| Line | Branch | Exec | Source |
|---|---|---|---|
| 1 | /* | ||
| 2 | - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | ||
| 3 | SLEPc - Scalable Library for Eigenvalue Problem Computations | ||
| 4 | Copyright (c) 2002-, Universitat Politecnica de Valencia, Spain | ||
| 5 | |||
| 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. | ||
| 13 | |||
| 14 | The problem is the 1-D heat equation with source term | ||
| 15 | |||
| 16 | y_t = y_xx + 1/(1+u^2) + psi | ||
| 17 | |||
| 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]. | ||
| 20 | |||
| 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 | */ | ||
| 25 | |||
| 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"; | ||
| 32 | |||
| 33 | #include <slepcmfn.h> | ||
| 34 | |||
| 35 | /* | ||
| 36 | BuildFNPhi: builds an FNCOMBINE object representing the phi_1 function | ||
| 37 | |||
| 38 | f(x) = (exp(x)-1)/x | ||
| 39 | |||
| 40 | with the following tree: | ||
| 41 | |||
| 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 | 10 | PetscErrorCode BuildFNPhi(FN fphi) | |
| 49 | { | ||
| 50 | 10 | FN fexp,faux,fconst,fpol; | |
| 51 | 10 | PetscScalar coeffs[2]; | |
| 52 | |||
| 53 |
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10 | PetscFunctionBeginUser; |
| 54 |
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10 | PetscCall(FNCreate(PETSC_COMM_WORLD,&fexp)); |
| 55 |
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10 | PetscCall(FNCreate(PETSC_COMM_WORLD,&fconst)); |
| 56 |
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10 | PetscCall(FNCreate(PETSC_COMM_WORLD,&faux)); |
| 57 |
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10 | PetscCall(FNCreate(PETSC_COMM_WORLD,&fpol)); |
| 58 | |||
| 59 |
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10 | PetscCall(FNSetType(fexp,FNEXP)); |
| 60 | |||
| 61 |
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10 | PetscCall(FNSetType(fconst,FNRATIONAL)); |
| 62 | 10 | coeffs[0] = -1.0; | |
| 63 |
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10 | PetscCall(FNRationalSetNumerator(fconst,1,coeffs)); |
| 64 | |||
| 65 |
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10 | PetscCall(FNSetType(faux,FNCOMBINE)); |
| 66 |
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10 | PetscCall(FNCombineSetChildren(faux,FN_COMBINE_ADD,fexp,fconst)); |
| 67 | |||
| 68 |
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10 | PetscCall(FNSetType(fpol,FNRATIONAL)); |
| 69 | 10 | coeffs[0] = 1.0; coeffs[1] = 0.0; | |
| 70 |
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10 | PetscCall(FNRationalSetNumerator(fpol,2,coeffs)); |
| 71 | |||
| 72 |
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10 | PetscCall(FNSetType(fphi,FNCOMBINE)); |
| 73 |
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10 | PetscCall(FNCombineSetChildren(fphi,FN_COMBINE_DIVIDE,faux,fpol)); |
| 74 | |||
| 75 |
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10 | PetscCall(FNDestroy(&faux)); |
| 76 |
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10 | PetscCall(FNDestroy(&fpol)); |
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10 | PetscCall(FNDestroy(&fconst)); |
| 78 |
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10 | PetscCall(FNDestroy(&fexp)); |
| 79 |
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2 | PetscFunctionReturn(PETSC_SUCCESS); |
| 80 | } | ||
| 81 | |||
| 82 | 20 | int main(int argc,char **argv) | |
| 83 | { | ||
| 84 | 20 | Mat L; | |
| 85 | 20 | Vec u,w,z,yex; | |
| 86 | 20 | MFN mfnexp,mfnphi; | |
| 87 | 20 | FN fexp,fphi; | |
| 88 | 20 | PetscBool combine=PETSC_FALSE; | |
| 89 | 20 | PetscInt i,k,Istart,Iend,n=199,steps; | |
| 90 | 20 | PetscReal t,tend=1.0,deltat=0.01,nrmd,nrmu,x,h; | |
| 91 | 20 | const PetscReal half=0.5; | |
| 92 | 20 | PetscScalar value,c,uval,*warray; | |
| 93 | 20 | const PetscScalar *uarray; | |
| 94 | |||
| 95 |
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20 | PetscFunctionBeginUser; |
| 96 |
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20 | PetscCall(SlepcInitialize(&argc,&argv,NULL,help)); |
| 97 | |||
| 98 |
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20 | PetscCall(PetscOptionsGetInt(NULL,NULL,"-n",&n,NULL)); |
| 99 |
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20 | PetscCall(PetscOptionsGetReal(NULL,NULL,"-tend",&tend,NULL)); |
| 100 |
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20 | PetscCall(PetscOptionsGetReal(NULL,NULL,"-deltat",&deltat,NULL)); |
| 101 |
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20 | PetscCall(PetscOptionsGetBool(NULL,NULL,"-combine",&combine,NULL)); |
| 102 | 20 | h = 1.0/(n+1.0); | |
| 103 | 20 | c = (n+1)*(n+1); | |
| 104 | |||
| 105 | 20 | steps = (PetscInt)(tend/deltat); | |
| 106 |
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20 | 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 |
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30 | 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)":"")); |
| 108 | |||
| 109 | /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | ||
| 110 | Build the 1-D Laplacian and various vectors | ||
| 111 | - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ | ||
| 112 |
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20 | PetscCall(MatCreate(PETSC_COMM_WORLD,&L)); |
| 113 |
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20 | PetscCall(MatSetSizes(L,PETSC_DECIDE,PETSC_DECIDE,n,n)); |
| 114 |
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20 | PetscCall(MatSetFromOptions(L)); |
| 115 |
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20 | PetscCall(MatGetOwnershipRange(L,&Istart,&Iend)); |
| 116 |
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2560 | for (i=Istart;i<Iend;i++) { |
| 117 |
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2540 | if (i>0) PetscCall(MatSetValue(L,i,i-1,c,INSERT_VALUES)); |
| 118 |
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2540 | if (i<n-1) PetscCall(MatSetValue(L,i,i+1,c,INSERT_VALUES)); |
| 119 |
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2540 | PetscCall(MatSetValue(L,i,i,-2.0*c,INSERT_VALUES)); |
| 120 | } | ||
| 121 |
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20 | PetscCall(MatAssemblyBegin(L,MAT_FINAL_ASSEMBLY)); |
| 122 |
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20 | PetscCall(MatAssemblyEnd(L,MAT_FINAL_ASSEMBLY)); |
| 123 |
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20 | PetscCall(MatCreateVecs(L,NULL,&u)); |
| 124 |
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20 | PetscCall(VecDuplicate(u,&yex)); |
| 125 |
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20 | PetscCall(VecDuplicate(u,&w)); |
| 126 |
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20 | PetscCall(VecDuplicate(u,&z)); |
| 127 | |||
| 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 |
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2560 | for (i=Istart;i<Iend;i++) { |
| 134 | 2540 | x = (i+1)*h; | |
| 135 | 2540 | value = x*(1.0-x)*PetscExpReal(tend); | |
| 136 |
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2540 | PetscCall(VecSetValue(yex,i,value,INSERT_VALUES)); |
| 137 | 2540 | value = PetscAbsReal(x-half)-half; | |
| 138 |
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2540 | PetscCall(VecSetValue(u,i,value,INSERT_VALUES)); |
| 139 | } | ||
| 140 |
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20 | PetscCall(VecAssemblyBegin(yex)); |
| 141 |
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20 | PetscCall(VecAssemblyBegin(u)); |
| 142 |
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20 | PetscCall(VecAssemblyEnd(yex)); |
| 143 |
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20 | PetscCall(VecAssemblyEnd(u)); |
| 144 |
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20 | PetscCall(VecViewFromOptions(yex,NULL,"-exact_sol")); |
| 145 |
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20 | PetscCall(VecViewFromOptions(u,NULL,"-initial_cond")); |
| 146 | |||
| 147 | /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | ||
| 148 | Create two MFN solvers, for exp() and phi_1() | ||
| 149 | - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ | ||
| 150 |
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20 | PetscCall(MFNCreate(PETSC_COMM_WORLD,&mfnexp)); |
| 151 |
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20 | PetscCall(MFNSetOperator(mfnexp,L)); |
| 152 |
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20 | PetscCall(MFNGetFN(mfnexp,&fexp)); |
| 153 |
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20 | PetscCall(FNSetType(fexp,FNEXP)); |
| 154 |
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20 | PetscCall(FNSetScale(fexp,deltat,1.0)); |
| 155 |
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20 | PetscCall(MFNSetErrorIfNotConverged(mfnexp,PETSC_TRUE)); |
| 156 |
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20 | PetscCall(MFNSetFromOptions(mfnexp)); |
| 157 | |||
| 158 |
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20 | PetscCall(MFNCreate(PETSC_COMM_WORLD,&mfnphi)); |
| 159 |
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20 | PetscCall(MFNSetOperator(mfnphi,L)); |
| 160 |
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20 | PetscCall(MFNGetFN(mfnphi,&fphi)); |
| 161 |
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20 | if (combine) PetscCall(BuildFNPhi(fphi)); |
| 162 | else { | ||
| 163 |
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10 | PetscCall(FNSetType(fphi,FNPHI)); |
| 164 |
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10 | PetscCall(FNPhiSetIndex(fphi,1)); |
| 165 | } | ||
| 166 |
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20 | PetscCall(FNSetScale(fphi,deltat,1.0)); |
| 167 |
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20 | PetscCall(MFNSetErrorIfNotConverged(mfnphi,PETSC_TRUE)); |
| 168 |
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20 | PetscCall(MFNSetFromOptions(mfnphi)); |
| 169 | |||
| 170 | /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | ||
| 171 | Solve the problem with the Norsett-Euler scheme | ||
| 172 | - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ | ||
| 173 | t = 0.0; | ||
| 174 |
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120 | for (k=0;k<steps;k++) { |
| 175 | |||
| 176 | /* evaluate nonlinear part */ | ||
| 177 |
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100 | PetscCall(VecGetArrayRead(u,&uarray)); |
| 178 |
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100 | PetscCall(VecGetArray(w,&warray)); |
| 179 |
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12800 | for (i=Istart;i<Iend;i++) { |
| 180 | 12700 | x = (i+1)*h; | |
| 181 | 12700 | uval = uarray[i-Istart]; | |
| 182 | 12700 | value = x*(1.0-x)*PetscExpReal(t); | |
| 183 | 12700 | value = value + 2.0*PetscExpReal(t) - 1.0/(1.0+value*value); | |
| 184 | 12700 | value = value + 1.0/(1.0+uval*uval); | |
| 185 | 12700 | warray[i-Istart] = deltat*value; | |
| 186 | } | ||
| 187 |
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100 | PetscCall(VecRestoreArrayRead(u,&uarray)); |
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100 | PetscCall(VecRestoreArray(w,&warray)); |
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100 | PetscCall(MFNSolve(mfnphi,w,z)); |
| 190 | |||
| 191 | /* evaluate linear part */ | ||
| 192 |
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100 | PetscCall(MFNSolve(mfnexp,u,u)); |
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100 | PetscCall(VecAXPY(u,1.0,z)); |
| 194 | 100 | t = t + deltat; | |
| 195 | |||
| 196 | } | ||
| 197 |
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20 | PetscCall(VecViewFromOptions(u,NULL,"-computed_sol")); |
| 198 | |||
| 199 | /* | ||
| 200 | Compare with exact solution and show error norm | ||
| 201 | */ | ||
| 202 |
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20 | PetscCall(VecCopy(u,z)); |
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20 | PetscCall(VecAXPY(z,-1.0,yex)); |
| 204 |
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20 | PetscCall(VecNorm(z,NORM_2,&nrmd)); |
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20 | PetscCall(VecNorm(u,NORM_2,&nrmu)); |
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20 | PetscCall(PetscPrintf(PETSC_COMM_WORLD," The relative error at t=%g is %.4f\n\n",(double)t,(double)(nrmd/nrmu))); |
| 207 | |||
| 208 | /* | ||
| 209 | Free work space | ||
| 210 | */ | ||
| 211 |
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20 | PetscCall(MFNDestroy(&mfnexp)); |
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20 | PetscCall(MFNDestroy(&mfnphi)); |
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20 | PetscCall(MatDestroy(&L)); |
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20 | PetscCall(VecDestroy(&u)); |
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20 | PetscCall(VecDestroy(&yex)); |
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20 | PetscCall(VecDestroy(&w)); |
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20 | PetscCall(VecDestroy(&z)); |
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20 | PetscCall(SlepcFinalize()); |
| 219 | return 0; | ||
| 220 | } | ||
| 221 | |||
| 222 | /*TEST | ||
| 223 | |||
| 224 | test: | ||
| 225 | suffix: 1 | ||
| 226 | args: -n 127 -tend 0.125 -mfn_tol 1e-3 -deltat 0.025 | ||
| 227 | timeoutfactor: 2 | ||
| 228 | |||
| 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 | ||
| 236 | |||
| 237 | TEST*/ | ||
| 238 |