Actual source code: ex39.c
slepc-3.17.0 2022-03-31
1: /*
2: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3: SLEPc - Scalable Library for Eigenvalue Problem Computations
4: Copyright (c) 2002-, Universitat Politecnica de Valencia, Spain
6: This file is part of SLEPc.
7: SLEPc is distributed under a 2-clause BSD license (see LICENSE).
8: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
9: */
10: /*
11: This example illustrates the use of Phi functions in exponential integrators.
12: In particular, it implements the Norsett-Euler scheme of stiff order 1.
14: The problem is the 1-D heat equation with source term
16: y_t = y_xx + 1/(1+u^2) + psi
18: where psi is chosen so that the exact solution is yex = x*(1-x)*exp(tend).
19: The space domain is [0,1] and the time interval is [0,tend].
21: [1] M. Hochbruck and A. Ostermann, "Explicit exponential Runge-Kutta
22: methods for semilinear parabolic problems", SIAM J. Numer. Anal. 43(3),
23: 1069-1090, 2005.
24: */
26: static char help[] = "Exponential integrator for the heat equation with source term.\n\n"
27: "The command line options are:\n"
28: " -n <idim>, where <idim> = dimension of the spatial discretization.\n"
29: " -tend <rval>, where <rval> = real value that corresponding to the final time.\n"
30: " -deltat <rval>, where <rval> = real value for the time increment.\n"
31: " -combine <bool>, to represent the phi function with FNCOMBINE instead of FNPHI.\n\n";
33: #include <slepcmfn.h>
35: /*
36: BuildFNPhi: builds an FNCOMBINE object representing the phi_1 function
38: f(x) = (exp(x)-1)/x
40: with the following tree:
42: f(x) f(x) (combined by division)
43: / \ p(x) = x (polynomial)
44: a(x) p(x) a(x) (combined by addition)
45: / \ e(x) = exp(x) (exponential)
46: e(x) c(x) c(x) = -1 (constant)
47: */
48: PetscErrorCode BuildFNPhi(FN fphi)
49: {
50: FN fexp,faux,fconst,fpol;
51: PetscScalar coeffs[2];
54: FNCreate(PETSC_COMM_WORLD,&fexp);
55: FNCreate(PETSC_COMM_WORLD,&fconst);
56: FNCreate(PETSC_COMM_WORLD,&faux);
57: FNCreate(PETSC_COMM_WORLD,&fpol);
59: FNSetType(fexp,FNEXP);
61: FNSetType(fconst,FNRATIONAL);
62: coeffs[0] = -1.0;
63: FNRationalSetNumerator(fconst,1,coeffs);
65: FNSetType(faux,FNCOMBINE);
66: FNCombineSetChildren(faux,FN_COMBINE_ADD,fexp,fconst);
68: FNSetType(fpol,FNRATIONAL);
69: coeffs[0] = 1.0; coeffs[1] = 0.0;
70: FNRationalSetNumerator(fpol,2,coeffs);
72: FNSetType(fphi,FNCOMBINE);
73: FNCombineSetChildren(fphi,FN_COMBINE_DIVIDE,faux,fpol);
75: FNDestroy(&faux);
76: FNDestroy(&fpol);
77: FNDestroy(&fconst);
78: FNDestroy(&fexp);
79: PetscFunctionReturn(0);
80: }
82: int main(int argc,char **argv)
83: {
84: Mat L;
85: Vec u,w,z,yex;
86: MFN mfnexp,mfnphi;
87: FN fexp,fphi;
88: PetscBool combine=PETSC_FALSE;
89: PetscInt i,k,Istart,Iend,n=199,steps;
90: PetscReal t,tend=1.0,deltat=0.01,nrmd,nrmu,x,h;
91: const PetscReal half=0.5;
92: PetscScalar value,c,uval,*warray;
93: const PetscScalar *uarray;
95: SlepcInitialize(&argc,&argv,(char*)0,help);
97: PetscOptionsGetInt(NULL,NULL,"-n",&n,NULL);
98: PetscOptionsGetReal(NULL,NULL,"-tend",&tend,NULL);
99: PetscOptionsGetReal(NULL,NULL,"-deltat",&deltat,NULL);
100: PetscOptionsGetBool(NULL,NULL,"-combine",&combine,NULL);
101: h = 1.0/(n+1.0);
102: c = (n+1)*(n+1);
104: steps = (PetscInt)(tend/deltat);
106: PetscPrintf(PETSC_COMM_WORLD,"\nHeat equation via phi functions, n=%" PetscInt_FMT ", tend=%g, deltat=%g%s\n\n",n,(double)tend,(double)deltat,combine?" (combine)":"");
108: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
109: Build the 1-D Laplacian and various vectors
110: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
111: MatCreate(PETSC_COMM_WORLD,&L);
112: MatSetSizes(L,PETSC_DECIDE,PETSC_DECIDE,n,n);
113: MatSetFromOptions(L);
114: MatSetUp(L);
115: MatGetOwnershipRange(L,&Istart,&Iend);
116: for (i=Istart;i<Iend;i++) {
117: if (i>0) MatSetValue(L,i,i-1,c,INSERT_VALUES);
118: if (i<n-1) MatSetValue(L,i,i+1,c,INSERT_VALUES);
119: MatSetValue(L,i,i,-2.0*c,INSERT_VALUES);
120: }
121: MatAssemblyBegin(L,MAT_FINAL_ASSEMBLY);
122: MatAssemblyEnd(L,MAT_FINAL_ASSEMBLY);
123: MatCreateVecs(L,NULL,&u);
124: VecDuplicate(u,&yex);
125: VecDuplicate(u,&w);
126: VecDuplicate(u,&z);
128: /*
129: Compute various vectors:
130: - the exact solution yex = x*(1-x)*exp(tend)
131: - the initial condition u = abs(x-0.5)-0.5
132: */
133: for (i=Istart;i<Iend;i++) {
134: x = (i+1)*h;
135: value = x*(1.0-x)*PetscExpReal(tend);
136: VecSetValue(yex,i,value,INSERT_VALUES);
137: value = PetscAbsReal(x-half)-half;
138: VecSetValue(u,i,value,INSERT_VALUES);
139: }
140: VecAssemblyBegin(yex);
141: VecAssemblyBegin(u);
142: VecAssemblyEnd(yex);
143: VecAssemblyEnd(u);
144: VecViewFromOptions(yex,NULL,"-exact_sol");
145: VecViewFromOptions(u,NULL,"-initial_cond");
147: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
148: Create two MFN solvers, for exp() and phi_1()
149: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
150: MFNCreate(PETSC_COMM_WORLD,&mfnexp);
151: MFNSetOperator(mfnexp,L);
152: MFNGetFN(mfnexp,&fexp);
153: FNSetType(fexp,FNEXP);
154: FNSetScale(fexp,deltat,1.0);
155: MFNSetErrorIfNotConverged(mfnexp,PETSC_TRUE);
156: MFNSetFromOptions(mfnexp);
158: MFNCreate(PETSC_COMM_WORLD,&mfnphi);
159: MFNSetOperator(mfnphi,L);
160: MFNGetFN(mfnphi,&fphi);
161: if (combine) BuildFNPhi(fphi);
162: else {
163: FNSetType(fphi,FNPHI);
164: FNPhiSetIndex(fphi,1);
165: }
166: FNSetScale(fphi,deltat,1.0);
167: MFNSetErrorIfNotConverged(mfnphi,PETSC_TRUE);
168: MFNSetFromOptions(mfnphi);
170: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
171: Solve the problem with the Norsett-Euler scheme
172: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
173: t = 0.0;
174: for (k=0;k<steps;k++) {
176: /* evaluate nonlinear part */
177: VecGetArrayRead(u,&uarray);
178: VecGetArray(w,&warray);
179: for (i=Istart;i<Iend;i++) {
180: x = (i+1)*h;
181: uval = uarray[i-Istart];
182: value = x*(1.0-x)*PetscExpReal(t);
183: value = value + 2.0*PetscExpReal(t) - 1.0/(1.0+value*value);
184: value = value + 1.0/(1.0+uval*uval);
185: warray[i-Istart] = deltat*value;
186: }
187: VecRestoreArrayRead(u,&uarray);
188: VecRestoreArray(w,&warray);
189: MFNSolve(mfnphi,w,z);
191: /* evaluate linear part */
192: MFNSolve(mfnexp,u,u);
193: VecAXPY(u,1.0,z);
194: t = t + deltat;
196: }
197: VecViewFromOptions(u,NULL,"-computed_sol");
199: /*
200: Compare with exact solution and show error norm
201: */
202: VecCopy(u,z);
203: VecAXPY(z,-1.0,yex);
204: VecNorm(z,NORM_2,&nrmd);
205: VecNorm(u,NORM_2,&nrmu);
206: PetscPrintf(PETSC_COMM_WORLD," The relative error at t=%g is %.4f\n\n",(double)t,(double)(nrmd/nrmu));
208: /*
209: Free work space
210: */
211: MFNDestroy(&mfnexp);
212: MFNDestroy(&mfnphi);
213: MatDestroy(&L);
214: VecDestroy(&u);
215: VecDestroy(&yex);
216: VecDestroy(&w);
217: VecDestroy(&z);
218: SlepcFinalize();
219: return 0;
220: }
222: /*TEST
224: test:
225: suffix: 1
226: args: -n 127 -tend 0.125 -mfn_tol 1e-3 -deltat 0.025
227: timeoutfactor: 2
229: test:
230: suffix: 2
231: args: -n 127 -tend 0.125 -mfn_tol 1e-3 -deltat 0.025 -combine
232: filter: sed -e "s/ (combine)//"
233: requires: !single
234: output_file: output/ex39_1.out
235: timeoutfactor: 2
237: TEST*/