Actual source code: ex39.c

slepc-3.22.1 2024-10-28
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  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];

 53:   PetscFunctionBeginUser;
 54:   PetscCall(FNCreate(PETSC_COMM_WORLD,&fexp));
 55:   PetscCall(FNCreate(PETSC_COMM_WORLD,&fconst));
 56:   PetscCall(FNCreate(PETSC_COMM_WORLD,&faux));
 57:   PetscCall(FNCreate(PETSC_COMM_WORLD,&fpol));

 59:   PetscCall(FNSetType(fexp,FNEXP));

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

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

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

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

 75:   PetscCall(FNDestroy(&faux));
 76:   PetscCall(FNDestroy(&fpol));
 77:   PetscCall(FNDestroy(&fconst));
 78:   PetscCall(FNDestroy(&fexp));
 79:   PetscFunctionReturn(PETSC_SUCCESS);
 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:   PetscFunctionBeginUser;
 96:   PetscCall(SlepcInitialize(&argc,&argv,NULL,help));

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

105:   steps = (PetscInt)(tend/deltat);
106:   PetscCheck(PetscAbsReal(tend-steps*deltat)<10*PETSC_MACHINE_EPSILON,PETSC_COMM_WORLD,PETSC_ERR_SUP,"This example requires tend being a multiple of deltat");
107:   PetscCall(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)":""));

109:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
110:                  Build the 1-D Laplacian and various vectors
111:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
112:   PetscCall(MatCreate(PETSC_COMM_WORLD,&L));
113:   PetscCall(MatSetSizes(L,PETSC_DECIDE,PETSC_DECIDE,n,n));
114:   PetscCall(MatSetFromOptions(L));
115:   PetscCall(MatGetOwnershipRange(L,&Istart,&Iend));
116:   for (i=Istart;i<Iend;i++) {
117:     if (i>0) PetscCall(MatSetValue(L,i,i-1,c,INSERT_VALUES));
118:     if (i<n-1) PetscCall(MatSetValue(L,i,i+1,c,INSERT_VALUES));
119:     PetscCall(MatSetValue(L,i,i,-2.0*c,INSERT_VALUES));
120:   }
121:   PetscCall(MatAssemblyBegin(L,MAT_FINAL_ASSEMBLY));
122:   PetscCall(MatAssemblyEnd(L,MAT_FINAL_ASSEMBLY));
123:   PetscCall(MatCreateVecs(L,NULL,&u));
124:   PetscCall(VecDuplicate(u,&yex));
125:   PetscCall(VecDuplicate(u,&w));
126:   PetscCall(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:     PetscCall(VecSetValue(yex,i,value,INSERT_VALUES));
137:     value = PetscAbsReal(x-half)-half;
138:     PetscCall(VecSetValue(u,i,value,INSERT_VALUES));
139:   }
140:   PetscCall(VecAssemblyBegin(yex));
141:   PetscCall(VecAssemblyBegin(u));
142:   PetscCall(VecAssemblyEnd(yex));
143:   PetscCall(VecAssemblyEnd(u));
144:   PetscCall(VecViewFromOptions(yex,NULL,"-exact_sol"));
145:   PetscCall(VecViewFromOptions(u,NULL,"-initial_cond"));

147:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
148:               Create two MFN solvers, for exp() and phi_1()
149:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
150:   PetscCall(MFNCreate(PETSC_COMM_WORLD,&mfnexp));
151:   PetscCall(MFNSetOperator(mfnexp,L));
152:   PetscCall(MFNGetFN(mfnexp,&fexp));
153:   PetscCall(FNSetType(fexp,FNEXP));
154:   PetscCall(FNSetScale(fexp,deltat,1.0));
155:   PetscCall(MFNSetErrorIfNotConverged(mfnexp,PETSC_TRUE));
156:   PetscCall(MFNSetFromOptions(mfnexp));

158:   PetscCall(MFNCreate(PETSC_COMM_WORLD,&mfnphi));
159:   PetscCall(MFNSetOperator(mfnphi,L));
160:   PetscCall(MFNGetFN(mfnphi,&fphi));
161:   if (combine) PetscCall(BuildFNPhi(fphi));
162:   else {
163:     PetscCall(FNSetType(fphi,FNPHI));
164:     PetscCall(FNPhiSetIndex(fphi,1));
165:   }
166:   PetscCall(FNSetScale(fphi,deltat,1.0));
167:   PetscCall(MFNSetErrorIfNotConverged(mfnphi,PETSC_TRUE));
168:   PetscCall(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:     PetscCall(VecGetArrayRead(u,&uarray));
178:     PetscCall(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:     PetscCall(VecRestoreArrayRead(u,&uarray));
188:     PetscCall(VecRestoreArray(w,&warray));
189:     PetscCall(MFNSolve(mfnphi,w,z));

191:     /* evaluate linear part */
192:     PetscCall(MFNSolve(mfnexp,u,u));
193:     PetscCall(VecAXPY(u,1.0,z));
194:     t = t + deltat;

196:   }
197:   PetscCall(VecViewFromOptions(u,NULL,"-computed_sol"));

199:   /*
200:      Compare with exact solution and show error norm
201:   */
202:   PetscCall(VecCopy(u,z));
203:   PetscCall(VecAXPY(z,-1.0,yex));
204:   PetscCall(VecNorm(z,NORM_2,&nrmd));
205:   PetscCall(VecNorm(u,NORM_2,&nrmu));
206:   PetscCall(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:   PetscCall(MFNDestroy(&mfnexp));
212:   PetscCall(MFNDestroy(&mfnphi));
213:   PetscCall(MatDestroy(&L));
214:   PetscCall(VecDestroy(&u));
215:   PetscCall(VecDestroy(&yex));
216:   PetscCall(VecDestroy(&w));
217:   PetscCall(VecDestroy(&z));
218:   PetscCall(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*/