1 2 static char help[] = "Solves the van der Pol equation.\n\ 3 Input parameters include:\n"; 4 5 /* 6 Concepts: TS^time-dependent nonlinear problems 7 Concepts: TS^van der Pol equation DAE equivalent 8 Processors: 1 9 */ 10 /* ------------------------------------------------------------------------ 11 12 This program solves the van der Pol DAE ODE equivalent 13 y' = z (1) 14 z' = \mu ((1-y^2)z-y) 15 on the domain 0 <= x <= 1, with the boundary conditions 16 y(0) = 2, y'(0) = - 2/3 +10/(81*\mu) - 292/(2187*\mu^2), 17 and 18 \mu = 10^6 ( y'(0) ~ -0.6666665432100101). 19 This is a nonlinear equation. The well prepared initial condition gives errors that are not dominated by the first few steps of the method when \mu is large. 20 21 Notes: 22 This code demonstrates the TS solver interface to an ODE -- RHSFunction for explicit form and IFunction for implicit form. 23 24 ------------------------------------------------------------------------- */ 25 26 #include <petscts.h> 27 28 typedef struct _n_User *User; 29 struct _n_User { 30 PetscReal mu; 31 PetscReal next_output; 32 }; 33 34 /* 35 User-defined routines 36 */ 37 static PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec X,Vec F,void *ctx) 38 { 39 PetscErrorCode ierr; 40 User user = (User)ctx; 41 PetscScalar *f; 42 const PetscScalar *x; 43 44 PetscFunctionBeginUser; 45 ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); 46 ierr = VecGetArray(F,&f);CHKERRQ(ierr); 47 f[0] = x[1]; 48 f[1] = user->mu*(1.-x[0]*x[0])*x[1]-x[0]; 49 ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); 50 ierr = VecRestoreArray(F,&f);CHKERRQ(ierr); 51 PetscFunctionReturn(0); 52 } 53 54 static PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx) 55 { 56 PetscErrorCode ierr; 57 User user = (User)ctx; 58 const PetscScalar *x,*xdot; 59 PetscScalar *f; 60 61 PetscFunctionBeginUser; 62 ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); 63 ierr = VecGetArrayRead(Xdot,&xdot);CHKERRQ(ierr); 64 ierr = VecGetArray(F,&f);CHKERRQ(ierr); 65 f[0] = xdot[0] - x[1]; 66 f[1] = xdot[1] - user->mu*((1.0-x[0]*x[0])*x[1] - x[0]); 67 ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); 68 ierr = VecRestoreArrayRead(Xdot,&xdot);CHKERRQ(ierr); 69 ierr = VecRestoreArray(F,&f);CHKERRQ(ierr); 70 PetscFunctionReturn(0); 71 } 72 73 static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat A,Mat B,void *ctx) 74 { 75 PetscErrorCode ierr; 76 User user = (User)ctx; 77 PetscInt rowcol[] = {0,1}; 78 const PetscScalar *x; 79 PetscScalar J[2][2]; 80 81 PetscFunctionBeginUser; 82 ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); 83 J[0][0] = a; J[0][1] = -1.0; 84 J[1][0] = user->mu*(2.0*x[0]*x[1] + 1.0); J[1][1] = a - user->mu*(1.0-x[0]*x[0]); 85 ierr = MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);CHKERRQ(ierr); 86 ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); 87 88 ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 89 ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 90 if (A != B) { 91 ierr = MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 92 ierr = MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 93 } 94 PetscFunctionReturn(0); 95 } 96 97 /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ 98 static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx) 99 { 100 PetscErrorCode ierr; 101 const PetscScalar *x; 102 PetscReal tfinal, dt; 103 User user = (User)ctx; 104 Vec interpolatedX; 105 106 PetscFunctionBeginUser; 107 ierr = TSGetTimeStep(ts,&dt);CHKERRQ(ierr); 108 ierr = TSGetMaxTime(ts,&tfinal);CHKERRQ(ierr); 109 110 while (user->next_output <= t && user->next_output <= tfinal) { 111 ierr = VecDuplicate(X,&interpolatedX);CHKERRQ(ierr); 112 ierr = TSInterpolate(ts,user->next_output,interpolatedX);CHKERRQ(ierr); 113 ierr = VecGetArrayRead(interpolatedX,&x);CHKERRQ(ierr); 114 ierr = PetscPrintf(PETSC_COMM_WORLD,"[%.1f] %D TS %.6f (dt = %.6f) X % 12.6e % 12.6e\n", 115 user->next_output,step,t,dt,(double)PetscRealPart(x[0]), 116 (double)PetscRealPart(x[1]));CHKERRQ(ierr); 117 ierr = VecRestoreArrayRead(interpolatedX,&x);CHKERRQ(ierr); 118 ierr = VecDestroy(&interpolatedX);CHKERRQ(ierr); 119 user->next_output += 0.1; 120 } 121 PetscFunctionReturn(0); 122 } 123 124 int main(int argc,char **argv) 125 { 126 TS ts; /* nonlinear solver */ 127 Vec x; /* solution, residual vectors */ 128 Mat A; /* Jacobian matrix */ 129 PetscInt steps; 130 PetscReal ftime = 0.5; 131 PetscBool monitor = PETSC_FALSE,implicitform = PETSC_TRUE; 132 PetscScalar *x_ptr; 133 PetscMPIInt size; 134 struct _n_User user; 135 PetscErrorCode ierr; 136 137 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 138 Initialize program 139 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 140 ierr = PetscInitialize(&argc,&argv,NULL,help);if (ierr) return ierr; 141 ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRMPI(ierr); 142 PetscCheckFalse(size != 1,PETSC_COMM_WORLD,PETSC_ERR_WRONG_MPI_SIZE,"This is a uniprocessor example only!"); 143 144 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 145 Set runtime options 146 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 147 user.next_output = 0.0; 148 user.mu = 1.0e3; 149 ierr = PetscOptionsGetBool(NULL,NULL,"-monitor",&monitor,NULL);CHKERRQ(ierr); 150 ierr = PetscOptionsGetBool(NULL,NULL,"-implicitform",&implicitform,NULL);CHKERRQ(ierr); 151 ierr = PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Physical parameters",NULL);CHKERRQ(ierr); 152 ierr = PetscOptionsReal("-mu","Stiffness parameter","<1.0e6>",user.mu,&user.mu,NULL);CHKERRQ(ierr); 153 ierr = PetscOptionsEnd();CHKERRQ(ierr); 154 155 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 156 Create necessary matrix and vectors, solve same ODE on every process 157 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 158 ierr = MatCreate(PETSC_COMM_WORLD,&A);CHKERRQ(ierr); 159 ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);CHKERRQ(ierr); 160 ierr = MatSetFromOptions(A);CHKERRQ(ierr); 161 ierr = MatSetUp(A);CHKERRQ(ierr); 162 163 ierr = MatCreateVecs(A,&x,NULL);CHKERRQ(ierr); 164 165 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 166 Create timestepping solver context 167 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 168 ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr); 169 if (implicitform) { 170 ierr = TSSetIFunction(ts,NULL,IFunction,&user);CHKERRQ(ierr); 171 ierr = TSSetIJacobian(ts,A,A,IJacobian,&user);CHKERRQ(ierr); 172 ierr = TSSetType(ts,TSBEULER);CHKERRQ(ierr); 173 } else { 174 ierr = TSSetRHSFunction(ts,NULL,RHSFunction,&user);CHKERRQ(ierr); 175 ierr = TSSetType(ts,TSRK);CHKERRQ(ierr); 176 } 177 ierr = TSSetMaxTime(ts,ftime);CHKERRQ(ierr); 178 ierr = TSSetTimeStep(ts,0.001);CHKERRQ(ierr); 179 ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);CHKERRQ(ierr); 180 if (monitor) { 181 ierr = TSMonitorSet(ts,Monitor,&user,NULL);CHKERRQ(ierr); 182 } 183 184 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 185 Set initial conditions 186 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 187 ierr = VecGetArray(x,&x_ptr);CHKERRQ(ierr); 188 x_ptr[0] = 2.0; 189 x_ptr[1] = -2.0/3.0 + 10.0/(81.0*user.mu) - 292.0/(2187.0*user.mu*user.mu); 190 ierr = VecRestoreArray(x,&x_ptr);CHKERRQ(ierr); 191 192 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 193 Set runtime options 194 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 195 ierr = TSSetFromOptions(ts);CHKERRQ(ierr); 196 197 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 198 Solve nonlinear system 199 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 200 ierr = TSSolve(ts,x);CHKERRQ(ierr); 201 ierr = TSGetSolveTime(ts,&ftime);CHKERRQ(ierr); 202 ierr = TSGetStepNumber(ts,&steps);CHKERRQ(ierr); 203 ierr = PetscPrintf(PETSC_COMM_WORLD,"steps %D, ftime %g\n",steps,(double)ftime);CHKERRQ(ierr); 204 ierr = VecView(x,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); 205 206 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 207 Free work space. All PETSc objects should be destroyed when they 208 are no longer needed. 209 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 210 ierr = MatDestroy(&A);CHKERRQ(ierr); 211 ierr = VecDestroy(&x);CHKERRQ(ierr); 212 ierr = TSDestroy(&ts);CHKERRQ(ierr); 213 214 ierr = PetscFinalize(); 215 return(ierr); 216 } 217 218 /*TEST 219 220 test: 221 requires: !single 222 args: -mu 1e6 223 224 test: 225 requires: !single 226 suffix: 2 227 args: -implicitform false -ts_type rk -ts_rk_type 5dp -ts_adapt_type dsp 228 229 test: 230 requires: !single 231 suffix: 3 232 args: -implicitform false -ts_type rk -ts_rk_type 5dp -ts_adapt_type dsp -ts_adapt_dsp_filter H0312 233 234 TEST*/ 235