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 55 static PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx) 56 { 57 PetscErrorCode ierr; 58 User user = (User)ctx; 59 const PetscScalar *x,*xdot; 60 PetscScalar *f; 61 62 PetscFunctionBeginUser; 63 ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); 64 ierr = VecGetArrayRead(Xdot,&xdot);CHKERRQ(ierr); 65 ierr = VecGetArray(F,&f);CHKERRQ(ierr); 66 f[0] = xdot[0] - x[1]; 67 f[1] = xdot[1] - user->mu*((1.0-x[0]*x[0])*x[1] - x[0]); 68 ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); 69 ierr = VecRestoreArrayRead(Xdot,&xdot);CHKERRQ(ierr); 70 ierr = VecRestoreArray(F,&f);CHKERRQ(ierr); 71 PetscFunctionReturn(0); 72 } 73 74 static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat A,Mat B,void *ctx) 75 { 76 PetscErrorCode ierr; 77 User user = (User)ctx; 78 PetscInt rowcol[] = {0,1}; 79 const PetscScalar *x; 80 PetscScalar J[2][2]; 81 82 PetscFunctionBeginUser; 83 ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); 84 J[0][0] = a; J[0][1] = -1.0; 85 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]); 86 ierr = MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);CHKERRQ(ierr); 87 ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); 88 89 ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 90 ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 91 if (A != B) { 92 ierr = MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 93 ierr = MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 94 } 95 PetscFunctionReturn(0); 96 } 97 98 /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ 99 static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx) 100 { 101 PetscErrorCode ierr; 102 const PetscScalar *x; 103 PetscReal tfinal, dt; 104 User user = (User)ctx; 105 Vec interpolatedX; 106 107 PetscFunctionBeginUser; 108 ierr = TSGetTimeStep(ts,&dt);CHKERRQ(ierr); 109 ierr = TSGetMaxTime(ts,&tfinal);CHKERRQ(ierr); 110 111 while (user->next_output <= t && user->next_output <= tfinal) { 112 ierr = VecDuplicate(X,&interpolatedX);CHKERRQ(ierr); 113 ierr = TSInterpolate(ts,user->next_output,interpolatedX);CHKERRQ(ierr); 114 ierr = VecGetArrayRead(interpolatedX,&x);CHKERRQ(ierr); 115 ierr = PetscPrintf(PETSC_COMM_WORLD,"[%.1f] %D TS %.6f (dt = %.6f) X % 12.6e % 12.6e\n", 116 user->next_output,step,t,dt,(double)PetscRealPart(x[0]), 117 (double)PetscRealPart(x[1]));CHKERRQ(ierr); 118 ierr = VecRestoreArrayRead(interpolatedX,&x);CHKERRQ(ierr); 119 ierr = VecDestroy(&interpolatedX);CHKERRQ(ierr); 120 user->next_output += 0.1; 121 } 122 PetscFunctionReturn(0); 123 } 124 125 int main(int argc,char **argv) 126 { 127 TS ts; /* nonlinear solver */ 128 Vec x; /* solution, residual vectors */ 129 Mat A; /* Jacobian matrix */ 130 PetscInt steps; 131 PetscReal ftime = 0.5; 132 PetscBool monitor = PETSC_FALSE,implicitform = PETSC_TRUE; 133 PetscScalar *x_ptr; 134 PetscMPIInt size; 135 struct _n_User user; 136 PetscErrorCode ierr; 137 138 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 139 Initialize program 140 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 141 ierr = PetscInitialize(&argc,&argv,NULL,help);if (ierr) return ierr; 142 ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRQ(ierr); 143 if (size != 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_WRONG_MPI_SIZE,"This is a uniprocessor example only!"); 144 145 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 146 Set runtime options 147 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 148 user.next_output = 0.0; 149 user.mu = 1.0e3; 150 ierr = PetscOptionsGetBool(NULL,NULL,"-monitor",&monitor,NULL);CHKERRQ(ierr); 151 ierr = PetscOptionsGetBool(NULL,NULL,"-implicitform",&implicitform,NULL);CHKERRQ(ierr); 152 ierr = PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Physical parameters",NULL);CHKERRQ(ierr); 153 ierr = PetscOptionsReal("-mu","Stiffness parameter","<1.0e6>",user.mu,&user.mu,NULL);CHKERRQ(ierr); 154 ierr = PetscOptionsEnd();CHKERRQ(ierr); 155 156 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 157 Create necessary matrix and vectors, solve same ODE on every process 158 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 159 ierr = MatCreate(PETSC_COMM_WORLD,&A);CHKERRQ(ierr); 160 ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);CHKERRQ(ierr); 161 ierr = MatSetFromOptions(A);CHKERRQ(ierr); 162 ierr = MatSetUp(A);CHKERRQ(ierr); 163 164 ierr = MatCreateVecs(A,&x,NULL);CHKERRQ(ierr); 165 166 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 167 Create timestepping solver context 168 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 169 ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr); 170 if (implicitform) { 171 ierr = TSSetIFunction(ts,NULL,IFunction,&user);CHKERRQ(ierr); 172 ierr = TSSetIJacobian(ts,A,A,IJacobian,&user);CHKERRQ(ierr); 173 ierr = TSSetType(ts,TSBEULER);CHKERRQ(ierr); 174 } else { 175 ierr = TSSetRHSFunction(ts,NULL,RHSFunction,&user);CHKERRQ(ierr); 176 ierr = TSSetType(ts,TSRK);CHKERRQ(ierr); 177 } 178 ierr = TSSetMaxTime(ts,ftime);CHKERRQ(ierr); 179 ierr = TSSetTimeStep(ts,0.001);CHKERRQ(ierr); 180 ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);CHKERRQ(ierr); 181 if (monitor) { 182 ierr = TSMonitorSet(ts,Monitor,&user,NULL);CHKERRQ(ierr); 183 } 184 185 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 186 Set initial conditions 187 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 188 ierr = VecGetArray(x,&x_ptr);CHKERRQ(ierr); 189 x_ptr[0] = 2.0; 190 x_ptr[1] = -2.0/3.0 + 10.0/(81.0*user.mu) - 292.0/(2187.0*user.mu*user.mu); 191 ierr = VecRestoreArray(x,&x_ptr);CHKERRQ(ierr); 192 193 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 194 Set runtime options 195 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 196 ierr = TSSetFromOptions(ts);CHKERRQ(ierr); 197 198 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 199 Solve nonlinear system 200 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 201 ierr = TSSolve(ts,x);CHKERRQ(ierr); 202 ierr = TSGetSolveTime(ts,&ftime);CHKERRQ(ierr); 203 ierr = TSGetStepNumber(ts,&steps);CHKERRQ(ierr); 204 ierr = PetscPrintf(PETSC_COMM_WORLD,"steps %D, ftime %g\n",steps,(double)ftime);CHKERRQ(ierr); 205 ierr = VecView(x,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); 206 207 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 208 Free work space. All PETSc objects should be destroyed when they 209 are no longer needed. 210 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 211 ierr = MatDestroy(&A);CHKERRQ(ierr); 212 ierr = VecDestroy(&x);CHKERRQ(ierr); 213 ierr = TSDestroy(&ts);CHKERRQ(ierr); 214 215 ierr = PetscFinalize(); 216 return(ierr); 217 } 218 219 /*TEST 220 221 test: 222 requires: !single 223 args: -mu 1e6 224 225 TEST*/ 226