Actual source code: ex74.c
petsc-3.12.0 2019-09-29
1: static char help[] = "\
2: Solves the constant-coefficient 1D Heat equation with an Implicit \n\
3: Runge-Kutta method using MatKAIJ. \n\
4: \n\
5: du d^2 u \n\
6: -- = a ----- ; 0 <= x <= 1; \n\
7: dt dx^2 \n\
8: \n\
9: with periodic boundary conditions \n\
10: \n\
11: 2nd order central discretization in space: \n\
12: \n\
13: [ d^2 u ] u_{i+1} - 2u_i + u_{i-1} \n\
14: [ ----- ] = ------------------------ \n\
15: [ dx^2 ]i h^2 \n\
16: \n\
17: i = grid index; h = x_{i+1}-x_i (Uniform) \n\
18: 0 <= i < n h = 1.0/n \n\
19: \n\
20: Thus, \n\
21: \n\
22: du \n\
23: -- = Ju; J = (a/h^2) tridiagonal(1,-2,1)_n \n\
24: dt \n\
25: \n\
26: Implicit Runge-Kutta method: \n\
27: \n\
28: U^(k) = u^n + dt \\sum_i a_{ki} JU^{i} \n\
29: u^{n+1} = u^n + dt \\sum_i b_i JU^{i} \n\
30: \n\
31: i = 1,...,s (s -> number of stages) \n\
32: \n\
33: At each time step, we solve \n\
34: \n\
35: [ 1 ] 1 \n\
36: [ -- I \\otimes A^{-1} - J \\otimes I ] U = -- u^n \\otimes A^{-1} \n\
37: [ dt ] dt \n\
38: \n\
39: where A is the Butcher tableaux of the implicit \n\
40: Runge-Kutta method, \n\
41: \n\
42: with MATKAIJ and KSP. \n\
43: \n\
44: Available IRK Methods: \n\
45: gauss n-stage Gauss method \n\
46: \n";
48: /*T
49: Concepts: MATKAIJ
50: Concepts: MAT
51: Concepts: KSP
52: T*/
54: /*
55: Include "petscksp.h" so that we can use KSP solvers. Note that this file
56: automatically includes:
57: petscsys.h - base PETSc routines
58: petscvec.h - vectors
59: petscmat.h - matrices
60: petscis.h - index sets
61: petscviewer.h - viewers
62: petscpc.h - preconditioners
63: */
64: #include <petscksp.h>
65: #include <petscdt.h>
67: /* define the IRK methods available */
68: #define IRKGAUSS "gauss"
70: typedef enum {
71: PHYSICS_DIFFUSION,
72: PHYSICS_ADVECTION
73: } PhysicsType;
74: const char *const PhysicsTypes[] = {"DIFFUSION","ADVECTION","PhysicsType","PHYSICS_",NULL};
76: typedef struct __context__ {
77: PetscReal a; /* diffusion coefficient */
78: PetscReal xmin,xmax; /* domain bounds */
79: PetscInt imax; /* number of grid points */
80: PetscInt niter; /* number of time iterations */
81: PetscReal dt; /* time step size */
82: PhysicsType physics_type;
83: } UserContext;
85: static PetscErrorCode ExactSolution(Vec,void*,PetscReal);
86: static PetscErrorCode RKCreate_Gauss(PetscInt,PetscScalar**,PetscScalar**,PetscReal**);
87: static PetscErrorCode Assemble_AdvDiff(MPI_Comm,UserContext*,Mat*);
89: #include <petsc/private/kernels/blockinvert.h>
91: int main(int argc, char **argv)
92: {
93: PetscErrorCode ierr;
94: Vec u,uex,rhs,z;
95: UserContext ctxt;
96: PetscInt nstages,is,ie,matis,matie,*ix,*ix2;
97: PetscInt n,i,s,t,total_its;
98: PetscScalar *A,*B,*At,*b,*zvals,one = 1.0;
99: PetscReal *c,err,time;
100: Mat Identity,J,TA,SC,R;
101: KSP ksp;
102: PetscFunctionList IRKList = NULL;
103: char irktype[256] = IRKGAUSS;
105: PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
106: PetscFunctionListAdd(&IRKList,IRKGAUSS,RKCreate_Gauss);
108: /* default value */
109: ctxt.a = 1.0;
110: ctxt.xmin = 0.0;
111: ctxt.xmax = 1.0;
112: ctxt.imax = 20;
113: ctxt.niter = 0;
114: ctxt.dt = 0.0;
115: ctxt.physics_type = PHYSICS_DIFFUSION;
117: PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"IRK options","");
118: PetscOptionsReal("-a","diffusion coefficient","<1.0>",ctxt.a,&ctxt.a,NULL);
119: PetscOptionsInt ("-imax","grid size","<20>",ctxt.imax,&ctxt.imax,NULL);
120: PetscOptionsReal("-xmin","xmin","<0.0>",ctxt.xmin,&ctxt.xmin,NULL);
121: PetscOptionsReal("-xmax","xmax","<1.0>",ctxt.xmax,&ctxt.xmax,NULL);
122: PetscOptionsInt ("-niter","number of time steps","<0>",ctxt.niter,&ctxt.niter,NULL);
123: PetscOptionsReal("-dt","time step size","<0.0>",ctxt.dt,&ctxt.dt,NULL);
124: PetscOptionsFList("-irk_type","IRK method family","",IRKList,irktype,irktype,sizeof(irktype),NULL);
125: nstages = 2;
126: PetscOptionsInt ("-irk_nstages","Number of stages in IRK method","",nstages,&nstages,NULL);
127: PetscOptionsEnum("-physics_type","Type of process to discretize","",PhysicsTypes,(PetscEnum)ctxt.physics_type,(PetscEnum*)&ctxt.physics_type,NULL);
128: PetscOptionsEnd();
130: /* allocate and initialize solution vector and exact solution */
131: VecCreate(PETSC_COMM_WORLD,&u);
132: VecSetSizes(u,PETSC_DECIDE,ctxt.imax);
133: VecSetFromOptions(u);
134: VecDuplicate(u,&uex);
135: /* initial solution */
136: ExactSolution(u ,&ctxt,0.0);
137: /* exact solution */
138: ExactSolution(uex,&ctxt,ctxt.dt*ctxt.niter);
140: { /* Create A,b,c */
141: PetscErrorCode (*irkcreate)(PetscInt,PetscScalar**,PetscScalar**,PetscReal**);
142: PetscFunctionListFind(IRKList,irktype,&irkcreate);
143: (*irkcreate)(nstages,&A,&b,&c);
144: }
145: { /* Invert A */
146: /* PETSc does not provide a routine to calculate the inverse of a general matrix.
147: * To get the inverse of A, we form a sequential BAIJ matrix from it, consisting of a single block with block size
148: * equal to the dimension of A, and then use MatInvertBlockDiagonal(). */
149: Mat A_baij;
150: PetscInt idxm[1]={0},idxn[1]={0};
151: const PetscScalar *A_inv;
152: MatCreateSeqBAIJ(PETSC_COMM_SELF,nstages,nstages,nstages,1,NULL,&A_baij);
153: MatSetOption(A_baij,MAT_ROW_ORIENTED,PETSC_FALSE);
154: MatSetValuesBlocked(A_baij,1,idxm,1,idxn,A,INSERT_VALUES);
155: MatAssemblyBegin(A_baij,MAT_FINAL_ASSEMBLY);
156: MatAssemblyEnd(A_baij,MAT_FINAL_ASSEMBLY);
157: MatInvertBlockDiagonal(A_baij,&A_inv);
158: PetscMemcpy(A,A_inv,nstages*nstages*sizeof(PetscScalar));
159: MatDestroy(&A_baij);
160: }
161: /* Scale (1/dt)*A^{-1} and (1/dt)*b */
162: for (s=0; s<nstages*nstages; s++) A[s] *= 1.0/ctxt.dt;
163: for (s=0; s<nstages; s++) b[s] *= (-ctxt.dt);
165: /* Compute row sums At and identity B */
166: PetscMalloc2(nstages,&At,PetscSqr(nstages),&B);
167: for (s=0; s<nstages; s++) {
168: At[s] = 0;
169: for (t=0; t<nstages; t++) {
170: At[s] += A[s+nstages*t]; /* Row sums of */
171: B[s+nstages*t] = 1.*(s == t); /* identity */
172: }
173: }
175: /* allocate and calculate the (-J) matrix */
176: switch (ctxt.physics_type) {
177: case PHYSICS_ADVECTION:
178: case PHYSICS_DIFFUSION:
179: Assemble_AdvDiff(PETSC_COMM_WORLD,&ctxt,&J);
180: }
181: MatCreate(PETSC_COMM_WORLD,&Identity);
182: MatSetType(Identity,MATAIJ);
183: MatGetOwnershipRange(J,&matis,&matie);
184: MatSetSizes(Identity,matie-matis,matie-matis,ctxt.imax,ctxt.imax);
185: MatSetUp(Identity);
186: for (i=matis; i<matie; i++) {
187: ierr= MatSetValues(Identity,1,&i,1,&i,&one,INSERT_VALUES);
188: }
189: MatAssemblyBegin(Identity,MAT_FINAL_ASSEMBLY);
190: MatAssemblyEnd (Identity,MAT_FINAL_ASSEMBLY);
192: /* Create the KAIJ matrix for solving the stages */
193: MatCreateKAIJ(J,nstages,nstages,A,B,&TA);
195: /* Create the KAIJ matrix for step completion */
196: MatCreateKAIJ(J,1,nstages,NULL,b,&SC);
198: /* Create the KAIJ matrix to create the R for solving the stages */
199: MatCreateKAIJ(Identity,nstages,1,NULL,At,&R);
201: /* Create and set options for KSP */
202: KSPCreate(PETSC_COMM_WORLD,&ksp);
203: KSPSetOperators(ksp,TA,TA);
204: KSPSetFromOptions(ksp);
206: /* Allocate work and right-hand-side vectors */
207: VecCreate(PETSC_COMM_WORLD,&z);
208: VecSetFromOptions(z);
209: VecSetSizes(z,PETSC_DECIDE,ctxt.imax*nstages);
210: VecDuplicate(z,&rhs);
212: VecGetOwnershipRange(u,&is,&ie);
213: PetscMalloc3(nstages,&ix,nstages,&zvals,ie-is,&ix2);
214: /* iterate in time */
215: for (n=0,time=0.,total_its=0; n<ctxt.niter; n++) {
216: PetscInt its;
218: /* compute and set the right hand side */
219: MatMult(R,u,rhs);
221: /* Solve the system */
222: KSPSolve(ksp,rhs,z);
223: KSPGetIterationNumber(ksp,&its);
224: total_its += its;
226: /* Update the solution */
227: MatMultAdd(SC,z,u,u);
229: /* time step complete */
230: time += ctxt.dt;
231: }
232: PetscFree3(ix,ix2,zvals);
234: /* Deallocate work and right-hand-side vectors */
235: VecDestroy(&z);
236: VecDestroy(&rhs);
238: /* Calculate error in final solution */
239: VecAYPX(uex,-1.0,u);
240: VecNorm(uex,NORM_2,&err);
241: err = PetscSqrtReal(err*err/((PetscReal)ctxt.imax));
242: PetscPrintf(PETSC_COMM_WORLD,"L2 norm of the numerical error = %g (time=%g)\n",(double)err,(double)time);
243: PetscPrintf(PETSC_COMM_WORLD,"Number of time steps: %D (%D Krylov iterations)\n",ctxt.niter,total_its);
245: /* Free up memory */
246: KSPDestroy(&ksp);
247: MatDestroy(&TA);
248: MatDestroy(&SC);
249: MatDestroy(&R);
250: MatDestroy(&J);
251: MatDestroy(&Identity);
252: PetscFree3(A,b,c);
253: PetscFree2(At,B);
254: VecDestroy(&uex);
255: VecDestroy(&u);
256: PetscFunctionListDestroy(&IRKList);
258: PetscFinalize();
259: return ierr;
260: }
262: PetscErrorCode ExactSolution(Vec u,void *c,PetscReal t)
263: {
264: UserContext *ctxt = (UserContext*) c;
265: PetscErrorCode ierr;
266: PetscInt i,is,ie;
267: PetscScalar *uarr;
268: PetscReal x,dx,a=ctxt->a,pi=PETSC_PI;
271: dx = (ctxt->xmax - ctxt->xmin)/((PetscReal) ctxt->imax);
272: VecGetOwnershipRange(u,&is,&ie);
273: VecGetArray(u,&uarr);
274: for(i=is; i<ie; i++) {
275: x = i * dx;
276: switch (ctxt->physics_type) {
277: case PHYSICS_DIFFUSION:
278: uarr[i-is] = PetscExpScalar(-4.0*pi*pi*a*t)*PetscSinScalar(2*pi*x);
279: break;
280: case PHYSICS_ADVECTION:
281: uarr[i-is] = PetscSinScalar(2*pi*(x - a*t));
282: break;
283: default: SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_SUP,"No support for physics type %s",PhysicsTypes[ctxt->physics_type]);
284: }
285: }
286: VecRestoreArray(u,&uarr);
287: return(0);
288: }
290: /* Arrays should be freed with PetscFree3(A,b,c) */
291: static PetscErrorCode RKCreate_Gauss(PetscInt nstages,PetscScalar **gauss_A,PetscScalar **gauss_b,PetscReal **gauss_c)
292: {
293: PetscErrorCode ierr;
294: PetscScalar *A,*G0,*G1;
295: PetscReal *b,*c;
296: PetscInt i,j;
297: Mat G0mat,G1mat,Amat;
300: PetscMalloc3(PetscSqr(nstages),&A,nstages,gauss_b,nstages,&c);
301: PetscMalloc3(nstages,&b,PetscSqr(nstages),&G0,PetscSqr(nstages),&G1);
302: PetscDTGaussQuadrature(nstages,0.,1.,c,b);
303: for (i=0; i<nstages; i++) (*gauss_b)[i] = b[i]; /* copy to possibly-complex array */
305: /* A^T = G0^{-1} G1 */
306: for (i=0; i<nstages; i++) {
307: for (j=0; j<nstages; j++) {
308: G0[i*nstages+j] = PetscPowRealInt(c[i],j);
309: G1[i*nstages+j] = PetscPowRealInt(c[i],j+1)/(j+1);
310: }
311: }
312: /* The arrays above are row-aligned, but we create dense matrices as the transpose */
313: MatCreateSeqDense(PETSC_COMM_SELF,nstages,nstages,G0,&G0mat);
314: MatCreateSeqDense(PETSC_COMM_SELF,nstages,nstages,G1,&G1mat);
315: MatCreateSeqDense(PETSC_COMM_SELF,nstages,nstages,A,&Amat);
316: MatLUFactor(G0mat,NULL,NULL,NULL);
317: MatMatSolve(G0mat,G1mat,Amat);
318: MatTranspose(Amat,MAT_INPLACE_MATRIX,&Amat);
320: MatDestroy(&G0mat);
321: MatDestroy(&G1mat);
322: MatDestroy(&Amat);
323: PetscFree3(b,G0,G1);
324: *gauss_A = A;
325: *gauss_c = c;
326: return(0);
327: }
329: static PetscErrorCode Assemble_AdvDiff(MPI_Comm comm,UserContext *user,Mat *J)
330: {
332: PetscInt matis,matie,i;
333: PetscReal dx,dx2;
336: dx = (user->xmax - user->xmin)/((PetscReal)user->imax); dx2 = dx*dx;
337: MatCreate(comm,J);
338: MatSetType(*J,MATAIJ);
339: MatSetSizes(*J,PETSC_DECIDE,PETSC_DECIDE,user->imax,user->imax);
340: MatSetUp(*J);
341: MatGetOwnershipRange(*J,&matis,&matie);
342: for (i=matis; i<matie; i++) {
343: PetscScalar values[3];
344: PetscInt col[3];
345: switch (user->physics_type) {
346: case PHYSICS_DIFFUSION:
347: values[0] = -user->a*1.0/dx2;
348: values[1] = user->a*2.0/dx2;
349: values[2] = -user->a*1.0/dx2;
350: break;
351: case PHYSICS_ADVECTION:
352: values[0] = -user->a*.5/dx;
353: values[1] = 0.;
354: values[2] = user->a*.5/dx;
355: break;
356: default: SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_SUP,"No support for physics type %s",PhysicsTypes[user->physics_type]);
357: }
358: /* periodic boundaries */
359: if (i == 0) {
360: col[0] = user->imax-1;
361: col[1] = i;
362: col[2] = i+1;
363: } else if (i == user->imax-1) {
364: col[0] = i-1;
365: col[1] = i;
366: col[2] = 0;
367: } else {
368: col[0] = i-1;
369: col[1] = i;
370: col[2] = i+1;
371: }
372: ierr= MatSetValues(*J,1,&i,3,col,values,INSERT_VALUES);
373: }
374: MatAssemblyBegin(*J,MAT_FINAL_ASSEMBLY);
375: MatAssemblyEnd (*J,MAT_FINAL_ASSEMBLY);
376: return(0);
377: }
379: /*TEST
380: test:
381: suffix: 1
382: args: -a 0.1 -dt .125 -niter 5 -imax 40 -ksp_monitor_short -pc_type pbjacobi -ksp_atol 1e-6 -irk_type gauss -irk_nstages 2
383: test:
384: suffix: 2
385: args: -a 0.1 -dt .125 -niter 5 -imax 40 -ksp_monitor_short -pc_type pbjacobi -ksp_atol 1e-6 -irk_type gauss -irk_nstages 4 -ksp_gmres_restart 100
386: test:
387: suffix: 3
388: requires: !single
389: args: -a 1 -dt .33 -niter 3 -imax 40 -ksp_monitor_short -pc_type pbjacobi -ksp_atol 1e-6 -irk_type gauss -irk_nstages 4 -ksp_gmres_restart 100 -physics_type advection
391: TEST*/