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 PetscErrorCode ierr; 88 User user = (User)ctx; 89 PetscScalar *f; 90 const PetscScalar *x; 91 92 PetscFunctionBeginUser; 93 ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); 94 ierr = VecGetArray(F,&f);CHKERRQ(ierr); 95 f[0] = (user->imex ? x[1] : 0); 96 f[1] = 0.0; 97 ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); 98 ierr = VecRestoreArray(F,&f);CHKERRQ(ierr); 99 PetscFunctionReturn(0); 100 } 101 102 static PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx) 103 { 104 PetscErrorCode ierr; 105 User user = (User)ctx; 106 const PetscScalar *x,*xdot; 107 PetscScalar *f; 108 109 PetscFunctionBeginUser; 110 ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); 111 ierr = VecGetArrayRead(Xdot,&xdot);CHKERRQ(ierr); 112 ierr = VecGetArray(F,&f);CHKERRQ(ierr); 113 f[0] = xdot[0] + (user->imex ? 0 : x[1]); 114 f[1] = xdot[1] - user->mu*((1. - x[0]*x[0])*x[1] - x[0]); 115 ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); 116 ierr = VecRestoreArrayRead(Xdot,&xdot);CHKERRQ(ierr); 117 ierr = VecRestoreArray(F,&f);CHKERRQ(ierr); 118 PetscFunctionReturn(0); 119 } 120 121 static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat A,Mat B,void *ctx) 122 { 123 PetscErrorCode ierr; 124 User user = (User)ctx; 125 PetscReal mu = user->mu; 126 PetscInt rowcol[] = {0,1}; 127 const PetscScalar *x; 128 PetscScalar J[2][2]; 129 130 PetscFunctionBeginUser; 131 ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); 132 J[0][0] = a; J[0][1] = (user->imex ? 0 : 1.); 133 J[1][0] = mu*(2.*x[0]*x[1]+1.); J[1][1] = a - mu*(1. - x[0]*x[0]); 134 ierr = MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);CHKERRQ(ierr); 135 ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); 136 137 ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 138 ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 139 if (A != B) { 140 ierr = MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 141 ierr = MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 142 } 143 PetscFunctionReturn(0); 144 } 145 146 static PetscErrorCode RegisterMyARK2(void) 147 { 148 PetscErrorCode ierr; 149 150 PetscFunctionBeginUser; 151 { 152 const PetscReal 153 A[3][3] = {{0,0,0}, 154 {0.41421356237309504880,0,0}, 155 {0.75,0.25,0}}, 156 At[3][3] = {{0,0,0}, 157 {0.12132034355964257320,0.29289321881345247560,0}, 158 {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}}, 159 *bembedt = NULL,*bembed = NULL; 160 ierr = TSARKIMEXRegister("myark2",2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembed,0,NULL,NULL);CHKERRQ(ierr); 161 } 162 PetscFunctionReturn(0); 163 } 164 165 /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ 166 static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx) 167 { 168 PetscErrorCode ierr; 169 const PetscScalar *x; 170 PetscReal tfinal, dt; 171 User user = (User)ctx; 172 Vec interpolatedX; 173 174 PetscFunctionBeginUser; 175 ierr = TSGetTimeStep(ts,&dt);CHKERRQ(ierr); 176 ierr = TSGetMaxTime(ts,&tfinal);CHKERRQ(ierr); 177 178 while (user->next_output <= t && user->next_output <= tfinal) { 179 ierr = VecDuplicate(X,&interpolatedX);CHKERRQ(ierr); 180 ierr = TSInterpolate(ts,user->next_output,interpolatedX);CHKERRQ(ierr); 181 ierr = VecGetArrayRead(interpolatedX,&x);CHKERRQ(ierr); 182 ierr = 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]));CHKERRQ(ierr); 183 ierr = VecRestoreArrayRead(interpolatedX,&x);CHKERRQ(ierr); 184 ierr = VecDestroy(&interpolatedX);CHKERRQ(ierr); 185 186 user->next_output += 0.1; 187 } 188 PetscFunctionReturn(0); 189 } 190 191 int main(int argc,char **argv) 192 { 193 TS ts; /* nonlinear solver */ 194 Vec x; /* solution, residual vectors */ 195 Mat A; /* Jacobian matrix */ 196 PetscInt steps; 197 PetscReal ftime = 0.5; 198 PetscBool monitor = PETSC_FALSE; 199 PetscScalar *x_ptr; 200 PetscMPIInt size; 201 struct _n_User user; 202 PetscErrorCode ierr; 203 204 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 205 Initialize program 206 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 207 ierr = PetscInitialize(&argc,&argv,NULL,help);if (ierr) return ierr; 208 ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRQ(ierr); 209 if (size != 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_WRONG_MPI_SIZE,"This is a uniprocessor example only!"); 210 211 ierr = RegisterMyARK2();CHKERRQ(ierr); 212 213 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 214 Set runtime options 215 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 216 user.mu = 1000.0; 217 user.imex = PETSC_TRUE; 218 user.next_output = 0.0; 219 220 ierr = PetscOptionsGetReal(NULL,NULL,"-mu",&user.mu,NULL);CHKERRQ(ierr); 221 ierr = PetscOptionsGetBool(NULL,NULL,"-imex",&user.imex,NULL);CHKERRQ(ierr); 222 ierr = PetscOptionsGetBool(NULL,NULL,"-monitor",&monitor,NULL);CHKERRQ(ierr); 223 224 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 225 Create necessary matrix and vectors, solve same ODE on every process 226 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 227 ierr = MatCreate(PETSC_COMM_WORLD,&A);CHKERRQ(ierr); 228 ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);CHKERRQ(ierr); 229 ierr = MatSetFromOptions(A);CHKERRQ(ierr); 230 ierr = MatSetUp(A);CHKERRQ(ierr); 231 ierr = MatCreateVecs(A,&x,NULL);CHKERRQ(ierr); 232 233 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 234 Create timestepping solver context 235 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 236 ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr); 237 ierr = TSSetType(ts,TSBEULER);CHKERRQ(ierr); 238 ierr = TSSetRHSFunction(ts,NULL,RHSFunction,&user);CHKERRQ(ierr); 239 ierr = TSSetIFunction(ts,NULL,IFunction,&user);CHKERRQ(ierr); 240 ierr = TSSetIJacobian(ts,A,A,IJacobian,&user);CHKERRQ(ierr); 241 ierr = TSSetMaxTime(ts,ftime);CHKERRQ(ierr); 242 ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);CHKERRQ(ierr); 243 if (monitor) { 244 ierr = TSMonitorSet(ts,Monitor,&user,NULL);CHKERRQ(ierr); 245 } 246 247 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 248 Set initial conditions 249 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 250 ierr = VecGetArray(x,&x_ptr);CHKERRQ(ierr); 251 x_ptr[0] = 2.0; 252 x_ptr[1] = -2.0/3.0 + 10.0/(81.0*user.mu) - 292.0/(2187.0*user.mu*user.mu); 253 ierr = VecRestoreArray(x,&x_ptr);CHKERRQ(ierr); 254 ierr = TSSetTimeStep(ts,0.01);CHKERRQ(ierr); 255 256 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 257 Set runtime options 258 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 259 ierr = TSSetFromOptions(ts);CHKERRQ(ierr); 260 261 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 262 Solve nonlinear system 263 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 264 ierr = TSSolve(ts,x);CHKERRQ(ierr); 265 ierr = TSGetSolveTime(ts,&ftime);CHKERRQ(ierr); 266 ierr = TSGetStepNumber(ts,&steps);CHKERRQ(ierr); 267 ierr = PetscPrintf(PETSC_COMM_WORLD,"mu %g, steps %D, ftime %g\n",(double)user.mu,steps,(double)ftime);CHKERRQ(ierr); 268 ierr = VecView(x,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); 269 270 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 271 Free work space. All PETSc objects should be destroyed when they 272 are no longer needed. 273 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 274 ierr = MatDestroy(&A);CHKERRQ(ierr); 275 ierr = VecDestroy(&x);CHKERRQ(ierr); 276 ierr = TSDestroy(&ts);CHKERRQ(ierr); 277 278 ierr = PetscFinalize(); 279 return ierr; 280 } 281 282 /*TEST 283 284 test: 285 args: -ts_type arkimex -ts_arkimex_type myark2 -ts_adapt_type none 286 requires: !single 287 288 TEST*/ 289