1 2 static char help[] = "Solves the van der Pol equation and demonstrate IMEX.\n\ 3 Input parameters include:\n\ 4 -mu : stiffness parameter\n\n"; 5 6 /* 7 Concepts: TS^time-dependent nonlinear problems 8 Concepts: TS^van der Pol equation 9 Processors: 1 10 */ 11 /* ------------------------------------------------------------------------ 12 13 This program solves the van der Pol equation 14 y'' - \mu ((1-y^2)*y' - y) = 0 (1) 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 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. 18 19 Notes: 20 This code demonstrates the TS solver interface to two variants of 21 linear problems, u_t = f(u,t), namely turning (1) into a system of 22 first order differential equations, 23 24 [ y' ] = [ z ] 25 [ z' ] [ \mu ((1 - y^2) z - y) ] 26 27 which then we can write as a vector equation 28 29 [ u_1' ] = [ u_2 ] (2) 30 [ u_2' ] [ \mu (1 - u_1^2) u_2 - u_1 ] 31 32 which is now in the desired form of u_t = f(u,t). One way that we 33 can split f(u,t) in (2) is to split by component, 34 35 [ u_1' ] = [ u_2 ] + [ 0 ] 36 [ u_2' ] [ 0 ] [ \mu ((1 - u_1^2) u_2 - u_1) ] 37 38 where 39 40 [ G(u,t) ] = [ u_2 ] 41 [ 0 ] 42 43 and 44 45 [ F(u',u,t) ] = [ u_1' ] - [ 0 ] 46 [ u_2' ] [ \mu ((1 - u_1^2) u_2 - u_1) ] 47 48 Using the definition of the Jacobian of F (from the PETSc user manual), 49 in the equation F(u',u,t) = G(u,t), 50 51 dF dF 52 J(F) = a * -- - -- 53 du' du 54 55 where d is the partial derivative. In this example, 56 57 dF [ 1 ; 0 ] 58 -- = [ ] 59 du' [ 0 ; 1 ] 60 61 dF [ 0 ; 0 ] 62 -- = [ ] 63 du [ -\mu (2*u_1*u_2 + 1); \mu (1 - u_1^2) ] 64 65 Hence, 66 67 [ a ; 0 ] 68 J(F) = [ ] 69 [ \mu (2*u_1*u_2 + 1); a - \mu (1 - u_1^2) ] 70 71 ------------------------------------------------------------------------- */ 72 73 #include <petscts.h> 74 75 typedef struct _n_User *User; 76 struct _n_User { 77 PetscReal mu; 78 PetscBool imex; 79 PetscReal next_output; 80 }; 81 82 /* 83 User-defined routines 84 */ 85 static PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec X,Vec F,void *ctx) 86 { 87 User user = (User)ctx; 88 PetscScalar *f; 89 const PetscScalar *x; 90 91 PetscFunctionBeginUser; 92 CHKERRQ(VecGetArrayRead(X,&x)); 93 CHKERRQ(VecGetArray(F,&f)); 94 f[0] = (user->imex ? x[1] : 0); 95 f[1] = 0.0; 96 CHKERRQ(VecRestoreArrayRead(X,&x)); 97 CHKERRQ(VecRestoreArray(F,&f)); 98 PetscFunctionReturn(0); 99 } 100 101 static PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx) 102 { 103 User user = (User)ctx; 104 const PetscScalar *x,*xdot; 105 PetscScalar *f; 106 107 PetscFunctionBeginUser; 108 CHKERRQ(VecGetArrayRead(X,&x)); 109 CHKERRQ(VecGetArrayRead(Xdot,&xdot)); 110 CHKERRQ(VecGetArray(F,&f)); 111 f[0] = xdot[0] + (user->imex ? 0 : x[1]); 112 f[1] = xdot[1] - user->mu*((1. - x[0]*x[0])*x[1] - x[0]); 113 CHKERRQ(VecRestoreArrayRead(X,&x)); 114 CHKERRQ(VecRestoreArrayRead(Xdot,&xdot)); 115 CHKERRQ(VecRestoreArray(F,&f)); 116 PetscFunctionReturn(0); 117 } 118 119 static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat A,Mat B,void *ctx) 120 { 121 User user = (User)ctx; 122 PetscReal mu = user->mu; 123 PetscInt rowcol[] = {0,1}; 124 const PetscScalar *x; 125 PetscScalar J[2][2]; 126 127 PetscFunctionBeginUser; 128 CHKERRQ(VecGetArrayRead(X,&x)); 129 J[0][0] = a; J[0][1] = (user->imex ? 0 : 1.); 130 J[1][0] = mu*(2.*x[0]*x[1]+1.); J[1][1] = a - mu*(1. - x[0]*x[0]); 131 CHKERRQ(MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES)); 132 CHKERRQ(VecRestoreArrayRead(X,&x)); 133 134 CHKERRQ(MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY)); 135 CHKERRQ(MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY)); 136 if (A != B) { 137 CHKERRQ(MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY)); 138 CHKERRQ(MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY)); 139 } 140 PetscFunctionReturn(0); 141 } 142 143 static PetscErrorCode RegisterMyARK2(void) 144 { 145 PetscFunctionBeginUser; 146 { 147 const PetscReal 148 A[3][3] = {{0,0,0}, 149 {0.41421356237309504880,0,0}, 150 {0.75,0.25,0}}, 151 At[3][3] = {{0,0,0}, 152 {0.12132034355964257320,0.29289321881345247560,0}, 153 {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}}, 154 *bembedt = NULL,*bembed = NULL; 155 CHKERRQ(TSARKIMEXRegister("myark2",2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembed,0,NULL,NULL)); 156 } 157 PetscFunctionReturn(0); 158 } 159 160 /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ 161 static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx) 162 { 163 const PetscScalar *x; 164 PetscReal tfinal, dt; 165 User user = (User)ctx; 166 Vec interpolatedX; 167 168 PetscFunctionBeginUser; 169 CHKERRQ(TSGetTimeStep(ts,&dt)); 170 CHKERRQ(TSGetMaxTime(ts,&tfinal)); 171 172 while (user->next_output <= t && user->next_output <= tfinal) { 173 CHKERRQ(VecDuplicate(X,&interpolatedX)); 174 CHKERRQ(TSInterpolate(ts,user->next_output,interpolatedX)); 175 CHKERRQ(VecGetArrayRead(interpolatedX,&x)); 176 CHKERRQ(PetscPrintf(PETSC_COMM_WORLD,"[%.1f] %D TS %.6f (dt = %.6f) X % 12.6e % 12.6e\n",user->next_output,step,t,dt,(double)PetscRealPart(x[0]),(double)PetscRealPart(x[1]))); 177 CHKERRQ(VecRestoreArrayRead(interpolatedX,&x)); 178 CHKERRQ(VecDestroy(&interpolatedX)); 179 180 user->next_output += 0.1; 181 } 182 PetscFunctionReturn(0); 183 } 184 185 int main(int argc,char **argv) 186 { 187 TS ts; /* nonlinear solver */ 188 Vec x; /* solution, residual vectors */ 189 Mat A; /* Jacobian matrix */ 190 PetscInt steps; 191 PetscReal ftime = 0.5; 192 PetscBool monitor = PETSC_FALSE; 193 PetscScalar *x_ptr; 194 PetscMPIInt size; 195 struct _n_User user; 196 PetscErrorCode ierr; 197 198 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 199 Initialize program 200 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 201 ierr = PetscInitialize(&argc,&argv,NULL,help);if (ierr) return ierr; 202 CHKERRMPI(MPI_Comm_size(PETSC_COMM_WORLD,&size)); 203 PetscCheck(size == 1,PETSC_COMM_WORLD,PETSC_ERR_WRONG_MPI_SIZE,"This is a uniprocessor example only!"); 204 205 CHKERRQ(RegisterMyARK2()); 206 207 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 208 Set runtime options 209 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 210 user.mu = 1000.0; 211 user.imex = PETSC_TRUE; 212 user.next_output = 0.0; 213 214 CHKERRQ(PetscOptionsGetReal(NULL,NULL,"-mu",&user.mu,NULL)); 215 CHKERRQ(PetscOptionsGetBool(NULL,NULL,"-imex",&user.imex,NULL)); 216 CHKERRQ(PetscOptionsGetBool(NULL,NULL,"-monitor",&monitor,NULL)); 217 218 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 219 Create necessary matrix and vectors, solve same ODE on every process 220 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 221 CHKERRQ(MatCreate(PETSC_COMM_WORLD,&A)); 222 CHKERRQ(MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2)); 223 CHKERRQ(MatSetFromOptions(A)); 224 CHKERRQ(MatSetUp(A)); 225 CHKERRQ(MatCreateVecs(A,&x,NULL)); 226 227 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 228 Create timestepping solver context 229 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 230 CHKERRQ(TSCreate(PETSC_COMM_WORLD,&ts)); 231 CHKERRQ(TSSetType(ts,TSBEULER)); 232 CHKERRQ(TSSetRHSFunction(ts,NULL,RHSFunction,&user)); 233 CHKERRQ(TSSetIFunction(ts,NULL,IFunction,&user)); 234 CHKERRQ(TSSetIJacobian(ts,A,A,IJacobian,&user)); 235 CHKERRQ(TSSetMaxTime(ts,ftime)); 236 CHKERRQ(TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER)); 237 if (monitor) { 238 CHKERRQ(TSMonitorSet(ts,Monitor,&user,NULL)); 239 } 240 241 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 242 Set initial conditions 243 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 244 CHKERRQ(VecGetArray(x,&x_ptr)); 245 x_ptr[0] = 2.0; 246 x_ptr[1] = -2.0/3.0 + 10.0/(81.0*user.mu) - 292.0/(2187.0*user.mu*user.mu); 247 CHKERRQ(VecRestoreArray(x,&x_ptr)); 248 CHKERRQ(TSSetTimeStep(ts,0.01)); 249 250 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 251 Set runtime options 252 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 253 CHKERRQ(TSSetFromOptions(ts)); 254 255 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 256 Solve nonlinear system 257 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 258 CHKERRQ(TSSolve(ts,x)); 259 CHKERRQ(TSGetSolveTime(ts,&ftime)); 260 CHKERRQ(TSGetStepNumber(ts,&steps)); 261 CHKERRQ(PetscPrintf(PETSC_COMM_WORLD,"mu %g, steps %D, ftime %g\n",(double)user.mu,steps,(double)ftime)); 262 CHKERRQ(VecView(x,PETSC_VIEWER_STDOUT_WORLD)); 263 264 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 265 Free work space. All PETSc objects should be destroyed when they 266 are no longer needed. 267 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 268 CHKERRQ(MatDestroy(&A)); 269 CHKERRQ(VecDestroy(&x)); 270 CHKERRQ(TSDestroy(&ts)); 271 272 ierr = PetscFinalize(); 273 return ierr; 274 } 275 276 /*TEST 277 278 test: 279 args: -ts_type arkimex -ts_arkimex_type myark2 -ts_adapt_type none 280 requires: !single 281 282 TEST*/ 283