0}},
454: Gamma[3][3] = {{0,0,0},
455: {(-3.0-1.732050807568877293527)/6.0,0.7886751345948128822546,0},
456: {(-3.0-1.732050807568877293527)/24.0,(-3.0-1.732050807568877293527)/8.0,0.7886751345948128822546}},
457: b[3] = {1./6.,1./6.,2./3.},
458: b2[3] = {1./4.,1./4.,1./2.};
459: PetscReal binterpt[3][2];
461: binterpt[0][0]=0.089316397477040902157517886164709;
462: binterpt[1][0]=-0.91068360252295909784248211383529;
463: binterpt[2][0]=1.8213672050459181956849642276706;
464: binterpt[0][1]=0.077350269189625764509148780501957;
465: binterpt[1][1]=1.077350269189625764509148780502;
466: binterpt[2][1]=-1.1547005383792515290182975610039;
468: TSRosWRegister(TSROSWASSP3P3S1C,3,3,&A[0][0],&Gamma[0][0],b,b2,2,&binterpt[0][0]);
469: }
471: {
472: const PetscReal
473: A[4][4] = {{0,0,0,0},
474: {1./2.,0,0,0},
475: {1./2.,1./2.,0,0},
476: {1./6.,1./6.,1./6.,0}},
477: Gamma[4][4] = {{1./2.,0,0,0},
478: {0.0,1./4.,0,0},
479: {-2.,-2./3.,2./3.,0},
480: {1./2.,5./36.,-2./9,0}},
481: b[4] = {1./6.,1./6.,1./6.,1./2.},
482: b2[4] = {1./8.,3./4.,1./8.,0};
483: PetscReal binterpt[4][3];
485: binterpt[0][0]=6.25;
486: binterpt[1][0]=-30.25;
487: binterpt[2][0]=1.75;
488: binterpt[3][0]=23.25;
489: binterpt[0][1]=-9.75;
490: binterpt[1][1]=58.75;
491: binterpt[2][1]=-3.25;
492: binterpt[3][1]=-45.75;
493: binterpt[0][2]=3.6666666666666666666666666666667;
494: binterpt[1][2]=-28.333333333333333333333333333333;
495: binterpt[2][2]=1.6666666666666666666666666666667;
496: binterpt[3][2]=23.;
498: TSRosWRegister(TSROSWLASSP3P4S2C,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
499: }
501: {
502: const PetscReal
503: A[4][4] = {{0,0,0,0},
504: {1./2.,0,0,0},
505: {1./2.,1./2.,0,0},
506: {1./6.,1./6.,1./6.,0}},
507: Gamma[4][4] = {{1./2.,0,0,0},
508: {0.0,3./4.,0,0},
509: {-2./3.,-23./9.,2./9.,0},
510: {1./18.,65./108.,-2./27,0}},
511: b[4] = {1./6.,1./6.,1./6.,1./2.},
512: b2[4] = {3./16.,10./16.,3./16.,0};
513: PetscReal binterpt[4][3];
515: binterpt[0][0]=1.6911764705882352941176470588235;
516: binterpt[1][0]=3.6813725490196078431372549019608;
517: binterpt[2][0]=0.23039215686274509803921568627451;
518: binterpt[3][0]=-4.6029411764705882352941176470588;
519: binterpt[0][1]=-0.95588235294117647058823529411765;
520: binterpt[1][1]=-6.2401960784313725490196078431373;
521: binterpt[2][1]=-0.31862745098039215686274509803922;
522: binterpt[3][1]=7.5147058823529411764705882352941;
523: binterpt[0][2]=-0.56862745098039215686274509803922;
524: binterpt[1][2]=2.7254901960784313725490196078431;
525: binterpt[2][2]=0.25490196078431372549019607843137;
526: binterpt[3][2]=-2.4117647058823529411764705882353;
528: TSRosWRegister(TSROSWLLSSP3P4S2C,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
529: }
531: {
532: PetscReal A[4][4],Gamma[4][4],b[4],b2[4];
533: PetscReal binterpt[4][3];
535: Gamma[0][0]=0.4358665215084589994160194475295062513822671686978816;
536: Gamma[0][1]=0; Gamma[0][2]=0; Gamma[0][3]=0;
537: Gamma[1][0]=-1.997527830934941248426324674704153457289527280554476;
538: Gamma[1][1]=0.4358665215084589994160194475295062513822671686978816;
539: Gamma[1][2]=0; Gamma[1][3]=0;
540: Gamma[2][0]=-1.007948511795029620852002345345404191008352770119903;
541: Gamma[2][1]=-0.004648958462629345562774289390054679806993396798458131;
542: Gamma[2][2]=0.4358665215084589994160194475295062513822671686978816;
543: Gamma[2][3]=0;
544: Gamma[3][0]=-0.6685429734233467180451604600279552604364311322650783;
545: Gamma[3][1]=0.6056625986449338476089525334450053439525178740492984;
546: Gamma[3][2]=-0.9717899277217721234705114616271378792182450260943198;
547: Gamma[3][3]=0;
549: A[0][0]=0; A[0][1]=0; A[0][2]=0; A[0][3]=0;
550: A[1][0]=0.8717330430169179988320388950590125027645343373957631;
551: A[1][1]=0; A[1][2]=0; A[1][3]=0;
552: A[2][0]=0.5275890119763004115618079766722914408876108660811028;
553: A[2][1]=0.07241098802369958843819203208518599088698057726988732;
554: A[2][2]=0; A[2][3]=0;
555: A[3][0]=0.3990960076760701320627260685975778145384666450351314;
556: A[3][1]=-0.4375576546135194437228463747348862825846903771419953;
557: A[3][2]=1.038461646937449311660120300601880176655352737312713;
558: A[3][3]=0;
560: b[0]=0.1876410243467238251612921333138006734899663569186926;
561: b[1]=-0.5952974735769549480478230473706443582188442040780541;
562: b[2]=0.9717899277217721234705114616271378792182450260943198;
563: b[3]=0.4358665215084589994160194475295062513822671686978816;
565: b2[0]=0.2147402862233891404862383521089097657790734483804460;
566: b2[1]=-0.4851622638849390928209050538171743017757490232519684;
567: b2[2]=0.8687250025203875511662123688667549217531982787600080;
568: b2[3]=0.4016969751411624011684543450940068201770721128357014;
570: binterpt[0][0]=2.2565812720167954547104627844105;
571: binterpt[1][0]=1.349166413351089573796243820819;
572: binterpt[2][0]=-2.4695174540533503758652847586647;
573: binterpt[3][0]=-0.13623023131453465264142184656474;
574: binterpt[0][1]=-3.0826699111559187902922463354557;
575: binterpt[1][1]=-2.4689115685996042534544925650515;
576: binterpt[2][1]=5.7428279814696677152129332773553;
577: binterpt[3][1]=-0.19124650171414467146619437684812;
578: binterpt[0][2]=1.0137296634858471607430756831148;
579: binterpt[1][2]=0.52444768167155973161042570784064;
580: binterpt[2][2]=-2.3015205996945452158771370439586;
581: binterpt[3][2]=0.76334325453713832352363565300308;
583: TSRosWRegister(TSROSWARK3,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
584: }
585: TSRosWRegisterRos4(TSROSWGRK4T,0.231,PETSC_DEFAULT,PETSC_DEFAULT,0,-0.1282612945269037e+01);
586: TSRosWRegisterRos4(TSROSWSHAMP4,0.5,PETSC_DEFAULT,PETSC_DEFAULT,0,125./108.);
587: TSRosWRegisterRos4(TSROSWVELDD4,0.22570811482256823492,PETSC_DEFAULT,PETSC_DEFAULT,0,-1.355958941201148);
588: TSRosWRegisterRos4(TSROSW4L,0.57282,PETSC_DEFAULT,PETSC_DEFAULT,0,-1.093502252409163);
589: return 0;
590: }
592: /*@C
593: TSRosWRegisterDestroy - Frees the list of schemes that were registered by TSRosWRegister().
595: Not Collective
597: Level: advanced
599: .seealso: TSRosWRegister(), TSRosWRegisterAll()
600: @*/
601: PetscErrorCode TSRosWRegisterDestroy(void)
602: {
603: RosWTableauLink link;
605: while ((link = RosWTableauList)) {
606: RosWTableau t = &link->tab;
607: RosWTableauList = link->next;
608: PetscFree5(t->A,t->Gamma,t->b,t->ASum,t->GammaSum);
609: PetscFree5(t->At,t->bt,t->GammaInv,t->GammaZeroDiag,t->GammaExplicitCorr);
610: PetscFree2(t->bembed,t->bembedt);
611: PetscFree(t->binterpt);
612: PetscFree(t->name);
613: PetscFree(link);
614: }
615: TSRosWRegisterAllCalled = PETSC_FALSE;
616: return 0;
617: }
619: /*@C
620: TSRosWInitializePackage - This function initializes everything in the TSRosW package. It is called
621: from TSInitializePackage().
623: Level: developer
625: .seealso: PetscInitialize()
626: @*/
627: PetscErrorCode TSRosWInitializePackage(void)
628: {
629: if (TSRosWPackageInitialized) return 0;
630: TSRosWPackageInitialized = PETSC_TRUE;
631: TSRosWRegisterAll();
632: PetscRegisterFinalize(TSRosWFinalizePackage);
633: return 0;
634: }
636: /*@C
637: TSRosWFinalizePackage - This function destroys everything in the TSRosW package. It is
638: called from PetscFinalize().
640: Level: developer
642: .seealso: PetscFinalize()
643: @*/
644: PetscErrorCode TSRosWFinalizePackage(void)
645: {
646: TSRosWPackageInitialized = PETSC_FALSE;
647: TSRosWRegisterDestroy();
648: return 0;
649: }
651: /*@C
652: TSRosWRegister - register a Rosenbrock W scheme by providing the entries in the Butcher tableau and optionally embedded approximations and interpolation
654: Not Collective, but the same schemes should be registered on all processes on which they will be used
656: Input Parameters:
657: + name - identifier for method
658: . order - approximation order of method
659: . s - number of stages, this is the dimension of the matrices below
660: . A - Table of propagated stage coefficients (dimension s*s, row-major), strictly lower triangular
661: . Gamma - Table of coefficients in implicit stage equations (dimension s*s, row-major), lower triangular with nonzero diagonal
662: . b - Step completion table (dimension s)
663: . bembed - Step completion table for a scheme of order one less (dimension s, NULL if no embedded scheme is available)
664: . pinterp - Order of the interpolation scheme, equal to the number of columns of binterpt
665: - binterpt - Coefficients of the interpolation formula (dimension s*pinterp)
667: Notes:
668: Several Rosenbrock W methods are provided, this function is only needed to create new methods.
670: Level: advanced
672: .seealso: TSRosW
673: @*/
674: PetscErrorCode TSRosWRegister(TSRosWType name,PetscInt order,PetscInt s,const PetscReal A[],const PetscReal Gamma[],const PetscReal b[],const PetscReal bembed[],
675: PetscInt pinterp,const PetscReal binterpt[])
676: {
677: RosWTableauLink link;
678: RosWTableau t;
679: PetscInt i,j,k;
680: PetscScalar *GammaInv;
688: TSRosWInitializePackage();
689: PetscNew(&link);
690: t = &link->tab;
691: PetscStrallocpy(name,&t->name);
692: t->order = order;
693: t->s = s;
694: PetscMalloc5(s*s,&t->A,s*s,&t->Gamma,s,&t->b,s,&t->ASum,s,&t->GammaSum);
695: PetscMalloc5(s*s,&t->At,s,&t->bt,s*s,&t->GammaInv,s,&t->GammaZeroDiag,s*s,&t->GammaExplicitCorr);
696: PetscArraycpy(t->A,A,s*s);
697: PetscArraycpy(t->Gamma,Gamma,s*s);
698: PetscArraycpy(t->GammaExplicitCorr,Gamma,s*s);
699: PetscArraycpy(t->b,b,s);
700: if (bembed) {
701: PetscMalloc2(s,&t->bembed,s,&t->bembedt);
702: PetscArraycpy(t->bembed,bembed,s);
703: }
704: for (i=0; i<s; i++) {
705: t->ASum[i] = 0;
706: t->GammaSum[i] = 0;
707: for (j=0; j<s; j++) {
708: t->ASum[i] += A[i*s+j];
709: t->GammaSum[i] += Gamma[i*s+j];
710: }
711: }
712: PetscMalloc1(s*s,&GammaInv); /* Need to use Scalar for inverse, then convert back to Real */
713: for (i=0; i<s*s; i++) GammaInv[i] = Gamma[i];
714: for (i=0; i<s; i++) {
715: if (Gamma[i*s+i] == 0.0) {
716: GammaInv[i*s+i] = 1.0;
717: t->GammaZeroDiag[i] = PETSC_TRUE;
718: } else {
719: t->GammaZeroDiag[i] = PETSC_FALSE;
720: }
721: }
723: switch (s) {
724: case 1: GammaInv[0] = 1./GammaInv[0]; break;
725: case 2: PetscKernel_A_gets_inverse_A_2(GammaInv,0,PETSC_FALSE,NULL); break;
726: case 3: PetscKernel_A_gets_inverse_A_3(GammaInv,0,PETSC_FALSE,NULL); break;
727: case 4: PetscKernel_A_gets_inverse_A_4(GammaInv,0,PETSC_FALSE,NULL); break;
728: case 5: {
729: PetscInt ipvt5[5];
730: MatScalar work5[5*5];
731: PetscKernel_A_gets_inverse_A_5(GammaInv,ipvt5,work5,0,PETSC_FALSE,NULL); break;
732: }
733: case 6: PetscKernel_A_gets_inverse_A_6(GammaInv,0,PETSC_FALSE,NULL); break;
734: case 7: PetscKernel_A_gets_inverse_A_7(GammaInv,0,PETSC_FALSE,NULL); break;
735: default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Not implemented for %D stages",s);
736: }
737: for (i=0; i<s*s; i++) t->GammaInv[i] = PetscRealPart(GammaInv[i]);
738: PetscFree(GammaInv);
740: for (i=0; i<s; i++) {
741: for (k=0; k<i+1; k++) {
742: t->GammaExplicitCorr[i*s+k]=(t->GammaExplicitCorr[i*s+k])*(t->GammaInv[k*s+k]);
743: for (j=k+1; j<i+1; j++) {
744: t->GammaExplicitCorr[i*s+k]+=(t->GammaExplicitCorr[i*s+j])*(t->GammaInv[j*s+k]);
745: }
746: }
747: }
749: for (i=0; i<s; i++) {
750: for (j=0; j<s; j++) {
751: t->At[i*s+j] = 0;
752: for (k=0; k<s; k++) {
753: t->At[i*s+j] += t->A[i*s+k] * t->GammaInv[k*s+j];
754: }
755: }
756: t->bt[i] = 0;
757: for (j=0; j<s; j++) {
758: t->bt[i] += t->b[j] * t->GammaInv[j*s+i];
759: }
760: if (bembed) {
761: t->bembedt[i] = 0;
762: for (j=0; j<s; j++) {
763: t->bembedt[i] += t->bembed[j] * t->GammaInv[j*s+i];
764: }
765: }
766: }
767: t->ccfl = 1.0; /* Fix this */
769: t->pinterp = pinterp;
770: PetscMalloc1(s*pinterp,&t->binterpt);
771: PetscArraycpy(t->binterpt,binterpt,s*pinterp);
772: link->next = RosWTableauList;
773: RosWTableauList = link;
774: return 0;
775: }
777: /*@C
778: TSRosWRegisterRos4 - register a fourth order Rosenbrock scheme by providing parameter choices
780: Not Collective, but the same schemes should be registered on all processes on which they will be used
782: Input Parameters:
783: + name - identifier for method
784: . gamma - leading coefficient (diagonal entry)
785: . a2 - design parameter, see Table 7.2 of Hairer&Wanner
786: . a3 - design parameter or PETSC_DEFAULT to satisfy one of the order five conditions (Eq 7.22)
787: . b3 - design parameter, see Table 7.2 of Hairer&Wanner
788: . beta43 - design parameter or PETSC_DEFAULT to use Equation 7.21 of Hairer&Wanner
789: - e4 - design parameter for embedded method, see coefficient E4 in ros4.f code from Hairer
791: Notes:
792: This routine encodes the design of fourth order Rosenbrock methods as described in Hairer and Wanner volume 2.
793: It is used here to implement several methods from the book and can be used to experiment with new methods.
794: It was written this way instead of by copying coefficients in order to provide better than double precision satisfaction of the order conditions.
796: Level: developer
798: .seealso: TSRosW, TSRosWRegister()
799: @*/
800: PetscErrorCode TSRosWRegisterRos4(TSRosWType name,PetscReal gamma,PetscReal a2,PetscReal a3,PetscReal b3,PetscReal e4)
801: {
802: /* Declare numeric constants so they can be quad precision without being truncated at double */
803: const PetscReal one = 1,two = 2,three = 3,four = 4,five = 5,six = 6,eight = 8,twelve = 12,twenty = 20,twentyfour = 24,
804: p32 = one/six - gamma + gamma*gamma,
805: p42 = one/eight - gamma/three,
806: p43 = one/twelve - gamma/three,
807: p44 = one/twentyfour - gamma/two + three/two*gamma*gamma - gamma*gamma*gamma,
808: p56 = one/twenty - gamma/four;
809: PetscReal a4,a32,a42,a43,b1,b2,b4,beta2p,beta3p,beta4p,beta32,beta42,beta43,beta32beta2p,beta4jbetajp;
810: PetscReal A[4][4],Gamma[4][4],b[4],bm[4];
811: PetscScalar M[3][3],rhs[3];
813: /* Step 1: choose Gamma (input) */
814: /* Step 2: choose a2,a3,a4; b1,b2,b3,b4 to satisfy order conditions */
815: if (a3 == PETSC_DEFAULT) a3 = (one/five - a2/four)/(one/four - a2/three); /* Eq 7.22 */
816: a4 = a3; /* consequence of 7.20 */
818: /* Solve order conditions 7.15a, 7.15c, 7.15e */
819: M[0][0] = one; M[0][1] = one; M[0][2] = one; /* 7.15a */
820: M[1][0] = 0.0; M[1][1] = a2*a2; M[1][2] = a4*a4; /* 7.15c */
821: M[2][0] = 0.0; M[2][1] = a2*a2*a2; M[2][2] = a4*a4*a4; /* 7.15e */
822: rhs[0] = one - b3;
823: rhs[1] = one/three - a3*a3*b3;
824: rhs[2] = one/four - a3*a3*a3*b3;
825: PetscKernel_A_gets_inverse_A_3(&M[0][0],0,PETSC_FALSE,NULL);
826: b1 = PetscRealPart(M[0][0]*rhs[0] + M[0][1]*rhs[1] + M[0][2]*rhs[2]);
827: b2 = PetscRealPart(M[1][0]*rhs[0] + M[1][1]*rhs[1] + M[1][2]*rhs[2]);
828: b4 = PetscRealPart(M[2][0]*rhs[0] + M[2][1]*rhs[1] + M[2][2]*rhs[2]);
830: /* Step 3 */
831: beta43 = (p56 - a2*p43) / (b4*a3*a3*(a3 - a2)); /* 7.21 */
832: beta32beta2p = p44 / (b4*beta43); /* 7.15h */
833: beta4jbetajp = (p32 - b3*beta32beta2p) / b4;
834: M[0][0] = b2; M[0][1] = b3; M[0][2] = b4;
835: M[1][0] = a4*a4*beta32beta2p-a3*a3*beta4jbetajp; M[1][1] = a2*a2*beta4jbetajp; M[1][2] = -a2*a2*beta32beta2p;
836: M[2][0] = b4*beta43*a3*a3-p43; M[2][1] = -b4*beta43*a2*a2; M[2][2] = 0;
837: rhs[0] = one/two - gamma; rhs[1] = 0; rhs[2] = -a2*a2*p32;
838: PetscKernel_A_gets_inverse_A_3(&M[0][0],0,PETSC_FALSE,NULL);
839: beta2p = PetscRealPart(M[0][0]*rhs[0] + M[0][1]*rhs[1] + M[0][2]*rhs[2]);
840: beta3p = PetscRealPart(M[1][0]*rhs[0] + M[1][1]*rhs[1] + M[1][2]*rhs[2]);
841: beta4p = PetscRealPart(M[2][0]*rhs[0] + M[2][1]*rhs[1] + M[2][2]*rhs[2]);
843: /* Step 4: back-substitute */
844: beta32 = beta32beta2p / beta2p;
845: beta42 = (beta4jbetajp - beta43*beta3p) / beta2p;
847: /* Step 5: 7.15f and 7.20, then 7.16 */
848: a43 = 0;
849: a32 = p42 / (b3*a3*beta2p + b4*a4*beta2p);
850: a42 = a32;
852: A[0][0] = 0; A[0][1] = 0; A[0][2] = 0; A[0][3] = 0;
853: A[1][0] = a2; A[1][1] = 0; A[1][2] = 0; A[1][3] = 0;
854: A[2][0] = a3-a32; A[2][1] = a32; A[2][2] = 0; A[2][3] = 0;
855: A[3][0] = a4-a43-a42; A[3][1] = a42; A[3][2] = a43; A[3][3] = 0;
856: Gamma[0][0] = gamma; Gamma[0][1] = 0; Gamma[0][2] = 0; Gamma[0][3] = 0;
857: Gamma[1][0] = beta2p-A[1][0]; Gamma[1][1] = gamma; Gamma[1][2] = 0; Gamma[1][3] = 0;
858: Gamma[2][0] = beta3p-beta32-A[2][0]; Gamma[2][1] = beta32-A[2][1]; Gamma[2][2] = gamma; Gamma[2][3] = 0;
859: Gamma[3][0] = beta4p-beta42-beta43-A[3][0]; Gamma[3][1] = beta42-A[3][1]; Gamma[3][2] = beta43-A[3][2]; Gamma[3][3] = gamma;
860: b[0] = b1; b[1] = b2; b[2] = b3; b[3] = b4;
862: /* Construct embedded formula using given e4. We are solving Equation 7.18. */
863: bm[3] = b[3] - e4*gamma; /* using definition of E4 */
864: bm[2] = (p32 - beta4jbetajp*bm[3]) / (beta32*beta2p); /* fourth row of 7.18 */
865: bm[1] = (one/two - gamma - beta3p*bm[2] - beta4p*bm[3]) / beta2p; /* second row */
866: bm[0] = one - bm[1] - bm[2] - bm[3]; /* first row */
868: {
869: const PetscReal misfit = a2*a2*bm[1] + a3*a3*bm[2] + a4*a4*bm[3] - one/three;
871: }
872: TSRosWRegister(name,4,4,&A[0][0],&Gamma[0][0],b,bm,0,NULL);
873: return 0;
874: }
876: /*
877: The step completion formula is
879: x1 = x0 + b^T Y
881: where Y is the multi-vector of stages corrections. This function can be called before or after ts->vec_sol has been
882: updated. Suppose we have a completion formula b and an embedded formula be of different order. We can write
884: x1e = x0 + be^T Y
885: = x1 - b^T Y + be^T Y
886: = x1 + (be - b)^T Y
888: so we can evaluate the method of different order even after the step has been optimistically completed.
889: */
890: static PetscErrorCode TSEvaluateStep_RosW(TS ts,PetscInt order,Vec U,PetscBool *done)
891: {
892: TS_RosW *ros = (TS_RosW*)ts->data;
893: RosWTableau tab = ros->tableau;
894: PetscScalar *w = ros->work;
895: PetscInt i;
897: if (order == tab->order) {
898: if (ros->status == TS_STEP_INCOMPLETE) { /* Use standard completion formula */
899: VecCopy(ts->vec_sol,U);
900: for (i=0; i<tab->s; i++) w[i] = tab->bt[i];
901: VecMAXPY(U,tab->s,w,ros->Y);
902: } else VecCopy(ts->vec_sol,U);
903: if (done) *done = PETSC_TRUE;
904: return 0;
905: } else if (order == tab->order-1) {
906: if (!tab->bembedt) goto unavailable;
907: if (ros->status == TS_STEP_INCOMPLETE) { /* Use embedded completion formula */
908: VecCopy(ts->vec_sol,U);
909: for (i=0; i<tab->s; i++) w[i] = tab->bembedt[i];
910: VecMAXPY(U,tab->s,w,ros->Y);
911: } else { /* Use rollback-and-recomplete formula (bembedt - bt) */
912: for (i=0; i<tab->s; i++) w[i] = tab->bembedt[i] - tab->bt[i];
913: VecCopy(ts->vec_sol,U);
914: VecMAXPY(U,tab->s,w,ros->Y);
915: }
916: if (done) *done = PETSC_TRUE;
917: return 0;
918: }
919: unavailable:
920: if (done) *done = PETSC_FALSE;
921: else SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_SUP,"Rosenbrock-W '%s' of order %D cannot evaluate step at order %D. Consider using -ts_adapt_type none or a different method that has an embedded estimate.",tab->name,tab->order,order);
922: return 0;
923: }
925: static PetscErrorCode TSRollBack_RosW(TS ts)
926: {
927: TS_RosW *ros = (TS_RosW*)ts->data;
929: VecCopy(ros->vec_sol_prev,ts->vec_sol);
930: return 0;
931: }
933: static PetscErrorCode TSStep_RosW(TS ts)
934: {
935: TS_RosW *ros = (TS_RosW*)ts->data;
936: RosWTableau tab = ros->tableau;
937: const PetscInt s = tab->s;
938: const PetscReal *At = tab->At,*Gamma = tab->Gamma,*ASum = tab->ASum,*GammaInv = tab->GammaInv;
939: const PetscReal *GammaExplicitCorr = tab->GammaExplicitCorr;
940: const PetscBool *GammaZeroDiag = tab->GammaZeroDiag;
941: PetscScalar *w = ros->work;
942: Vec *Y = ros->Y,Ydot = ros->Ydot,Zdot = ros->Zdot,Zstage = ros->Zstage;
943: SNES snes;
944: TSAdapt adapt;
945: PetscInt i,j,its,lits;
946: PetscInt rejections = 0;
947: PetscBool stageok,accept = PETSC_TRUE;
948: PetscReal next_time_step = ts->time_step;
949: PetscInt lag;
951: if (!ts->steprollback) {
952: VecCopy(ts->vec_sol,ros->vec_sol_prev);
953: }
955: ros->status = TS_STEP_INCOMPLETE;
956: while (!ts->reason && ros->status != TS_STEP_COMPLETE) {
957: const PetscReal h = ts->time_step;
958: for (i=0; i<s; i++) {
959: ros->stage_time = ts->ptime + h*ASum[i];
960: TSPreStage(ts,ros->stage_time);
961: if (GammaZeroDiag[i]) {
962: ros->stage_explicit = PETSC_TRUE;
963: ros->scoeff = 1.;
964: } else {
965: ros->stage_explicit = PETSC_FALSE;
966: ros->scoeff = 1./Gamma[i*s+i];
967: }
969: VecCopy(ts->vec_sol,Zstage);
970: for (j=0; j<i; j++) w[j] = At[i*s+j];
971: VecMAXPY(Zstage,i,w,Y);
973: for (j=0; j<i; j++) w[j] = 1./h * GammaInv[i*s+j];
974: VecZeroEntries(Zdot);
975: VecMAXPY(Zdot,i,w,Y);
977: /* Initial guess taken from last stage */
978: VecZeroEntries(Y[i]);
980: if (!ros->stage_explicit) {
981: TSGetSNES(ts,&snes);
982: if (!ros->recompute_jacobian && !i) {
983: SNESGetLagJacobian(snes,&lag);
984: if (lag == 1) { /* use did not set a nontrivial lag, so lag over all stages */
985: SNESSetLagJacobian(snes,-2); /* Recompute the Jacobian on this solve, but not again for the rest of the stages */
986: }
987: }
988: SNESSolve(snes,NULL,Y[i]);
989: if (!ros->recompute_jacobian && i == s-1 && lag == 1) {
990: SNESSetLagJacobian(snes,lag); /* Set lag back to 1 so we know user did not set it */
991: }
992: SNESGetIterationNumber(snes,&its);
993: SNESGetLinearSolveIterations(snes,&lits);
994: ts->snes_its += its; ts->ksp_its += lits;
995: } else {
996: Mat J,Jp;
997: VecZeroEntries(Ydot); /* Evaluate Y[i]=G(t,Ydot=0,Zstage) */
998: TSComputeIFunction(ts,ros->stage_time,Zstage,Ydot,Y[i],PETSC_FALSE);
999: VecScale(Y[i],-1.0);
1000: VecAXPY(Y[i],-1.0,Zdot); /*Y[i] = F(Zstage)-Zdot[=GammaInv*Y]*/
1002: VecZeroEntries(Zstage); /* Zstage = GammaExplicitCorr[i,j] * Y[j] */
1003: for (j=0; j<i; j++) w[j] = GammaExplicitCorr[i*s+j];
1004: VecMAXPY(Zstage,i,w,Y);
1006: /* Y[i] = Y[i] + Jac*Zstage[=Jac*GammaExplicitCorr[i,j] * Y[j]] */
1007: TSGetIJacobian(ts,&J,&Jp,NULL,NULL);
1008: TSComputeIJacobian(ts,ros->stage_time,ts->vec_sol,Ydot,0,J,Jp,PETSC_FALSE);
1009: MatMult(J,Zstage,Zdot);
1010: VecAXPY(Y[i],-1.0,Zdot);
1011: ts->ksp_its += 1;
1013: VecScale(Y[i],h);
1014: }
1015: TSPostStage(ts,ros->stage_time,i,Y);
1016: TSGetAdapt(ts,&adapt);
1017: TSAdaptCheckStage(adapt,ts,ros->stage_time,Y[i],&stageok);
1018: if (!stageok) goto reject_step;
1019: }
1021: ros->status = TS_STEP_INCOMPLETE;
1022: TSEvaluateStep_RosW(ts,tab->order,ts->vec_sol,NULL);
1023: ros->status = TS_STEP_PENDING;
1024: TSGetAdapt(ts,&adapt);
1025: TSAdaptCandidatesClear(adapt);
1026: TSAdaptCandidateAdd(adapt,tab->name,tab->order,1,tab->ccfl,(PetscReal)tab->s,PETSC_TRUE);
1027: TSAdaptChoose(adapt,ts,ts->time_step,NULL,&next_time_step,&accept);
1028: ros->status = accept ? TS_STEP_COMPLETE : TS_STEP_INCOMPLETE;
1029: if (!accept) { /* Roll back the current step */
1030: TSRollBack_RosW(ts);
1031: ts->time_step = next_time_step;
1032: goto reject_step;
1033: }
1035: ts->ptime += ts->time_step;
1036: ts->time_step = next_time_step;
1037: break;
1039: reject_step:
1040: ts->reject++; accept = PETSC_FALSE;
1041: if (!ts->reason && ++rejections > ts->max_reject && ts->max_reject >= 0) {
1042: ts->reason = TS_DIVERGED_STEP_REJECTED;
1043: PetscInfo(ts,"Step=%D, step rejections %D greater than current TS allowed, stopping solve\n",ts->steps,rejections);
1044: }
1045: }
1046: return 0;
1047: }
1049: static PetscErrorCode TSInterpolate_RosW(TS ts,PetscReal itime,Vec U)
1050: {
1051: TS_RosW *ros = (TS_RosW*)ts->data;
1052: PetscInt s = ros->tableau->s,pinterp = ros->tableau->pinterp,i,j;
1053: PetscReal h;
1054: PetscReal tt,t;
1055: PetscScalar *bt;
1056: const PetscReal *Bt = ros->tableau->binterpt;
1057: const PetscReal *GammaInv = ros->tableau->GammaInv;
1058: PetscScalar *w = ros->work;
1059: Vec *Y = ros->Y;
1063: switch (ros->status) {
1064: case TS_STEP_INCOMPLETE:
1065: case TS_STEP_PENDING:
1066: h = ts->time_step;
1067: t = (itime - ts->ptime)/h;
1068: break;
1069: case TS_STEP_COMPLETE:
1070: h = ts->ptime - ts->ptime_prev;
1071: t = (itime - ts->ptime)/h + 1; /* In the interval [0,1] */
1072: break;
1073: default: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_PLIB,"Invalid TSStepStatus");
1074: }
1075: PetscMalloc1(s,&bt);
1076: for (i=0; i<s; i++) bt[i] = 0;
1077: for (j=0,tt=t; j<pinterp; j++,tt*=t) {
1078: for (i=0; i<s; i++) {
1079: bt[i] += Bt[i*pinterp+j] * tt;
1080: }
1081: }
1083: /* y(t+tt*h) = y(t) + Sum bt(tt) * GammaInv * Ydot */
1084: /* U <- 0*/
1085: VecZeroEntries(U);
1086: /* U <- Sum bt_i * GammaInv(i,1:i) * Y(1:i) */
1087: for (j=0; j<s; j++) w[j] = 0;
1088: for (j=0; j<s; j++) {
1089: for (i=j; i<s; i++) {
1090: w[j] += bt[i]*GammaInv[i*s+j];
1091: }
1092: }
1093: VecMAXPY(U,i,w,Y);
1094: /* U <- y(t) + U */
1095: VecAXPY(U,1,ros->vec_sol_prev);
1097: PetscFree(bt);
1098: return 0;
1099: }
1101: /*------------------------------------------------------------*/
1103: static PetscErrorCode TSRosWTableauReset(TS ts)
1104: {
1105: TS_RosW *ros = (TS_RosW*)ts->data;
1106: RosWTableau tab = ros->tableau;
1108: if (!tab) return 0;
1109: VecDestroyVecs(tab->s,&ros->Y);
1110: PetscFree(ros->work);
1111: return 0;
1112: }
1114: static PetscErrorCode TSReset_RosW(TS ts)
1115: {
1116: TS_RosW *ros = (TS_RosW*)ts->data;
1118: TSRosWTableauReset(ts);
1119: VecDestroy(&ros->Ydot);
1120: VecDestroy(&ros->Ystage);
1121: VecDestroy(&ros->Zdot);
1122: VecDestroy(&ros->Zstage);
1123: VecDestroy(&ros->vec_sol_prev);
1124: return 0;
1125: }
1127: static PetscErrorCode TSRosWGetVecs(TS ts,DM dm,Vec *Ydot,Vec *Zdot,Vec *Ystage,Vec *Zstage)
1128: {
1129: TS_RosW *rw = (TS_RosW*)ts->data;
1131: if (Ydot) {
1132: if (dm && dm != ts->dm) {
1133: DMGetNamedGlobalVector(dm,"TSRosW_Ydot",Ydot);
1134: } else *Ydot = rw->Ydot;
1135: }
1136: if (Zdot) {
1137: if (dm && dm != ts->dm) {
1138: DMGetNamedGlobalVector(dm,"TSRosW_Zdot",Zdot);
1139: } else *Zdot = rw->Zdot;
1140: }
1141: if (Ystage) {
1142: if (dm && dm != ts->dm) {
1143: DMGetNamedGlobalVector(dm,"TSRosW_Ystage",Ystage);
1144: } else *Ystage = rw->Ystage;
1145: }
1146: if (Zstage) {
1147: if (dm && dm != ts->dm) {
1148: DMGetNamedGlobalVector(dm,"TSRosW_Zstage",Zstage);
1149: } else *Zstage = rw->Zstage;
1150: }
1151: return 0;
1152: }
1154: static PetscErrorCode TSRosWRestoreVecs(TS ts,DM dm,Vec *Ydot,Vec *Zdot, Vec *Ystage, Vec *Zstage)
1155: {
1156: if (Ydot) {
1157: if (dm && dm != ts->dm) {
1158: DMRestoreNamedGlobalVector(dm,"TSRosW_Ydot",Ydot);
1159: }
1160: }
1161: if (Zdot) {
1162: if (dm && dm != ts->dm) {
1163: DMRestoreNamedGlobalVector(dm,"TSRosW_Zdot",Zdot);
1164: }
1165: }
1166: if (Ystage) {
1167: if (dm && dm != ts->dm) {
1168: DMRestoreNamedGlobalVector(dm,"TSRosW_Ystage",Ystage);
1169: }
1170: }
1171: if (Zstage) {
1172: if (dm && dm != ts->dm) {
1173: DMRestoreNamedGlobalVector(dm,"TSRosW_Zstage",Zstage);
1174: }
1175: }
1176: return 0;
1177: }
1179: static PetscErrorCode DMCoarsenHook_TSRosW(DM fine,DM coarse,void *ctx)
1180: {
1181: return 0;
1182: }
1184: static PetscErrorCode DMRestrictHook_TSRosW(DM fine,Mat restrct,Vec rscale,Mat inject,DM coarse,void *ctx)
1185: {
1186: TS ts = (TS)ctx;
1187: Vec Ydot,Zdot,Ystage,Zstage;
1188: Vec Ydotc,Zdotc,Ystagec,Zstagec;
1190: TSRosWGetVecs(ts,fine,&Ydot,&Ystage,&Zdot,&Zstage);
1191: TSRosWGetVecs(ts,coarse,&Ydotc,&Ystagec,&Zdotc,&Zstagec);
1192: MatRestrict(restrct,Ydot,Ydotc);
1193: VecPointwiseMult(Ydotc,rscale,Ydotc);
1194: MatRestrict(restrct,Ystage,Ystagec);
1195: VecPointwiseMult(Ystagec,rscale,Ystagec);
1196: MatRestrict(restrct,Zdot,Zdotc);
1197: VecPointwiseMult(Zdotc,rscale,Zdotc);
1198: MatRestrict(restrct,Zstage,Zstagec);
1199: VecPointwiseMult(Zstagec,rscale,Zstagec);
1200: TSRosWRestoreVecs(ts,fine,&Ydot,&Ystage,&Zdot,&Zstage);
1201: TSRosWRestoreVecs(ts,coarse,&Ydotc,&Ystagec,&Zdotc,&Zstagec);
1202: return 0;
1203: }
1205: static PetscErrorCode DMSubDomainHook_TSRosW(DM fine,DM coarse,void *ctx)
1206: {
1207: return 0;
1208: }
1210: static PetscErrorCode DMSubDomainRestrictHook_TSRosW(DM dm,VecScatter gscat,VecScatter lscat,DM subdm,void *ctx)
1211: {
1212: TS ts = (TS)ctx;
1213: Vec Ydot,Zdot,Ystage,Zstage;
1214: Vec Ydots,Zdots,Ystages,Zstages;
1216: TSRosWGetVecs(ts,dm,&Ydot,&Ystage,&Zdot,&Zstage);
1217: TSRosWGetVecs(ts,subdm,&Ydots,&Ystages,&Zdots,&Zstages);
1219: VecScatterBegin(gscat,Ydot,Ydots,INSERT_VALUES,SCATTER_FORWARD);
1220: VecScatterEnd(gscat,Ydot,Ydots,INSERT_VALUES,SCATTER_FORWARD);
1222: VecScatterBegin(gscat,Ystage,Ystages,INSERT_VALUES,SCATTER_FORWARD);
1223: VecScatterEnd(gscat,Ystage,Ystages,INSERT_VALUES,SCATTER_FORWARD);
1225: VecScatterBegin(gscat,Zdot,Zdots,INSERT_VALUES,SCATTER_FORWARD);
1226: VecScatterEnd(gscat,Zdot,Zdots,INSERT_VALUES,SCATTER_FORWARD);
1228: VecScatterBegin(gscat,Zstage,Zstages,INSERT_VALUES,SCATTER_FORWARD);
1229: VecScatterEnd(gscat,Zstage,Zstages,INSERT_VALUES,SCATTER_FORWARD);
1231: TSRosWRestoreVecs(ts,dm,&Ydot,&Ystage,&Zdot,&Zstage);
1232: TSRosWRestoreVecs(ts,subdm,&Ydots,&Ystages,&Zdots,&Zstages);
1233: return 0;
1234: }
1236: /*
1237: This defines the nonlinear equation that is to be solved with SNES
1238: G(U) = F[t0+Theta*dt, U, (U-U0)*shift] = 0
1239: */
1240: static PetscErrorCode SNESTSFormFunction_RosW(SNES snes,Vec U,Vec F,TS ts)
1241: {
1242: TS_RosW *ros = (TS_RosW*)ts->data;
1243: Vec Ydot,Zdot,Ystage,Zstage;
1244: PetscReal shift = ros->scoeff / ts->time_step;
1245: DM dm,dmsave;
1247: SNESGetDM(snes,&dm);
1248: TSRosWGetVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
1249: VecWAXPY(Ydot,shift,U,Zdot); /* Ydot = shift*U + Zdot */
1250: VecWAXPY(Ystage,1.0,U,Zstage); /* Ystage = U + Zstage */
1251: dmsave = ts->dm;
1252: ts->dm = dm;
1253: TSComputeIFunction(ts,ros->stage_time,Ystage,Ydot,F,PETSC_FALSE);
1254: ts->dm = dmsave;
1255: TSRosWRestoreVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
1256: return 0;
1257: }
1259: static PetscErrorCode SNESTSFormJacobian_RosW(SNES snes,Vec U,Mat A,Mat B,TS ts)
1260: {
1261: TS_RosW *ros = (TS_RosW*)ts->data;
1262: Vec Ydot,Zdot,Ystage,Zstage;
1263: PetscReal shift = ros->scoeff / ts->time_step;
1264: DM dm,dmsave;
1266: /* ros->Ydot and ros->Ystage have already been computed in SNESTSFormFunction_RosW (SNES guarantees this) */
1267: SNESGetDM(snes,&dm);
1268: TSRosWGetVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
1269: dmsave = ts->dm;
1270: ts->dm = dm;
1271: TSComputeIJacobian(ts,ros->stage_time,Ystage,Ydot,shift,A,B,PETSC_TRUE);
1272: ts->dm = dmsave;
1273: TSRosWRestoreVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
1274: return 0;
1275: }
1277: static PetscErrorCode TSRosWTableauSetUp(TS ts)
1278: {
1279: TS_RosW *ros = (TS_RosW*)ts->data;
1280: RosWTableau tab = ros->tableau;
1282: VecDuplicateVecs(ts->vec_sol,tab->s,&ros->Y);
1283: PetscMalloc1(tab->s,&ros->work);
1284: return 0;
1285: }
1287: static PetscErrorCode TSSetUp_RosW(TS ts)
1288: {
1289: TS_RosW *ros = (TS_RosW*)ts->data;
1290: DM dm;
1291: SNES snes;
1292: TSRHSJacobian rhsjacobian;
1294: TSRosWTableauSetUp(ts);
1295: VecDuplicate(ts->vec_sol,&ros->Ydot);
1296: VecDuplicate(ts->vec_sol,&ros->Ystage);
1297: VecDuplicate(ts->vec_sol,&ros->Zdot);
1298: VecDuplicate(ts->vec_sol,&ros->Zstage);
1299: VecDuplicate(ts->vec_sol,&ros->vec_sol_prev);
1300: TSGetDM(ts,&dm);
1301: DMCoarsenHookAdd(dm,DMCoarsenHook_TSRosW,DMRestrictHook_TSRosW,ts);
1302: DMSubDomainHookAdd(dm,DMSubDomainHook_TSRosW,DMSubDomainRestrictHook_TSRosW,ts);
1303: /* Rosenbrock methods are linearly implicit, so set that unless the user has specifically asked for something else */
1304: TSGetSNES(ts,&snes);
1305: if (!((PetscObject)snes)->type_name) {
1306: SNESSetType(snes,SNESKSPONLY);
1307: }
1308: DMTSGetRHSJacobian(dm,&rhsjacobian,NULL);
1309: if (rhsjacobian == TSComputeRHSJacobianConstant) {
1310: Mat Amat,Pmat;
1312: /* Set the SNES matrix to be different from the RHS matrix because there is no way to reconstruct shift*M-J */
1313: SNESGetJacobian(snes,&Amat,&Pmat,NULL,NULL);
1314: if (Amat && Amat == ts->Arhs) {
1315: if (Amat == Pmat) {
1316: MatDuplicate(ts->Arhs,MAT_COPY_VALUES,&Amat);
1317: SNESSetJacobian(snes,Amat,Amat,NULL,NULL);
1318: } else {
1319: MatDuplicate(ts->Arhs,MAT_COPY_VALUES,&Amat);
1320: SNESSetJacobian(snes,Amat,NULL,NULL,NULL);
1321: if (Pmat && Pmat == ts->Brhs) {
1322: MatDuplicate(ts->Brhs,MAT_COPY_VALUES,&Pmat);
1323: SNESSetJacobian(snes,NULL,Pmat,NULL,NULL);
1324: MatDestroy(&Pmat);
1325: }
1326: }
1327: MatDestroy(&Amat);
1328: }
1329: }
1330: return 0;
1331: }
1332: /*------------------------------------------------------------*/
1334: static PetscErrorCode TSSetFromOptions_RosW(PetscOptionItems *PetscOptionsObject,TS ts)
1335: {
1336: TS_RosW *ros = (TS_RosW*)ts->data;
1337: SNES snes;
1339: PetscOptionsHead(PetscOptionsObject,"RosW ODE solver options");
1340: {
1341: RosWTableauLink link;
1342: PetscInt count,choice;
1343: PetscBool flg;
1344: const char **namelist;
1346: for (link=RosWTableauList,count=0; link; link=link->next,count++) ;
1347: PetscMalloc1(count,(char***)&namelist);
1348: for (link=RosWTableauList,count=0; link; link=link->next,count++) namelist[count] = link->tab.name;
1349: PetscOptionsEList("-ts_rosw_type","Family of Rosenbrock-W method","TSRosWSetType",(const char*const*)namelist,count,ros->tableau->name,&choice,&flg);
1350: if (flg) TSRosWSetType(ts,namelist[choice]);
1351: PetscFree(namelist);
1353: PetscOptionsBool("-ts_rosw_recompute_jacobian","Recompute the Jacobian at each stage","TSRosWSetRecomputeJacobian",ros->recompute_jacobian,&ros->recompute_jacobian,NULL);
1354: }
1355: PetscOptionsTail();
1356: /* Rosenbrock methods are linearly implicit, so set that unless the user has specifically asked for something else */
1357: TSGetSNES(ts,&snes);
1358: if (!((PetscObject)snes)->type_name) {
1359: SNESSetType(snes,SNESKSPONLY);
1360: }
1361: return 0;
1362: }
1364: static PetscErrorCode TSView_RosW(TS ts,PetscViewer viewer)
1365: {
1366: TS_RosW *ros = (TS_RosW*)ts->data;
1367: PetscBool iascii;
1369: PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
1370: if (iascii) {
1371: RosWTableau tab = ros->tableau;
1372: TSRosWType rostype;
1373: char buf[512];
1374: PetscInt i;
1375: PetscReal abscissa[512];
1376: TSRosWGetType(ts,&rostype);
1377: PetscViewerASCIIPrintf(viewer," Rosenbrock-W %s\n",rostype);
1378: PetscFormatRealArray(buf,sizeof(buf),"% 8.6f",tab->s,tab->ASum);
1379: PetscViewerASCIIPrintf(viewer," Abscissa of A = %s\n",buf);
1380: for (i=0; i<tab->s; i++) abscissa[i] = tab->ASum[i] + tab->Gamma[i];
1381: PetscFormatRealArray(buf,sizeof(buf),"% 8.6f",tab->s,abscissa);
1382: PetscViewerASCIIPrintf(viewer," Abscissa of A+Gamma = %s\n",buf);
1383: }
1384: return 0;
1385: }
1387: static PetscErrorCode TSLoad_RosW(TS ts,PetscViewer viewer)
1388: {
1389: SNES snes;
1390: TSAdapt adapt;
1392: TSGetAdapt(ts,&adapt);
1393: TSAdaptLoad(adapt,viewer);
1394: TSGetSNES(ts,&snes);
1395: SNESLoad(snes,viewer);
1396: /* function and Jacobian context for SNES when used with TS is always ts object */
1397: SNESSetFunction(snes,NULL,NULL,ts);
1398: SNESSetJacobian(snes,NULL,NULL,NULL,ts);
1399: return 0;
1400: }
1402: /*@C
1403: TSRosWSetType - Set the type of Rosenbrock-W scheme
1405: Logically collective
1407: Input Parameters:
1408: + ts - timestepping context
1409: - roswtype - type of Rosenbrock-W scheme
1411: Level: beginner
1413: .seealso: TSRosWGetType(), TSROSW, TSROSW2M, TSROSW2P, TSROSWRA3PW, TSROSWRA34PW2, TSROSWRODAS3, TSROSWSANDU3, TSROSWASSP3P3S1C, TSROSWLASSP3P4S2C, TSROSWLLSSP3P4S2C, TSROSWARK3
1414: @*/
1415: PetscErrorCode TSRosWSetType(TS ts,TSRosWType roswtype)
1416: {
1419: PetscTryMethod(ts,"TSRosWSetType_C",(TS,TSRosWType),(ts,roswtype));
1420: return 0;
1421: }
1423: /*@C
1424: TSRosWGetType - Get the type of Rosenbrock-W scheme
1426: Logically collective
1428: Input Parameter:
1429: . ts - timestepping context
1431: Output Parameter:
1432: . rostype - type of Rosenbrock-W scheme
1434: Level: intermediate
1436: .seealso: TSRosWGetType()
1437: @*/
1438: PetscErrorCode TSRosWGetType(TS ts,TSRosWType *rostype)
1439: {
1441: PetscUseMethod(ts,"TSRosWGetType_C",(TS,TSRosWType*),(ts,rostype));
1442: return 0;
1443: }
1445: /*@C
1446: TSRosWSetRecomputeJacobian - Set whether to recompute the Jacobian at each stage. The default is to update the Jacobian once per step.
1448: Logically collective
1450: Input Parameters:
1451: + ts - timestepping context
1452: - flg - PETSC_TRUE to recompute the Jacobian at each stage
1454: Level: intermediate
1456: .seealso: TSRosWGetType()
1457: @*/
1458: PetscErrorCode TSRosWSetRecomputeJacobian(TS ts,PetscBool flg)
1459: {
1461: PetscTryMethod(ts,"TSRosWSetRecomputeJacobian_C",(TS,PetscBool),(ts,flg));
1462: return 0;
1463: }
1465: static PetscErrorCode TSRosWGetType_RosW(TS ts,TSRosWType *rostype)
1466: {
1467: TS_RosW *ros = (TS_RosW*)ts->data;
1469: *rostype = ros->tableau->name;
1470: return 0;
1471: }
1473: static PetscErrorCode TSRosWSetType_RosW(TS ts,TSRosWType rostype)
1474: {
1475: TS_RosW *ros = (TS_RosW*)ts->data;
1476: PetscBool match;
1477: RosWTableauLink link;
1479: if (ros->tableau) {
1480: PetscStrcmp(ros->tableau->name,rostype,&match);
1481: if (match) return 0;
1482: }
1483: for (link = RosWTableauList; link; link=link->next) {
1484: PetscStrcmp(link->tab.name,rostype,&match);
1485: if (match) {
1486: if (ts->setupcalled) TSRosWTableauReset(ts);
1487: ros->tableau = &link->tab;
1488: if (ts->setupcalled) TSRosWTableauSetUp(ts);
1489: ts->default_adapt_type = ros->tableau->bembed ? TSADAPTBASIC : TSADAPTNONE;
1490: return 0;
1491: }
1492: }
1493: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_ARG_UNKNOWN_TYPE,"Could not find '%s'",rostype);
1494: }
1496: static PetscErrorCode TSRosWSetRecomputeJacobian_RosW(TS ts,PetscBool flg)
1497: {
1498: TS_RosW *ros = (TS_RosW*)ts->data;
1500: ros->recompute_jacobian = flg;
1501: return 0;
1502: }
1504: static PetscErrorCode TSDestroy_RosW(TS ts)
1505: {
1506: TSReset_RosW(ts);
1507: if (ts->dm) {
1508: DMCoarsenHookRemove(ts->dm,DMCoarsenHook_TSRosW,DMRestrictHook_TSRosW,ts);
1509: DMSubDomainHookRemove(ts->dm,DMSubDomainHook_TSRosW,DMSubDomainRestrictHook_TSRosW,ts);
1510: }
1511: PetscFree(ts->data);
1512: PetscObjectComposeFunction((PetscObject)ts,"TSRosWGetType_C",NULL);
1513: PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetType_C",NULL);
1514: PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetRecomputeJacobian_C",NULL);
1515: return 0;
1516: }
1518: /* ------------------------------------------------------------ */
1519: /*MC
1520: TSROSW - ODE solver using Rosenbrock-W schemes
1522: These methods are intended for problems with well-separated time scales, especially when a slow scale is strongly
1523: nonlinear such that it is expensive to solve with a fully implicit method. The user should provide the stiff part
1524: of the equation using TSSetIFunction() and the non-stiff part with TSSetRHSFunction().
1526: Notes:
1527: This method currently only works with autonomous ODE and DAE.
1529: Consider trying TSARKIMEX if the stiff part is strongly nonlinear.
1531: Since this uses a single linear solve per time-step if you wish to lag the jacobian or preconditioner computation you must use also -snes_lag_jacobian_persists true or -snes_lag_jacobian_preconditioner true
1533: Developer Notes:
1534: Rosenbrock-W methods are typically specified for autonomous ODE
1536: $ udot = f(u)
1538: by the stage equations
1540: $ k_i = h f(u_0 + sum_j alpha_ij k_j) + h J sum_j gamma_ij k_j
1542: and step completion formula
1544: $ u_1 = u_0 + sum_j b_j k_j
1546: with step size h and coefficients alpha_ij, gamma_ij, and b_i. Implementing the method in this form would require f(u)
1547: and the Jacobian J to be available, in addition to the shifted matrix I - h gamma_ii J. Following Hairer and Wanner,
1548: we define new variables for the stage equations
1550: $ y_i = gamma_ij k_j
1552: The k_j can be recovered because Gamma is invertible. Let C be the lower triangular part of Gamma^{-1} and define
1554: $ A = Alpha Gamma^{-1}, bt^T = b^T Gamma^{-1}
1556: to rewrite the method as
1558: $ [M/(h gamma_ii) - J] y_i = f(u_0 + sum_j a_ij y_j) + M sum_j (c_ij/h) y_j
1559: $ u_1 = u_0 + sum_j bt_j y_j
1561: where we have introduced the mass matrix M. Continue by defining
1563: $ ydot_i = 1/(h gamma_ii) y_i - sum_j (c_ij/h) y_j
1565: or, more compactly in tensor notation
1567: $ Ydot = 1/h (Gamma^{-1} \otimes I) Y .
1569: Note that Gamma^{-1} is lower triangular. With this definition of Ydot in terms of known quantities and the current
1570: stage y_i, the stage equations reduce to performing one Newton step (typically with a lagged Jacobian) on the
1571: equation
1573: $ g(u_0 + sum_j a_ij y_j + y_i, ydot_i) = 0
1575: with initial guess y_i = 0.
1577: Level: beginner
1579: .seealso: TSCreate(), TS, TSSetType(), TSRosWSetType(), TSRosWRegister(), TSROSWTHETA1, TSROSWTHETA2, TSROSW2M, TSROSW2P, TSROSWRA3PW, TSROSWRA34PW2, TSROSWRODAS3,
1580: TSROSWSANDU3, TSROSWASSP3P3S1C, TSROSWLASSP3P4S2C, TSROSWLLSSP3P4S2C, TSROSWGRK4T, TSROSWSHAMP4, TSROSWVELDD4, TSROSW4L
1581: M*/
1582: PETSC_EXTERN PetscErrorCode TSCreate_RosW(TS ts)
1583: {
1584: TS_RosW *ros;
1586: TSRosWInitializePackage();
1588: ts->ops->reset = TSReset_RosW;
1589: ts->ops->destroy = TSDestroy_RosW;
1590: ts->ops->view = TSView_RosW;
1591: ts->ops->load = TSLoad_RosW;
1592: ts->ops->setup = TSSetUp_RosW;
1593: ts->ops->step = TSStep_RosW;
1594: ts->ops->interpolate = TSInterpolate_RosW;
1595: ts->ops->evaluatestep = TSEvaluateStep_RosW;
1596: ts->ops->rollback = TSRollBack_RosW;
1597: ts->ops->setfromoptions = TSSetFromOptions_RosW;
1598: ts->ops->snesfunction = SNESTSFormFunction_RosW;
1599: ts->ops->snesjacobian = SNESTSFormJacobian_RosW;
1601: ts->usessnes = PETSC_TRUE;
1603: PetscNewLog(ts,&ros);
1604: ts->data = (void*)ros;
1606: PetscObjectComposeFunction((PetscObject)ts,"TSRosWGetType_C",TSRosWGetType_RosW);
1607: PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetType_C",TSRosWSetType_RosW);
1608: PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetRecomputeJacobian_C",TSRosWSetRecomputeJacobian_RosW);
1610: TSRosWSetType(ts,TSRosWDefault);
1611: return 0;
1612: }