Actual source code: rosw.c
petsc-3.8.4 2018-03-24
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: . 1. - 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: . 1. - 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: . 1. - 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: . 1. - 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: + 1. - Kaps and Rentrop, Generalized Runge Kutta methods of order four with stepsize control for stiff ordinary differential equations, 1979.
225: - 2. - 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: + 1. - Shampine, Implementation of Rosenbrock methods, 1982.
245: - 2. - 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: + 1. - van Veldhuizen, D stability and Kaps Rentrop methods, 1984.
265: - 2. - 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: . 1. - 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: .keywords: TS, TSRosW, register, all
302: .seealso: TSRosWRegisterDestroy()
303: @*/
304: PetscErrorCode TSRosWRegisterAll(void)
305: {
309: if (TSRosWRegisterAllCalled) return(0);
310: TSRosWRegisterAllCalled = PETSC_TRUE;
312: {
313: const PetscReal A = 0;
314: const PetscReal Gamma = 1;
315: const PetscReal b = 1;
316: const PetscReal binterpt=1;
318: TSRosWRegister(TSROSWTHETA1,1,1,&A,&Gamma,&b,NULL,1,&binterpt);
319: }
321: {
322: const PetscReal A = 0;
323: const PetscReal Gamma = 0.5;
324: const PetscReal b = 1;
325: const PetscReal binterpt=1;
327: TSRosWRegister(TSROSWTHETA2,2,1,&A,&Gamma,&b,NULL,1,&binterpt);
328: }
330: {
331: /*const PetscReal g = 1. + 1./PetscSqrtReal(2.0); Direct evaluation: 1.707106781186547524401. Used for setting up arrays of values known at compile time below. */
332: const PetscReal
333: A[2][2] = {{0,0}, {1.,0}},
334: Gamma[2][2] = {{1.707106781186547524401,0}, {-2.*1.707106781186547524401,1.707106781186547524401}},
335: b[2] = {0.5,0.5},
336: b1[2] = {1.0,0.0};
337: PetscReal binterpt[2][2];
338: binterpt[0][0] = 1.707106781186547524401 - 1.0;
339: binterpt[1][0] = 2.0 - 1.707106781186547524401;
340: binterpt[0][1] = 1.707106781186547524401 - 1.5;
341: binterpt[1][1] = 1.5 - 1.707106781186547524401;
343: TSRosWRegister(TSROSW2P,2,2,&A[0][0],&Gamma[0][0],b,b1,2,&binterpt[0][0]);
344: }
345: {
346: /*const PetscReal g = 1. - 1./PetscSqrtReal(2.0); Direct evaluation: 0.2928932188134524755992. Used for setting up arrays of values known at compile time below. */
347: const PetscReal
348: A[2][2] = {{0,0}, {1.,0}},
349: Gamma[2][2] = {{0.2928932188134524755992,0}, {-2.*0.2928932188134524755992,0.2928932188134524755992}},
350: b[2] = {0.5,0.5},
351: b1[2] = {1.0,0.0};
352: PetscReal binterpt[2][2];
353: binterpt[0][0] = 0.2928932188134524755992 - 1.0;
354: binterpt[1][0] = 2.0 - 0.2928932188134524755992;
355: binterpt[0][1] = 0.2928932188134524755992 - 1.5;
356: binterpt[1][1] = 1.5 - 0.2928932188134524755992;
358: TSRosWRegister(TSROSW2M,2,2,&A[0][0],&Gamma[0][0],b,b1,2,&binterpt[0][0]);
359: }
360: {
361: /*const PetscReal g = 7.8867513459481287e-01; Directly written in-place below */
362: PetscReal binterpt[3][2];
363: const PetscReal
364: A[3][3] = {{0,0,0},
365: {1.5773502691896257e+00,0,0},
366: {0.5,0,0}},
367: Gamma[3][3] = {{7.8867513459481287e-01,0,0},
368: {-1.5773502691896257e+00,7.8867513459481287e-01,0},
369: {-6.7075317547305480e-01,-1.7075317547305482e-01,7.8867513459481287e-01}},
370: b[3] = {1.0566243270259355e-01,4.9038105676657971e-02,8.4529946162074843e-01},
371: b2[3] = {-1.7863279495408180e-01,1./3.,8.4529946162074843e-01};
373: binterpt[0][0] = -0.8094010767585034;
374: binterpt[1][0] = -0.5;
375: binterpt[2][0] = 2.3094010767585034;
376: binterpt[0][1] = 0.9641016151377548;
377: binterpt[1][1] = 0.5;
378: binterpt[2][1] = -1.4641016151377548;
380: TSRosWRegister(TSROSWRA3PW,3,3,&A[0][0],&Gamma[0][0],b,b2,2,&binterpt[0][0]);
381: }
382: {
383: PetscReal binterpt[4][3];
384: /*const PetscReal g = 4.3586652150845900e-01; Directly written in-place below */
385: const PetscReal
386: A[4][4] = {{0,0,0,0},
387: {8.7173304301691801e-01,0,0,0},
388: {8.4457060015369423e-01,-1.1299064236484185e-01,0,0},
389: {0,0,1.,0}},
390: Gamma[4][4] = {{4.3586652150845900e-01,0,0,0},
391: {-8.7173304301691801e-01,4.3586652150845900e-01,0,0},
392: {-9.0338057013044082e-01,5.4180672388095326e-02,4.3586652150845900e-01,0},
393: {2.4212380706095346e-01,-1.2232505839045147e+00,5.4526025533510214e-01,4.3586652150845900e-01}},
394: b[4] = {2.4212380706095346e-01,-1.2232505839045147e+00,1.5452602553351020e+00,4.3586652150845900e-01},
395: b2[4] = {3.7810903145819369e-01,-9.6042292212423178e-02,5.0000000000000000e-01,2.1793326075422950e-01};
397: binterpt[0][0]=1.0564298455794094;
398: binterpt[1][0]=2.296429974281067;
399: binterpt[2][0]=-1.307599564525376;
400: binterpt[3][0]=-1.045260255335102;
401: binterpt[0][1]=-1.3864882699759573;
402: binterpt[1][1]=-8.262611700275677;
403: binterpt[2][1]=7.250979895056055;
404: binterpt[3][1]=2.398120075195581;
405: binterpt[0][2]=0.5721822314575016;
406: binterpt[1][2]=4.742931142090097;
407: binterpt[2][2]=-4.398120075195578;
408: binterpt[3][2]=-0.9169932983520199;
410: TSRosWRegister(TSROSWRA34PW2,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
411: }
412: {
413: /* const PetscReal g = 0.5; Directly written in-place below */
414: const PetscReal
415: A[4][4] = {{0,0,0,0},
416: {0,0,0,0},
417: {1.,0,0,0},
418: {0.75,-0.25,0.5,0}},
419: Gamma[4][4] = {{0.5,0,0,0},
420: {1.,0.5,0,0},
421: {-0.25,-0.25,0.5,0},
422: {1./12,1./12,-2./3,0.5}},
423: b[4] = {5./6,-1./6,-1./6,0.5},
424: b2[4] = {0.75,-0.25,0.5,0};
426: TSRosWRegister(TSROSWRODAS3,3,4,&A[0][0],&Gamma[0][0],b,b2,0,NULL);
427: }
428: {
429: /*const PetscReal g = 0.43586652150845899941601945119356; Directly written in-place below */
430: const PetscReal
431: A[3][3] = {{0,0,0},
432: {0.43586652150845899941601945119356,0,0},
433: {0.43586652150845899941601945119356,0,0}},
434: Gamma[3][3] = {{0.43586652150845899941601945119356,0,0},
435: {-0.19294655696029095575009695436041,0.43586652150845899941601945119356,0},
436: {0,1.74927148125794685173529749738960,0.43586652150845899941601945119356}},
437: b[3] = {-0.75457412385404315829818998646589,1.94100407061964420292840123379419,-0.18642994676560104463021124732829},
438: b2[3] = {-1.53358745784149585370766523913002,2.81745131148625772213931745457622,-0.28386385364476186843165221544619};
440: PetscReal binterpt[3][2];
441: binterpt[0][0] = 3.793692883777660870425141387941;
442: binterpt[1][0] = -2.918692883777660870425141387941;
443: binterpt[2][0] = 0.125;
444: binterpt[0][1] = -0.725741064379812106687651020584;
445: binterpt[1][1] = 0.559074397713145440020984353917;
446: binterpt[2][1] = 0.16666666666666666666666666666667;
448: TSRosWRegister(TSROSWSANDU3,3,3,&A[0][0],&Gamma[0][0],b,b2,2,&binterpt[0][0]);
449: }
450: {
451: /*const PetscReal s3 = PetscSqrtReal(3.),g = (3.0+s3)/6.0;
452: * Direct evaluation: s3 = 1.732050807568877293527;
453: * g = 0.7886751345948128822546;
454: * Values are directly inserted below to ensure availability at compile time (compiler warnings otherwise...) */
455: const PetscReal
456: A[3][3] = {{0,0,0},
457: {1,0,0},
458: {0.25,0.25,0}},
459: Gamma[3][3] = {{0,0,0},
460: {(-3.0-1.732050807568877293527)/6.0,0.7886751345948128822546,0},
461: {(-3.0-1.732050807568877293527)/24.0,(-3.0-1.732050807568877293527)/8.0,0.7886751345948128822546}},
462: b[3] = {1./6.,1./6.,2./3.},
463: b2[3] = {1./4.,1./4.,1./2.};
464: PetscReal binterpt[3][2];
466: binterpt[0][0]=0.089316397477040902157517886164709;
467: binterpt[1][0]=-0.91068360252295909784248211383529;
468: binterpt[2][0]=1.8213672050459181956849642276706;
469: binterpt[0][1]=0.077350269189625764509148780501957;
470: binterpt[1][1]=1.077350269189625764509148780502;
471: binterpt[2][1]=-1.1547005383792515290182975610039;
473: TSRosWRegister(TSROSWASSP3P3S1C,3,3,&A[0][0],&Gamma[0][0],b,b2,2,&binterpt[0][0]);
474: }
476: {
477: const PetscReal
478: A[4][4] = {{0,0,0,0},
479: {1./2.,0,0,0},
480: {1./2.,1./2.,0,0},
481: {1./6.,1./6.,1./6.,0}},
482: Gamma[4][4] = {{1./2.,0,0,0},
483: {0.0,1./4.,0,0},
484: {-2.,-2./3.,2./3.,0},
485: {1./2.,5./36.,-2./9,0}},
486: b[4] = {1./6.,1./6.,1./6.,1./2.},
487: b2[4] = {1./8.,3./4.,1./8.,0};
488: PetscReal binterpt[4][3];
490: binterpt[0][0]=6.25;
491: binterpt[1][0]=-30.25;
492: binterpt[2][0]=1.75;
493: binterpt[3][0]=23.25;
494: binterpt[0][1]=-9.75;
495: binterpt[1][1]=58.75;
496: binterpt[2][1]=-3.25;
497: binterpt[3][1]=-45.75;
498: binterpt[0][2]=3.6666666666666666666666666666667;
499: binterpt[1][2]=-28.333333333333333333333333333333;
500: binterpt[2][2]=1.6666666666666666666666666666667;
501: binterpt[3][2]=23.;
503: TSRosWRegister(TSROSWLASSP3P4S2C,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
504: }
506: {
507: const PetscReal
508: A[4][4] = {{0,0,0,0},
509: {1./2.,0,0,0},
510: {1./2.,1./2.,0,0},
511: {1./6.,1./6.,1./6.,0}},
512: Gamma[4][4] = {{1./2.,0,0,0},
513: {0.0,3./4.,0,0},
514: {-2./3.,-23./9.,2./9.,0},
515: {1./18.,65./108.,-2./27,0}},
516: b[4] = {1./6.,1./6.,1./6.,1./2.},
517: b2[4] = {3./16.,10./16.,3./16.,0};
518: PetscReal binterpt[4][3];
520: binterpt[0][0]=1.6911764705882352941176470588235;
521: binterpt[1][0]=3.6813725490196078431372549019608;
522: binterpt[2][0]=0.23039215686274509803921568627451;
523: binterpt[3][0]=-4.6029411764705882352941176470588;
524: binterpt[0][1]=-0.95588235294117647058823529411765;
525: binterpt[1][1]=-6.2401960784313725490196078431373;
526: binterpt[2][1]=-0.31862745098039215686274509803922;
527: binterpt[3][1]=7.5147058823529411764705882352941;
528: binterpt[0][2]=-0.56862745098039215686274509803922;
529: binterpt[1][2]=2.7254901960784313725490196078431;
530: binterpt[2][2]=0.25490196078431372549019607843137;
531: binterpt[3][2]=-2.4117647058823529411764705882353;
533: TSRosWRegister(TSROSWLLSSP3P4S2C,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
534: }
536: {
537: PetscReal A[4][4],Gamma[4][4],b[4],b2[4];
538: PetscReal binterpt[4][3];
540: Gamma[0][0]=0.4358665215084589994160194475295062513822671686978816;
541: Gamma[0][1]=0; Gamma[0][2]=0; Gamma[0][3]=0;
542: Gamma[1][0]=-1.997527830934941248426324674704153457289527280554476;
543: Gamma[1][1]=0.4358665215084589994160194475295062513822671686978816;
544: Gamma[1][2]=0; Gamma[1][3]=0;
545: Gamma[2][0]=-1.007948511795029620852002345345404191008352770119903;
546: Gamma[2][1]=-0.004648958462629345562774289390054679806993396798458131;
547: Gamma[2][2]=0.4358665215084589994160194475295062513822671686978816;
548: Gamma[2][3]=0;
549: Gamma[3][0]=-0.6685429734233467180451604600279552604364311322650783;
550: Gamma[3][1]=0.6056625986449338476089525334450053439525178740492984;
551: Gamma[3][2]=-0.9717899277217721234705114616271378792182450260943198;
552: Gamma[3][3]=0;
554: A[0][0]=0; A[0][1]=0; A[0][2]=0; A[0][3]=0;
555: A[1][0]=0.8717330430169179988320388950590125027645343373957631;
556: A[1][1]=0; A[1][2]=0; A[1][3]=0;
557: A[2][0]=0.5275890119763004115618079766722914408876108660811028;
558: A[2][1]=0.07241098802369958843819203208518599088698057726988732;
559: A[2][2]=0; A[2][3]=0;
560: A[3][0]=0.3990960076760701320627260685975778145384666450351314;
561: A[3][1]=-0.4375576546135194437228463747348862825846903771419953;
562: A[3][2]=1.038461646937449311660120300601880176655352737312713;
563: A[3][3]=0;
565: b[0]=0.1876410243467238251612921333138006734899663569186926;
566: b[1]=-0.5952974735769549480478230473706443582188442040780541;
567: b[2]=0.9717899277217721234705114616271378792182450260943198;
568: b[3]=0.4358665215084589994160194475295062513822671686978816;
570: b2[0]=0.2147402862233891404862383521089097657790734483804460;
571: b2[1]=-0.4851622638849390928209050538171743017757490232519684;
572: b2[2]=0.8687250025203875511662123688667549217531982787600080;
573: b2[3]=0.4016969751411624011684543450940068201770721128357014;
575: binterpt[0][0]=2.2565812720167954547104627844105;
576: binterpt[1][0]=1.349166413351089573796243820819;
577: binterpt[2][0]=-2.4695174540533503758652847586647;
578: binterpt[3][0]=-0.13623023131453465264142184656474;
579: binterpt[0][1]=-3.0826699111559187902922463354557;
580: binterpt[1][1]=-2.4689115685996042534544925650515;
581: binterpt[2][1]=5.7428279814696677152129332773553;
582: binterpt[3][1]=-0.19124650171414467146619437684812;
583: binterpt[0][2]=1.0137296634858471607430756831148;
584: binterpt[1][2]=0.52444768167155973161042570784064;
585: binterpt[2][2]=-2.3015205996945452158771370439586;
586: binterpt[3][2]=0.76334325453713832352363565300308;
588: TSRosWRegister(TSROSWARK3,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
589: }
590: TSRosWRegisterRos4(TSROSWGRK4T,0.231,PETSC_DEFAULT,PETSC_DEFAULT,0,-0.1282612945269037e+01);
591: TSRosWRegisterRos4(TSROSWSHAMP4,0.5,PETSC_DEFAULT,PETSC_DEFAULT,0,125./108.);
592: TSRosWRegisterRos4(TSROSWVELDD4,0.22570811482256823492,PETSC_DEFAULT,PETSC_DEFAULT,0,-1.355958941201148);
593: TSRosWRegisterRos4(TSROSW4L,0.57282,PETSC_DEFAULT,PETSC_DEFAULT,0,-1.093502252409163);
594: return(0);
595: }
599: /*@C
600: TSRosWRegisterDestroy - Frees the list of schemes that were registered by TSRosWRegister().
602: Not Collective
604: Level: advanced
606: .keywords: TSRosW, register, destroy
607: .seealso: TSRosWRegister(), TSRosWRegisterAll()
608: @*/
609: PetscErrorCode TSRosWRegisterDestroy(void)
610: {
611: PetscErrorCode ierr;
612: RosWTableauLink link;
615: while ((link = RosWTableauList)) {
616: RosWTableau t = &link->tab;
617: RosWTableauList = link->next;
618: PetscFree5(t->A,t->Gamma,t->b,t->ASum,t->GammaSum);
619: PetscFree5(t->At,t->bt,t->GammaInv,t->GammaZeroDiag,t->GammaExplicitCorr);
620: PetscFree2(t->bembed,t->bembedt);
621: PetscFree(t->binterpt);
622: PetscFree(t->name);
623: PetscFree(link);
624: }
625: TSRosWRegisterAllCalled = PETSC_FALSE;
626: return(0);
627: }
629: /*@C
630: TSRosWInitializePackage - This function initializes everything in the TSRosW package. It is called
631: from PetscDLLibraryRegister() when using dynamic libraries, and on the first call to TSCreate_RosW()
632: when using static libraries.
634: Level: developer
636: .keywords: TS, TSRosW, initialize, package
637: .seealso: PetscInitialize()
638: @*/
639: PetscErrorCode TSRosWInitializePackage(void)
640: {
644: if (TSRosWPackageInitialized) return(0);
645: TSRosWPackageInitialized = PETSC_TRUE;
646: TSRosWRegisterAll();
647: PetscRegisterFinalize(TSRosWFinalizePackage);
648: return(0);
649: }
651: /*@C
652: TSRosWFinalizePackage - This function destroys everything in the TSRosW package. It is
653: called from PetscFinalize().
655: Level: developer
657: .keywords: Petsc, destroy, package
658: .seealso: PetscFinalize()
659: @*/
660: PetscErrorCode TSRosWFinalizePackage(void)
661: {
665: TSRosWPackageInitialized = PETSC_FALSE;
666: TSRosWRegisterDestroy();
667: return(0);
668: }
670: /*@C
671: TSRosWRegister - register a Rosenbrock W scheme by providing the entries in the Butcher tableau and optionally embedded approximations and interpolation
673: Not Collective, but the same schemes should be registered on all processes on which they will be used
675: Input Parameters:
676: + name - identifier for method
677: . order - approximation order of method
678: . s - number of stages, this is the dimension of the matrices below
679: . A - Table of propagated stage coefficients (dimension s*s, row-major), strictly lower triangular
680: . Gamma - Table of coefficients in implicit stage equations (dimension s*s, row-major), lower triangular with nonzero diagonal
681: . b - Step completion table (dimension s)
682: . bembed - Step completion table for a scheme of order one less (dimension s, NULL if no embedded scheme is available)
683: . pinterp - Order of the interpolation scheme, equal to the number of columns of binterpt
684: - binterpt - Coefficients of the interpolation formula (dimension s*pinterp)
686: Notes:
687: Several Rosenbrock W methods are provided, this function is only needed to create new methods.
689: Level: advanced
691: .keywords: TS, register
693: .seealso: TSRosW
694: @*/
695: PetscErrorCode TSRosWRegister(TSRosWType name,PetscInt order,PetscInt s,const PetscReal A[],const PetscReal Gamma[],const PetscReal b[],const PetscReal bembed[],
696: PetscInt pinterp,const PetscReal binterpt[])
697: {
698: PetscErrorCode ierr;
699: RosWTableauLink link;
700: RosWTableau t;
701: PetscInt i,j,k;
702: PetscScalar *GammaInv;
711: PetscCalloc1(1,&link);
712: t = &link->tab;
713: PetscStrallocpy(name,&t->name);
714: t->order = order;
715: t->s = s;
716: PetscMalloc5(s*s,&t->A,s*s,&t->Gamma,s,&t->b,s,&t->ASum,s,&t->GammaSum);
717: PetscMalloc5(s*s,&t->At,s,&t->bt,s*s,&t->GammaInv,s,&t->GammaZeroDiag,s*s,&t->GammaExplicitCorr);
718: PetscMemcpy(t->A,A,s*s*sizeof(A[0]));
719: PetscMemcpy(t->Gamma,Gamma,s*s*sizeof(Gamma[0]));
720: PetscMemcpy(t->GammaExplicitCorr,Gamma,s*s*sizeof(Gamma[0]));
721: PetscMemcpy(t->b,b,s*sizeof(b[0]));
722: if (bembed) {
723: PetscMalloc2(s,&t->bembed,s,&t->bembedt);
724: PetscMemcpy(t->bembed,bembed,s*sizeof(bembed[0]));
725: }
726: for (i=0; i<s; i++) {
727: t->ASum[i] = 0;
728: t->GammaSum[i] = 0;
729: for (j=0; j<s; j++) {
730: t->ASum[i] += A[i*s+j];
731: t->GammaSum[i] += Gamma[i*s+j];
732: }
733: }
734: PetscMalloc1(s*s,&GammaInv); /* Need to use Scalar for inverse, then convert back to Real */
735: for (i=0; i<s*s; i++) GammaInv[i] = Gamma[i];
736: for (i=0; i<s; i++) {
737: if (Gamma[i*s+i] == 0.0) {
738: GammaInv[i*s+i] = 1.0;
739: t->GammaZeroDiag[i] = PETSC_TRUE;
740: } else {
741: t->GammaZeroDiag[i] = PETSC_FALSE;
742: }
743: }
745: switch (s) {
746: case 1: GammaInv[0] = 1./GammaInv[0]; break;
747: case 2: PetscKernel_A_gets_inverse_A_2(GammaInv,0,PETSC_FALSE,NULL); break;
748: case 3: PetscKernel_A_gets_inverse_A_3(GammaInv,0,PETSC_FALSE,NULL); break;
749: case 4: PetscKernel_A_gets_inverse_A_4(GammaInv,0,PETSC_FALSE,NULL); break;
750: case 5: {
751: PetscInt ipvt5[5];
752: MatScalar work5[5*5];
753: PetscKernel_A_gets_inverse_A_5(GammaInv,ipvt5,work5,0,PETSC_FALSE,NULL); break;
754: }
755: case 6: PetscKernel_A_gets_inverse_A_6(GammaInv,0,PETSC_FALSE,NULL); break;
756: case 7: PetscKernel_A_gets_inverse_A_7(GammaInv,0,PETSC_FALSE,NULL); break;
757: default: SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_SUP,"Not implemented for %D stages",s);
758: }
759: for (i=0; i<s*s; i++) t->GammaInv[i] = PetscRealPart(GammaInv[i]);
760: PetscFree(GammaInv);
762: for (i=0; i<s; i++) {
763: for (k=0; k<i+1; k++) {
764: t->GammaExplicitCorr[i*s+k]=(t->GammaExplicitCorr[i*s+k])*(t->GammaInv[k*s+k]);
765: for (j=k+1; j<i+1; j++) {
766: t->GammaExplicitCorr[i*s+k]+=(t->GammaExplicitCorr[i*s+j])*(t->GammaInv[j*s+k]);
767: }
768: }
769: }
771: for (i=0; i<s; i++) {
772: for (j=0; j<s; j++) {
773: t->At[i*s+j] = 0;
774: for (k=0; k<s; k++) {
775: t->At[i*s+j] += t->A[i*s+k] * t->GammaInv[k*s+j];
776: }
777: }
778: t->bt[i] = 0;
779: for (j=0; j<s; j++) {
780: t->bt[i] += t->b[j] * t->GammaInv[j*s+i];
781: }
782: if (bembed) {
783: t->bembedt[i] = 0;
784: for (j=0; j<s; j++) {
785: t->bembedt[i] += t->bembed[j] * t->GammaInv[j*s+i];
786: }
787: }
788: }
789: t->ccfl = 1.0; /* Fix this */
791: t->pinterp = pinterp;
792: PetscMalloc1(s*pinterp,&t->binterpt);
793: PetscMemcpy(t->binterpt,binterpt,s*pinterp*sizeof(binterpt[0]));
794: link->next = RosWTableauList;
795: RosWTableauList = link;
796: return(0);
797: }
799: /*@C
800: TSRosWRegisterRos4 - register a fourth order Rosenbrock scheme by providing paramter choices
802: Not Collective, but the same schemes should be registered on all processes on which they will be used
804: Input Parameters:
805: + name - identifier for method
806: . gamma - leading coefficient (diagonal entry)
807: . a2 - design parameter, see Table 7.2 of Hairer&Wanner
808: . a3 - design parameter or PETSC_DEFAULT to satisfy one of the order five conditions (Eq 7.22)
809: . b3 - design parameter, see Table 7.2 of Hairer&Wanner
810: . beta43 - design parameter or PETSC_DEFAULT to use Equation 7.21 of Hairer&Wanner
811: . e4 - design parameter for embedded method, see coefficient E4 in ros4.f code from Hairer
813: Notes:
814: This routine encodes the design of fourth order Rosenbrock methods as described in Hairer and Wanner volume 2.
815: It is used here to implement several methods from the book and can be used to experiment with new methods.
816: It was written this way instead of by copying coefficients in order to provide better than double precision satisfaction of the order conditions.
818: Level: developer
820: .keywords: TS, register
822: .seealso: TSRosW, TSRosWRegister()
823: @*/
824: PetscErrorCode TSRosWRegisterRos4(TSRosWType name,PetscReal gamma,PetscReal a2,PetscReal a3,PetscReal b3,PetscReal e4)
825: {
827: /* Declare numeric constants so they can be quad precision without being truncated at double */
828: const PetscReal one = 1,two = 2,three = 3,four = 4,five = 5,six = 6,eight = 8,twelve = 12,twenty = 20,twentyfour = 24,
829: p32 = one/six - gamma + gamma*gamma,
830: p42 = one/eight - gamma/three,
831: p43 = one/twelve - gamma/three,
832: p44 = one/twentyfour - gamma/two + three/two*gamma*gamma - gamma*gamma*gamma,
833: p56 = one/twenty - gamma/four;
834: PetscReal a4,a32,a42,a43,b1,b2,b4,beta2p,beta3p,beta4p,beta32,beta42,beta43,beta32beta2p,beta4jbetajp;
835: PetscReal A[4][4],Gamma[4][4],b[4],bm[4];
836: PetscScalar M[3][3],rhs[3];
839: /* Step 1: choose Gamma (input) */
840: /* Step 2: choose a2,a3,a4; b1,b2,b3,b4 to satisfy order conditions */
841: if (a3 == PETSC_DEFAULT) a3 = (one/five - a2/four)/(one/four - a2/three); /* Eq 7.22 */
842: a4 = a3; /* consequence of 7.20 */
844: /* Solve order conditions 7.15a, 7.15c, 7.15e */
845: M[0][0] = one; M[0][1] = one; M[0][2] = one; /* 7.15a */
846: M[1][0] = 0.0; M[1][1] = a2*a2; M[1][2] = a4*a4; /* 7.15c */
847: M[2][0] = 0.0; M[2][1] = a2*a2*a2; M[2][2] = a4*a4*a4; /* 7.15e */
848: rhs[0] = one - b3;
849: rhs[1] = one/three - a3*a3*b3;
850: rhs[2] = one/four - a3*a3*a3*b3;
851: PetscKernel_A_gets_inverse_A_3(&M[0][0],0,PETSC_FALSE,NULL);
852: b1 = PetscRealPart(M[0][0]*rhs[0] + M[0][1]*rhs[1] + M[0][2]*rhs[2]);
853: b2 = PetscRealPart(M[1][0]*rhs[0] + M[1][1]*rhs[1] + M[1][2]*rhs[2]);
854: b4 = PetscRealPart(M[2][0]*rhs[0] + M[2][1]*rhs[1] + M[2][2]*rhs[2]);
856: /* Step 3 */
857: beta43 = (p56 - a2*p43) / (b4*a3*a3*(a3 - a2)); /* 7.21 */
858: beta32beta2p = p44 / (b4*beta43); /* 7.15h */
859: beta4jbetajp = (p32 - b3*beta32beta2p) / b4;
860: M[0][0] = b2; M[0][1] = b3; M[0][2] = b4;
861: M[1][0] = a4*a4*beta32beta2p-a3*a3*beta4jbetajp; M[1][1] = a2*a2*beta4jbetajp; M[1][2] = -a2*a2*beta32beta2p;
862: M[2][0] = b4*beta43*a3*a3-p43; M[2][1] = -b4*beta43*a2*a2; M[2][2] = 0;
863: rhs[0] = one/two - gamma; rhs[1] = 0; rhs[2] = -a2*a2*p32;
864: PetscKernel_A_gets_inverse_A_3(&M[0][0],0,PETSC_FALSE,NULL);
865: beta2p = PetscRealPart(M[0][0]*rhs[0] + M[0][1]*rhs[1] + M[0][2]*rhs[2]);
866: beta3p = PetscRealPart(M[1][0]*rhs[0] + M[1][1]*rhs[1] + M[1][2]*rhs[2]);
867: beta4p = PetscRealPart(M[2][0]*rhs[0] + M[2][1]*rhs[1] + M[2][2]*rhs[2]);
869: /* Step 4: back-substitute */
870: beta32 = beta32beta2p / beta2p;
871: beta42 = (beta4jbetajp - beta43*beta3p) / beta2p;
873: /* Step 5: 7.15f and 7.20, then 7.16 */
874: a43 = 0;
875: a32 = p42 / (b3*a3*beta2p + b4*a4*beta2p);
876: a42 = a32;
878: A[0][0] = 0; A[0][1] = 0; A[0][2] = 0; A[0][3] = 0;
879: A[1][0] = a2; A[1][1] = 0; A[1][2] = 0; A[1][3] = 0;
880: A[2][0] = a3-a32; A[2][1] = a32; A[2][2] = 0; A[2][3] = 0;
881: A[3][0] = a4-a43-a42; A[3][1] = a42; A[3][2] = a43; A[3][3] = 0;
882: Gamma[0][0] = gamma; Gamma[0][1] = 0; Gamma[0][2] = 0; Gamma[0][3] = 0;
883: Gamma[1][0] = beta2p-A[1][0]; Gamma[1][1] = gamma; Gamma[1][2] = 0; Gamma[1][3] = 0;
884: Gamma[2][0] = beta3p-beta32-A[2][0]; Gamma[2][1] = beta32-A[2][1]; Gamma[2][2] = gamma; Gamma[2][3] = 0;
885: 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;
886: b[0] = b1; b[1] = b2; b[2] = b3; b[3] = b4;
888: /* Construct embedded formula using given e4. We are solving Equation 7.18. */
889: bm[3] = b[3] - e4*gamma; /* using definition of E4 */
890: bm[2] = (p32 - beta4jbetajp*bm[3]) / (beta32*beta2p); /* fourth row of 7.18 */
891: bm[1] = (one/two - gamma - beta3p*bm[2] - beta4p*bm[3]) / beta2p; /* second row */
892: bm[0] = one - bm[1] - bm[2] - bm[3]; /* first row */
894: {
895: const PetscReal misfit = a2*a2*bm[1] + a3*a3*bm[2] + a4*a4*bm[3] - one/three;
896: if (PetscAbs(misfit) > PETSC_SMALL) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Assumptions violated, could not construct a third order embedded method");
897: }
898: TSRosWRegister(name,4,4,&A[0][0],&Gamma[0][0],b,bm,0,NULL);
899: return(0);
900: }
902: /*
903: The step completion formula is
905: x1 = x0 + b^T Y
907: where Y is the multi-vector of stages corrections. This function can be called before or after ts->vec_sol has been
908: updated. Suppose we have a completion formula b and an embedded formula be of different order. We can write
910: x1e = x0 + be^T Y
911: = x1 - b^T Y + be^T Y
912: = x1 + (be - b)^T Y
914: so we can evaluate the method of different order even after the step has been optimistically completed.
915: */
916: static PetscErrorCode TSEvaluateStep_RosW(TS ts,PetscInt order,Vec U,PetscBool *done)
917: {
918: TS_RosW *ros = (TS_RosW*)ts->data;
919: RosWTableau tab = ros->tableau;
920: PetscScalar *w = ros->work;
921: PetscInt i;
925: if (order == tab->order) {
926: if (ros->status == TS_STEP_INCOMPLETE) { /* Use standard completion formula */
927: VecCopy(ts->vec_sol,U);
928: for (i=0; i<tab->s; i++) w[i] = tab->bt[i];
929: VecMAXPY(U,tab->s,w,ros->Y);
930: } else {VecCopy(ts->vec_sol,U);}
931: if (done) *done = PETSC_TRUE;
932: return(0);
933: } else if (order == tab->order-1) {
934: if (!tab->bembedt) goto unavailable;
935: if (ros->status == TS_STEP_INCOMPLETE) { /* Use embedded completion formula */
936: VecCopy(ts->vec_sol,U);
937: for (i=0; i<tab->s; i++) w[i] = tab->bembedt[i];
938: VecMAXPY(U,tab->s,w,ros->Y);
939: } else { /* Use rollback-and-recomplete formula (bembedt - bt) */
940: for (i=0; i<tab->s; i++) w[i] = tab->bembedt[i] - tab->bt[i];
941: VecCopy(ts->vec_sol,U);
942: VecMAXPY(U,tab->s,w,ros->Y);
943: }
944: if (done) *done = PETSC_TRUE;
945: return(0);
946: }
947: unavailable:
948: if (done) *done = PETSC_FALSE;
949: else SETERRQ3(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);
950: return(0);
951: }
953: static PetscErrorCode TSRollBack_RosW(TS ts)
954: {
955: TS_RosW *ros = (TS_RosW*)ts->data;
959: VecCopy(ros->vec_sol_prev,ts->vec_sol);
960: return(0);
961: }
963: static PetscErrorCode TSStep_RosW(TS ts)
964: {
965: TS_RosW *ros = (TS_RosW*)ts->data;
966: RosWTableau tab = ros->tableau;
967: const PetscInt s = tab->s;
968: const PetscReal *At = tab->At,*Gamma = tab->Gamma,*ASum = tab->ASum,*GammaInv = tab->GammaInv;
969: const PetscReal *GammaExplicitCorr = tab->GammaExplicitCorr;
970: const PetscBool *GammaZeroDiag = tab->GammaZeroDiag;
971: PetscScalar *w = ros->work;
972: Vec *Y = ros->Y,Ydot = ros->Ydot,Zdot = ros->Zdot,Zstage = ros->Zstage;
973: SNES snes;
974: TSAdapt adapt;
975: PetscInt i,j,its,lits;
976: PetscInt rejections = 0;
977: PetscBool stageok,accept = PETSC_TRUE;
978: PetscReal next_time_step = ts->time_step;
979: PetscErrorCode ierr;
982: if (!ts->steprollback) {
983: VecCopy(ts->vec_sol,ros->vec_sol_prev);
984: }
986: ros->status = TS_STEP_INCOMPLETE;
987: while (!ts->reason && ros->status != TS_STEP_COMPLETE) {
988: const PetscReal h = ts->time_step;
989: for (i=0; i<s; i++) {
990: ros->stage_time = ts->ptime + h*ASum[i];
991: TSPreStage(ts,ros->stage_time);
992: if (GammaZeroDiag[i]) {
993: ros->stage_explicit = PETSC_TRUE;
994: ros->scoeff = 1.;
995: } else {
996: ros->stage_explicit = PETSC_FALSE;
997: ros->scoeff = 1./Gamma[i*s+i];
998: }
1000: VecCopy(ts->vec_sol,Zstage);
1001: for (j=0; j<i; j++) w[j] = At[i*s+j];
1002: VecMAXPY(Zstage,i,w,Y);
1004: for (j=0; j<i; j++) w[j] = 1./h * GammaInv[i*s+j];
1005: VecZeroEntries(Zdot);
1006: VecMAXPY(Zdot,i,w,Y);
1008: /* Initial guess taken from last stage */
1009: VecZeroEntries(Y[i]);
1011: if (!ros->stage_explicit) {
1012: TSGetSNES(ts,&snes);
1013: if (!ros->recompute_jacobian && !i) {
1014: SNESSetLagJacobian(snes,-2); /* Recompute the Jacobian on this solve, but not again */
1015: }
1016: SNESSolve(snes,NULL,Y[i]);
1017: SNESGetIterationNumber(snes,&its);
1018: SNESGetLinearSolveIterations(snes,&lits);
1019: ts->snes_its += its; ts->ksp_its += lits;
1020: } else {
1021: Mat J,Jp;
1022: VecZeroEntries(Ydot); /* Evaluate Y[i]=G(t,Ydot=0,Zstage) */
1023: TSComputeIFunction(ts,ros->stage_time,Zstage,Ydot,Y[i],PETSC_FALSE);
1024: VecScale(Y[i],-1.0);
1025: VecAXPY(Y[i],-1.0,Zdot); /*Y[i] = F(Zstage)-Zdot[=GammaInv*Y]*/
1027: VecZeroEntries(Zstage); /* Zstage = GammaExplicitCorr[i,j] * Y[j] */
1028: for (j=0; j<i; j++) w[j] = GammaExplicitCorr[i*s+j];
1029: VecMAXPY(Zstage,i,w,Y);
1031: /* Y[i] = Y[i] + Jac*Zstage[=Jac*GammaExplicitCorr[i,j] * Y[j]] */
1032: TSGetIJacobian(ts,&J,&Jp,NULL,NULL);
1033: TSComputeIJacobian(ts,ros->stage_time,ts->vec_sol,Ydot,0,J,Jp,PETSC_FALSE);
1034: MatMult(J,Zstage,Zdot);
1035: VecAXPY(Y[i],-1.0,Zdot);
1036: ts->ksp_its += 1;
1038: VecScale(Y[i],h);
1039: }
1040: TSPostStage(ts,ros->stage_time,i,Y);
1041: TSGetAdapt(ts,&adapt);
1042: TSAdaptCheckStage(adapt,ts,ros->stage_time,Y[i],&stageok);
1043: if (!stageok) goto reject_step;
1044: }
1046: ros->status = TS_STEP_INCOMPLETE;
1047: TSEvaluateStep_RosW(ts,tab->order,ts->vec_sol,NULL);
1048: ros->status = TS_STEP_PENDING;
1049: TSGetAdapt(ts,&adapt);
1050: TSAdaptCandidatesClear(adapt);
1051: TSAdaptCandidateAdd(adapt,tab->name,tab->order,1,tab->ccfl,(PetscReal)tab->s,PETSC_TRUE);
1052: TSAdaptChoose(adapt,ts,ts->time_step,NULL,&next_time_step,&accept);
1053: ros->status = accept ? TS_STEP_COMPLETE : TS_STEP_INCOMPLETE;
1054: if (!accept) { /* Roll back the current step */
1055: TSRollBack_RosW(ts);
1056: ts->time_step = next_time_step;
1057: goto reject_step;
1058: }
1060: ts->ptime += ts->time_step;
1061: ts->time_step = next_time_step;
1062: break;
1064: reject_step:
1065: ts->reject++; accept = PETSC_FALSE;
1066: if (!ts->reason && ++rejections > ts->max_reject && ts->max_reject >= 0) {
1067: ts->reason = TS_DIVERGED_STEP_REJECTED;
1068: PetscInfo2(ts,"Step=%D, step rejections %D greater than current TS allowed, stopping solve\n",ts->steps,rejections);
1069: }
1070: }
1071: return(0);
1072: }
1074: static PetscErrorCode TSInterpolate_RosW(TS ts,PetscReal itime,Vec U)
1075: {
1076: TS_RosW *ros = (TS_RosW*)ts->data;
1077: PetscInt s = ros->tableau->s,pinterp = ros->tableau->pinterp,i,j;
1078: PetscReal h;
1079: PetscReal tt,t;
1080: PetscScalar *bt;
1081: const PetscReal *Bt = ros->tableau->binterpt;
1082: PetscErrorCode ierr;
1083: const PetscReal *GammaInv = ros->tableau->GammaInv;
1084: PetscScalar *w = ros->work;
1085: Vec *Y = ros->Y;
1088: if (!Bt) SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_SUP,"TSRosW %s does not have an interpolation formula",ros->tableau->name);
1090: switch (ros->status) {
1091: case TS_STEP_INCOMPLETE:
1092: case TS_STEP_PENDING:
1093: h = ts->time_step;
1094: t = (itime - ts->ptime)/h;
1095: break;
1096: case TS_STEP_COMPLETE:
1097: h = ts->ptime - ts->ptime_prev;
1098: t = (itime - ts->ptime)/h + 1; /* In the interval [0,1] */
1099: break;
1100: default: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_PLIB,"Invalid TSStepStatus");
1101: }
1102: PetscMalloc1(s,&bt);
1103: for (i=0; i<s; i++) bt[i] = 0;
1104: for (j=0,tt=t; j<pinterp; j++,tt*=t) {
1105: for (i=0; i<s; i++) {
1106: bt[i] += Bt[i*pinterp+j] * tt;
1107: }
1108: }
1110: /* y(t+tt*h) = y(t) + Sum bt(tt) * GammaInv * Ydot */
1111: /* U <- 0*/
1112: VecZeroEntries(U);
1113: /* U <- Sum bt_i * GammaInv(i,1:i) * Y(1:i) */
1114: for (j=0; j<s; j++) w[j] = 0;
1115: for (j=0; j<s; j++) {
1116: for (i=j; i<s; i++) {
1117: w[j] += bt[i]*GammaInv[i*s+j];
1118: }
1119: }
1120: VecMAXPY(U,i,w,Y);
1121: /* U <- y(t) + U */
1122: VecAXPY(U,1,ros->vec_sol_prev);
1124: PetscFree(bt);
1125: return(0);
1126: }
1128: /*------------------------------------------------------------*/
1130: static PetscErrorCode TSRosWTableauReset(TS ts)
1131: {
1132: TS_RosW *ros = (TS_RosW*)ts->data;
1133: RosWTableau tab = ros->tableau;
1137: if (!tab) return(0);
1138: VecDestroyVecs(tab->s,&ros->Y);
1139: PetscFree(ros->work);
1140: return(0);
1141: }
1143: static PetscErrorCode TSReset_RosW(TS ts)
1144: {
1145: TS_RosW *ros = (TS_RosW*)ts->data;
1149: TSRosWTableauReset(ts);
1150: VecDestroy(&ros->Ydot);
1151: VecDestroy(&ros->Ystage);
1152: VecDestroy(&ros->Zdot);
1153: VecDestroy(&ros->Zstage);
1154: VecDestroy(&ros->vec_sol_prev);
1155: return(0);
1156: }
1158: static PetscErrorCode TSRosWGetVecs(TS ts,DM dm,Vec *Ydot,Vec *Zdot,Vec *Ystage,Vec *Zstage)
1159: {
1160: TS_RosW *rw = (TS_RosW*)ts->data;
1164: if (Ydot) {
1165: if (dm && dm != ts->dm) {
1166: DMGetNamedGlobalVector(dm,"TSRosW_Ydot",Ydot);
1167: } else *Ydot = rw->Ydot;
1168: }
1169: if (Zdot) {
1170: if (dm && dm != ts->dm) {
1171: DMGetNamedGlobalVector(dm,"TSRosW_Zdot",Zdot);
1172: } else *Zdot = rw->Zdot;
1173: }
1174: if (Ystage) {
1175: if (dm && dm != ts->dm) {
1176: DMGetNamedGlobalVector(dm,"TSRosW_Ystage",Ystage);
1177: } else *Ystage = rw->Ystage;
1178: }
1179: if (Zstage) {
1180: if (dm && dm != ts->dm) {
1181: DMGetNamedGlobalVector(dm,"TSRosW_Zstage",Zstage);
1182: } else *Zstage = rw->Zstage;
1183: }
1184: return(0);
1185: }
1188: static PetscErrorCode TSRosWRestoreVecs(TS ts,DM dm,Vec *Ydot,Vec *Zdot, Vec *Ystage, Vec *Zstage)
1189: {
1193: if (Ydot) {
1194: if (dm && dm != ts->dm) {
1195: DMRestoreNamedGlobalVector(dm,"TSRosW_Ydot",Ydot);
1196: }
1197: }
1198: if (Zdot) {
1199: if (dm && dm != ts->dm) {
1200: DMRestoreNamedGlobalVector(dm,"TSRosW_Zdot",Zdot);
1201: }
1202: }
1203: if (Ystage) {
1204: if (dm && dm != ts->dm) {
1205: DMRestoreNamedGlobalVector(dm,"TSRosW_Ystage",Ystage);
1206: }
1207: }
1208: if (Zstage) {
1209: if (dm && dm != ts->dm) {
1210: DMRestoreNamedGlobalVector(dm,"TSRosW_Zstage",Zstage);
1211: }
1212: }
1213: return(0);
1214: }
1216: static PetscErrorCode DMCoarsenHook_TSRosW(DM fine,DM coarse,void *ctx)
1217: {
1219: return(0);
1220: }
1222: static PetscErrorCode DMRestrictHook_TSRosW(DM fine,Mat restrct,Vec rscale,Mat inject,DM coarse,void *ctx)
1223: {
1224: TS ts = (TS)ctx;
1226: Vec Ydot,Zdot,Ystage,Zstage;
1227: Vec Ydotc,Zdotc,Ystagec,Zstagec;
1230: TSRosWGetVecs(ts,fine,&Ydot,&Ystage,&Zdot,&Zstage);
1231: TSRosWGetVecs(ts,coarse,&Ydotc,&Ystagec,&Zdotc,&Zstagec);
1232: MatRestrict(restrct,Ydot,Ydotc);
1233: VecPointwiseMult(Ydotc,rscale,Ydotc);
1234: MatRestrict(restrct,Ystage,Ystagec);
1235: VecPointwiseMult(Ystagec,rscale,Ystagec);
1236: MatRestrict(restrct,Zdot,Zdotc);
1237: VecPointwiseMult(Zdotc,rscale,Zdotc);
1238: MatRestrict(restrct,Zstage,Zstagec);
1239: VecPointwiseMult(Zstagec,rscale,Zstagec);
1240: TSRosWRestoreVecs(ts,fine,&Ydot,&Ystage,&Zdot,&Zstage);
1241: TSRosWRestoreVecs(ts,coarse,&Ydotc,&Ystagec,&Zdotc,&Zstagec);
1242: return(0);
1243: }
1246: static PetscErrorCode DMSubDomainHook_TSRosW(DM fine,DM coarse,void *ctx)
1247: {
1249: return(0);
1250: }
1252: static PetscErrorCode DMSubDomainRestrictHook_TSRosW(DM dm,VecScatter gscat,VecScatter lscat,DM subdm,void *ctx)
1253: {
1254: TS ts = (TS)ctx;
1256: Vec Ydot,Zdot,Ystage,Zstage;
1257: Vec Ydots,Zdots,Ystages,Zstages;
1260: TSRosWGetVecs(ts,dm,&Ydot,&Ystage,&Zdot,&Zstage);
1261: TSRosWGetVecs(ts,subdm,&Ydots,&Ystages,&Zdots,&Zstages);
1263: VecScatterBegin(gscat,Ydot,Ydots,INSERT_VALUES,SCATTER_FORWARD);
1264: VecScatterEnd(gscat,Ydot,Ydots,INSERT_VALUES,SCATTER_FORWARD);
1266: VecScatterBegin(gscat,Ystage,Ystages,INSERT_VALUES,SCATTER_FORWARD);
1267: VecScatterEnd(gscat,Ystage,Ystages,INSERT_VALUES,SCATTER_FORWARD);
1269: VecScatterBegin(gscat,Zdot,Zdots,INSERT_VALUES,SCATTER_FORWARD);
1270: VecScatterEnd(gscat,Zdot,Zdots,INSERT_VALUES,SCATTER_FORWARD);
1272: VecScatterBegin(gscat,Zstage,Zstages,INSERT_VALUES,SCATTER_FORWARD);
1273: VecScatterEnd(gscat,Zstage,Zstages,INSERT_VALUES,SCATTER_FORWARD);
1275: TSRosWRestoreVecs(ts,dm,&Ydot,&Ystage,&Zdot,&Zstage);
1276: TSRosWRestoreVecs(ts,subdm,&Ydots,&Ystages,&Zdots,&Zstages);
1277: return(0);
1278: }
1280: /*
1281: This defines the nonlinear equation that is to be solved with SNES
1282: G(U) = F[t0+Theta*dt, U, (U-U0)*shift] = 0
1283: */
1284: static PetscErrorCode SNESTSFormFunction_RosW(SNES snes,Vec U,Vec F,TS ts)
1285: {
1286: TS_RosW *ros = (TS_RosW*)ts->data;
1288: Vec Ydot,Zdot,Ystage,Zstage;
1289: PetscReal shift = ros->scoeff / ts->time_step;
1290: DM dm,dmsave;
1293: SNESGetDM(snes,&dm);
1294: TSRosWGetVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
1295: VecWAXPY(Ydot,shift,U,Zdot); /* Ydot = shift*U + Zdot */
1296: VecWAXPY(Ystage,1.0,U,Zstage); /* Ystage = U + Zstage */
1297: dmsave = ts->dm;
1298: ts->dm = dm;
1299: TSComputeIFunction(ts,ros->stage_time,Ystage,Ydot,F,PETSC_FALSE);
1300: ts->dm = dmsave;
1301: TSRosWRestoreVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
1302: return(0);
1303: }
1305: static PetscErrorCode SNESTSFormJacobian_RosW(SNES snes,Vec U,Mat A,Mat B,TS ts)
1306: {
1307: TS_RosW *ros = (TS_RosW*)ts->data;
1308: Vec Ydot,Zdot,Ystage,Zstage;
1309: PetscReal shift = ros->scoeff / ts->time_step;
1311: DM dm,dmsave;
1314: /* ros->Ydot and ros->Ystage have already been computed in SNESTSFormFunction_RosW (SNES guarantees this) */
1315: SNESGetDM(snes,&dm);
1316: TSRosWGetVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
1317: dmsave = ts->dm;
1318: ts->dm = dm;
1319: TSComputeIJacobian(ts,ros->stage_time,Ystage,Ydot,shift,A,B,PETSC_TRUE);
1320: ts->dm = dmsave;
1321: TSRosWRestoreVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
1322: return(0);
1323: }
1325: static PetscErrorCode TSRosWTableauSetUp(TS ts)
1326: {
1327: TS_RosW *ros = (TS_RosW*)ts->data;
1328: RosWTableau tab = ros->tableau;
1332: VecDuplicateVecs(ts->vec_sol,tab->s,&ros->Y);
1333: PetscMalloc1(tab->s,&ros->work);
1334: return(0);
1335: }
1337: static PetscErrorCode TSSetUp_RosW(TS ts)
1338: {
1339: TS_RosW *ros = (TS_RosW*)ts->data;
1341: DM dm;
1342: SNES snes;
1345: TSRosWTableauSetUp(ts);
1346: VecDuplicate(ts->vec_sol,&ros->Ydot);
1347: VecDuplicate(ts->vec_sol,&ros->Ystage);
1348: VecDuplicate(ts->vec_sol,&ros->Zdot);
1349: VecDuplicate(ts->vec_sol,&ros->Zstage);
1350: VecDuplicate(ts->vec_sol,&ros->vec_sol_prev);
1351: TSGetDM(ts,&dm);
1352: DMCoarsenHookAdd(dm,DMCoarsenHook_TSRosW,DMRestrictHook_TSRosW,ts);
1353: DMSubDomainHookAdd(dm,DMSubDomainHook_TSRosW,DMSubDomainRestrictHook_TSRosW,ts);
1354: /* Rosenbrock methods are linearly implicit, so set that unless the user has specifically asked for something else */
1355: TSGetSNES(ts,&snes);
1356: if (!((PetscObject)snes)->type_name) {
1357: SNESSetType(snes,SNESKSPONLY);
1358: }
1359: return(0);
1360: }
1361: /*------------------------------------------------------------*/
1363: static PetscErrorCode TSSetFromOptions_RosW(PetscOptionItems *PetscOptionsObject,TS ts)
1364: {
1365: TS_RosW *ros = (TS_RosW*)ts->data;
1367: SNES snes;
1370: PetscOptionsHead(PetscOptionsObject,"RosW ODE solver options");
1371: {
1372: RosWTableauLink link;
1373: PetscInt count,choice;
1374: PetscBool flg;
1375: const char **namelist;
1377: for (link=RosWTableauList,count=0; link; link=link->next,count++) ;
1378: PetscMalloc1(count,&namelist);
1379: for (link=RosWTableauList,count=0; link; link=link->next,count++) namelist[count] = link->tab.name;
1380: PetscOptionsEList("-ts_rosw_type","Family of Rosenbrock-W method","TSRosWSetType",(const char*const*)namelist,count,ros->tableau->name,&choice,&flg);
1381: if (flg) {TSRosWSetType(ts,namelist[choice]);}
1382: PetscFree(namelist);
1384: PetscOptionsBool("-ts_rosw_recompute_jacobian","Recompute the Jacobian at each stage","TSRosWSetRecomputeJacobian",ros->recompute_jacobian,&ros->recompute_jacobian,NULL);
1385: }
1386: PetscOptionsTail();
1387: /* Rosenbrock methods are linearly implicit, so set that unless the user has specifically asked for something else */
1388: TSGetSNES(ts,&snes);
1389: if (!((PetscObject)snes)->type_name) {
1390: SNESSetType(snes,SNESKSPONLY);
1391: }
1392: return(0);
1393: }
1395: static PetscErrorCode PetscFormatRealArray(char buf[],size_t len,const char *fmt,PetscInt n,const PetscReal x[])
1396: {
1398: PetscInt i;
1399: size_t left,count;
1400: char *p;
1403: for (i=0,p=buf,left=len; i<n; i++) {
1404: PetscSNPrintfCount(p,left,fmt,&count,(double)x[i]);
1405: if (count >= left) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Insufficient space in buffer");
1406: left -= count;
1407: p += count;
1408: *p++ = ' ';
1409: }
1410: p[i ? 0 : -1] = 0;
1411: return(0);
1412: }
1414: static PetscErrorCode TSView_RosW(TS ts,PetscViewer viewer)
1415: {
1416: TS_RosW *ros = (TS_RosW*)ts->data;
1417: PetscBool iascii;
1421: PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
1422: if (iascii) {
1423: RosWTableau tab = ros->tableau;
1424: TSRosWType rostype;
1425: char buf[512];
1426: PetscInt i;
1427: PetscReal abscissa[512];
1428: TSRosWGetType(ts,&rostype);
1429: PetscViewerASCIIPrintf(viewer," Rosenbrock-W %s\n",rostype);
1430: PetscFormatRealArray(buf,sizeof(buf),"% 8.6f",tab->s,tab->ASum);
1431: PetscViewerASCIIPrintf(viewer," Abscissa of A = %s\n",buf);
1432: for (i=0; i<tab->s; i++) abscissa[i] = tab->ASum[i] + tab->Gamma[i];
1433: PetscFormatRealArray(buf,sizeof(buf),"% 8.6f",tab->s,abscissa);
1434: PetscViewerASCIIPrintf(viewer," Abscissa of A+Gamma = %s\n",buf);
1435: }
1436: return(0);
1437: }
1439: static PetscErrorCode TSLoad_RosW(TS ts,PetscViewer viewer)
1440: {
1442: SNES snes;
1443: TSAdapt adapt;
1446: TSGetAdapt(ts,&adapt);
1447: TSAdaptLoad(adapt,viewer);
1448: TSGetSNES(ts,&snes);
1449: SNESLoad(snes,viewer);
1450: /* function and Jacobian context for SNES when used with TS is always ts object */
1451: SNESSetFunction(snes,NULL,NULL,ts);
1452: SNESSetJacobian(snes,NULL,NULL,NULL,ts);
1453: return(0);
1454: }
1456: /*@C
1457: TSRosWSetType - Set the type of Rosenbrock-W scheme
1459: Logically collective
1461: Input Parameter:
1462: + ts - timestepping context
1463: - roswtype - type of Rosenbrock-W scheme
1465: Level: beginner
1467: .seealso: TSRosWGetType(), TSROSW, TSROSW2M, TSROSW2P, TSROSWRA3PW, TSROSWRA34PW2, TSROSWRODAS3, TSROSWSANDU3, TSROSWASSP3P3S1C, TSROSWLASSP3P4S2C, TSROSWLLSSP3P4S2C, TSROSWARK3
1468: @*/
1469: PetscErrorCode TSRosWSetType(TS ts,TSRosWType roswtype)
1470: {
1476: PetscTryMethod(ts,"TSRosWSetType_C",(TS,TSRosWType),(ts,roswtype));
1477: return(0);
1478: }
1480: /*@C
1481: TSRosWGetType - Get the type of Rosenbrock-W scheme
1483: Logically collective
1485: Input Parameter:
1486: . ts - timestepping context
1488: Output Parameter:
1489: . rostype - type of Rosenbrock-W scheme
1491: Level: intermediate
1493: .seealso: TSRosWGetType()
1494: @*/
1495: PetscErrorCode TSRosWGetType(TS ts,TSRosWType *rostype)
1496: {
1501: PetscUseMethod(ts,"TSRosWGetType_C",(TS,TSRosWType*),(ts,rostype));
1502: return(0);
1503: }
1505: /*@C
1506: TSRosWSetRecomputeJacobian - Set whether to recompute the Jacobian at each stage. The default is to update the Jacobian once per step.
1508: Logically collective
1510: Input Parameter:
1511: + ts - timestepping context
1512: - flg - PETSC_TRUE to recompute the Jacobian at each stage
1514: Level: intermediate
1516: .seealso: TSRosWGetType()
1517: @*/
1518: PetscErrorCode TSRosWSetRecomputeJacobian(TS ts,PetscBool flg)
1519: {
1524: PetscTryMethod(ts,"TSRosWSetRecomputeJacobian_C",(TS,PetscBool),(ts,flg));
1525: return(0);
1526: }
1528: static PetscErrorCode TSRosWGetType_RosW(TS ts,TSRosWType *rostype)
1529: {
1530: TS_RosW *ros = (TS_RosW*)ts->data;
1533: *rostype = ros->tableau->name;
1534: return(0);
1535: }
1537: static PetscErrorCode TSRosWSetType_RosW(TS ts,TSRosWType rostype)
1538: {
1539: TS_RosW *ros = (TS_RosW*)ts->data;
1540: PetscErrorCode ierr;
1541: PetscBool match;
1542: RosWTableauLink link;
1545: if (ros->tableau) {
1546: PetscStrcmp(ros->tableau->name,rostype,&match);
1547: if (match) return(0);
1548: }
1549: for (link = RosWTableauList; link; link=link->next) {
1550: PetscStrcmp(link->tab.name,rostype,&match);
1551: if (match) {
1552: if (ts->setupcalled) {TSRosWTableauReset(ts);}
1553: ros->tableau = &link->tab;
1554: if (ts->setupcalled) {TSRosWTableauSetUp(ts);}
1555: ts->default_adapt_type = ros->tableau->bembed ? TSADAPTBASIC : TSADAPTNONE;
1556: return(0);
1557: }
1558: }
1559: SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_ARG_UNKNOWN_TYPE,"Could not find '%s'",rostype);
1560: return(0);
1561: }
1563: static PetscErrorCode TSRosWSetRecomputeJacobian_RosW(TS ts,PetscBool flg)
1564: {
1565: TS_RosW *ros = (TS_RosW*)ts->data;
1568: ros->recompute_jacobian = flg;
1569: return(0);
1570: }
1572: static PetscErrorCode TSDestroy_RosW(TS ts)
1573: {
1577: TSReset_RosW(ts);
1578: if (ts->dm) {
1579: DMCoarsenHookRemove(ts->dm,DMCoarsenHook_TSRosW,DMRestrictHook_TSRosW,ts);
1580: DMSubDomainHookRemove(ts->dm,DMSubDomainHook_TSRosW,DMSubDomainRestrictHook_TSRosW,ts);
1581: }
1582: PetscFree(ts->data);
1583: PetscObjectComposeFunction((PetscObject)ts,"TSRosWGetType_C",NULL);
1584: PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetType_C",NULL);
1585: PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetRecomputeJacobian_C",NULL);
1586: return(0);
1587: }
1589: /* ------------------------------------------------------------ */
1590: /*MC
1591: TSROSW - ODE solver using Rosenbrock-W schemes
1593: These methods are intended for problems with well-separated time scales, especially when a slow scale is strongly
1594: nonlinear such that it is expensive to solve with a fully implicit method. The user should provide the stiff part
1595: of the equation using TSSetIFunction() and the non-stiff part with TSSetRHSFunction().
1597: Notes:
1598: This method currently only works with autonomous ODE and DAE.
1600: Consider trying TSARKIMEX if the stiff part is strongly nonlinear.
1602: Developer notes:
1603: Rosenbrock-W methods are typically specified for autonomous ODE
1605: $ udot = f(u)
1607: by the stage equations
1609: $ k_i = h f(u_0 + sum_j alpha_ij k_j) + h J sum_j gamma_ij k_j
1611: and step completion formula
1613: $ u_1 = u_0 + sum_j b_j k_j
1615: with step size h and coefficients alpha_ij, gamma_ij, and b_i. Implementing the method in this form would require f(u)
1616: and the Jacobian J to be available, in addition to the shifted matrix I - h gamma_ii J. Following Hairer and Wanner,
1617: we define new variables for the stage equations
1619: $ y_i = gamma_ij k_j
1621: The k_j can be recovered because Gamma is invertible. Let C be the lower triangular part of Gamma^{-1} and define
1623: $ A = Alpha Gamma^{-1}, bt^T = b^T Gamma^{-1}
1625: to rewrite the method as
1627: $ [M/(h gamma_ii) - J] y_i = f(u_0 + sum_j a_ij y_j) + M sum_j (c_ij/h) y_j
1628: $ u_1 = u_0 + sum_j bt_j y_j
1630: where we have introduced the mass matrix M. Continue by defining
1632: $ ydot_i = 1/(h gamma_ii) y_i - sum_j (c_ij/h) y_j
1634: or, more compactly in tensor notation
1636: $ Ydot = 1/h (Gamma^{-1} \otimes I) Y .
1638: Note that Gamma^{-1} is lower triangular. With this definition of Ydot in terms of known quantities and the current
1639: stage y_i, the stage equations reduce to performing one Newton step (typically with a lagged Jacobian) on the
1640: equation
1642: $ g(u_0 + sum_j a_ij y_j + y_i, ydot_i) = 0
1644: with initial guess y_i = 0.
1646: Level: beginner
1648: .seealso: TSCreate(), TS, TSSetType(), TSRosWSetType(), TSRosWRegister(), TSROSWTHETA1, TSROSWTHETA2, TSROSW2M, TSROSW2P, TSROSWRA3PW, TSROSWRA34PW2, TSROSWRODAS3,
1649: TSROSWSANDU3, TSROSWASSP3P3S1C, TSROSWLASSP3P4S2C, TSROSWLLSSP3P4S2C, TSROSWGRK4T, TSROSWSHAMP4, TSROSWVELDD4, TSROSW4L
1650: M*/
1651: PETSC_EXTERN PetscErrorCode TSCreate_RosW(TS ts)
1652: {
1653: TS_RosW *ros;
1657: TSRosWInitializePackage();
1659: ts->ops->reset = TSReset_RosW;
1660: ts->ops->destroy = TSDestroy_RosW;
1661: ts->ops->view = TSView_RosW;
1662: ts->ops->load = TSLoad_RosW;
1663: ts->ops->setup = TSSetUp_RosW;
1664: ts->ops->step = TSStep_RosW;
1665: ts->ops->interpolate = TSInterpolate_RosW;
1666: ts->ops->evaluatestep = TSEvaluateStep_RosW;
1667: ts->ops->rollback = TSRollBack_RosW;
1668: ts->ops->setfromoptions = TSSetFromOptions_RosW;
1669: ts->ops->snesfunction = SNESTSFormFunction_RosW;
1670: ts->ops->snesjacobian = SNESTSFormJacobian_RosW;
1672: ts->usessnes = PETSC_TRUE;
1674: PetscNewLog(ts,&ros);
1675: ts->data = (void*)ros;
1677: PetscObjectComposeFunction((PetscObject)ts,"TSRosWGetType_C",TSRosWGetType_RosW);
1678: PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetType_C",TSRosWSetType_RosW);
1679: PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetRecomputeJacobian_C",TSRosWSetRecomputeJacobian_RosW);
1681: TSRosWSetType(ts,TSRosWDefault);
1682: return(0);
1683: }