Actual source code: ex39.c

slepc-3.16.1 2021-11-17
Report Typos and Errors
  1: /*
  2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  4:    Copyright (c) 2002-2021, 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: {
 51:   FN             fexp,faux,fconst,fpol;
 52:   PetscScalar    coeffs[2];

 55:   FNCreate(PETSC_COMM_WORLD,&fexp);
 56:   FNCreate(PETSC_COMM_WORLD,&fconst);
 57:   FNCreate(PETSC_COMM_WORLD,&faux);
 58:   FNCreate(PETSC_COMM_WORLD,&fpol);

 60:   FNSetType(fexp,FNEXP);

 62:   FNSetType(fconst,FNRATIONAL);
 63:   coeffs[0] = -1.0;
 64:   FNRationalSetNumerator(fconst,1,coeffs);

 66:   FNSetType(faux,FNCOMBINE);
 67:   FNCombineSetChildren(faux,FN_COMBINE_ADD,fexp,fconst);

 69:   FNSetType(fpol,FNRATIONAL);
 70:   coeffs[0] = 1.0; coeffs[1] = 0.0;
 71:   FNRationalSetNumerator(fpol,2,coeffs);

 73:   FNSetType(fphi,FNCOMBINE);
 74:   FNCombineSetChildren(fphi,FN_COMBINE_DIVIDE,faux,fpol);

 76:   FNDestroy(&faux);
 77:   FNDestroy(&fpol);
 78:   FNDestroy(&fconst);
 79:   FNDestroy(&fexp);
 80:   return(0);
 81: }

 83: int main(int argc,char **argv)
 84: {
 85:   Mat               L;
 86:   Vec               u,w,z,yex;
 87:   MFN               mfnexp,mfnphi;
 88:   FN                fexp,fphi;
 89:   PetscBool         combine=PETSC_FALSE;
 90:   PetscInt          i,k,Istart,Iend,n=199,steps;
 91:   PetscReal         t,tend=1.0,deltat=0.01,nrmd,nrmu,x,h;
 92:   const PetscReal   half=0.5;
 93:   PetscScalar       value,c,uval,*warray;
 94:   const PetscScalar *uarray;
 95:   PetscErrorCode    ierr;

 97:   SlepcInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;

 99:   PetscOptionsGetInt(NULL,NULL,"-n",&n,NULL);
100:   PetscOptionsGetReal(NULL,NULL,"-tend",&tend,NULL);
101:   PetscOptionsGetReal(NULL,NULL,"-deltat",&deltat,NULL);
102:   PetscOptionsGetBool(NULL,NULL,"-combine",&combine,NULL);
103:   h = 1.0/(n+1.0);
104:   c = (n+1)*(n+1);

106:   steps = (PetscInt)(tend/deltat);
107:   if (PetscAbsReal(tend-steps*deltat)>10*PETSC_MACHINE_EPSILON) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_SUP,"This example requires tend being a multiple of deltat");
108:   PetscPrintf(PETSC_COMM_WORLD,"\nHeat equation via phi functions, n=%D, tend=%g, deltat=%g%s\n\n",n,(double)tend,(double)deltat,combine?" (combine)":"");

110:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
111:                  Build the 1-D Laplacian and various vectors
112:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
113:   MatCreate(PETSC_COMM_WORLD,&L);
114:   MatSetSizes(L,PETSC_DECIDE,PETSC_DECIDE,n,n);
115:   MatSetFromOptions(L);
116:   MatSetUp(L);
117:   MatGetOwnershipRange(L,&Istart,&Iend);
118:   for (i=Istart;i<Iend;i++) {
119:     if (i>0) { MatSetValue(L,i,i-1,c,INSERT_VALUES); }
120:     if (i<n-1) { MatSetValue(L,i,i+1,c,INSERT_VALUES); }
121:     MatSetValue(L,i,i,-2.0*c,INSERT_VALUES);
122:   }
123:   MatAssemblyBegin(L,MAT_FINAL_ASSEMBLY);
124:   MatAssemblyEnd(L,MAT_FINAL_ASSEMBLY);
125:   MatCreateVecs(L,NULL,&u);
126:   VecDuplicate(u,&yex);
127:   VecDuplicate(u,&w);
128:   VecDuplicate(u,&z);

130:   /*
131:      Compute various vectors:
132:      - the exact solution yex = x*(1-x)*exp(tend)
133:      - the initial condition u = abs(x-0.5)-0.5
134:   */
135:   for (i=Istart;i<Iend;i++) {
136:     x = (i+1)*h;
137:     value = x*(1.0-x)*PetscExpReal(tend);
138:     VecSetValue(yex,i,value,INSERT_VALUES);
139:     value = PetscAbsReal(x-half)-half;
140:     VecSetValue(u,i,value,INSERT_VALUES);
141:   }
142:   VecAssemblyBegin(yex);
143:   VecAssemblyBegin(u);
144:   VecAssemblyEnd(yex);
145:   VecAssemblyEnd(u);
146:   VecViewFromOptions(yex,NULL,"-exact_sol");
147:   VecViewFromOptions(u,NULL,"-initial_cond");

149:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
150:               Create two MFN solvers, for exp() and phi_1()
151:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
152:   MFNCreate(PETSC_COMM_WORLD,&mfnexp);
153:   MFNSetOperator(mfnexp,L);
154:   MFNGetFN(mfnexp,&fexp);
155:   FNSetType(fexp,FNEXP);
156:   FNSetScale(fexp,deltat,1.0);
157:   MFNSetErrorIfNotConverged(mfnexp,PETSC_TRUE);
158:   MFNSetFromOptions(mfnexp);

160:   MFNCreate(PETSC_COMM_WORLD,&mfnphi);
161:   MFNSetOperator(mfnphi,L);
162:   MFNGetFN(mfnphi,&fphi);
163:   if (combine) {
164:     BuildFNPhi(fphi);
165:   } else {
166:     FNSetType(fphi,FNPHI);
167:     FNPhiSetIndex(fphi,1);
168:   }
169:   FNSetScale(fphi,deltat,1.0);
170:   MFNSetErrorIfNotConverged(mfnphi,PETSC_TRUE);
171:   MFNSetFromOptions(mfnphi);

173:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
174:              Solve the problem with the Norsett-Euler scheme
175:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
176:   t = 0.0;
177:   for (k=0;k<steps;k++) {

179:     /* evaluate nonlinear part */
180:     VecGetArrayRead(u,&uarray);
181:     VecGetArray(w,&warray);
182:     for (i=Istart;i<Iend;i++) {
183:       x = (i+1)*h;
184:       uval = uarray[i-Istart];
185:       value = x*(1.0-x)*PetscExpReal(t);
186:       value = value + 2.0*PetscExpReal(t) - 1.0/(1.0+value*value);
187:       value = value + 1.0/(1.0+uval*uval);
188:       warray[i-Istart] = deltat*value;
189:     }
190:     VecRestoreArrayRead(u,&uarray);
191:     VecRestoreArray(w,&warray);
192:     MFNSolve(mfnphi,w,z);

194:     /* evaluate linear part */
195:     MFNSolve(mfnexp,u,u);
196:     VecAXPY(u,1.0,z);
197:     t = t + deltat;

199:   }
200:   VecViewFromOptions(u,NULL,"-computed_sol");

202:   /*
203:      Compare with exact solution and show error norm
204:   */
205:   VecCopy(u,z);
206:   VecAXPY(z,-1.0,yex);
207:   VecNorm(z,NORM_2,&nrmd);
208:   VecNorm(u,NORM_2,&nrmu);
209:   PetscPrintf(PETSC_COMM_WORLD," The relative error at t=%g is %.4f\n\n",(double)t,(double)(nrmd/nrmu));

211:   /*
212:      Free work space
213:   */
214:   MFNDestroy(&mfnexp);
215:   MFNDestroy(&mfnphi);
216:   MatDestroy(&L);
217:   VecDestroy(&u);
218:   VecDestroy(&yex);
219:   VecDestroy(&w);
220:   VecDestroy(&z);
221:   SlepcFinalize();
222:   return ierr;
223: }

225: /*TEST

227:    test:
228:       suffix: 1
229:       args: -n 127 -tend 0.125 -mfn_tol 1e-3 -deltat 0.025
230:       timeoutfactor: 2

232:    test:
233:       suffix: 2
234:       args: -n 127 -tend 0.125 -mfn_tol 1e-3 -deltat 0.025 -combine
235:       filter: sed -e "s/ (combine)//"
236:       requires: !single
237:       output_file: output/ex39_1.out
238:       timeoutfactor: 2

240: TEST*/