static char help[] = "Solves the van der Pol equation.\n\ Input parameters include:\n"; /* Concepts: TS^time-dependent nonlinear problems Concepts: TS^van der Pol equation DAE equivalent Processors: 1 */ /* ------------------------------------------------------------------------ This program solves the van der Pol DAE ODE equivalent y' = z (1) z' = mu[(1-y^2)z-y] on the domain 0 <= x <= 1, with the boundary conditions y(0) = 2, y'(0) = -6.666665432100101e-01, and mu = 10^6. This is a nonlinear equation. Notes: This code demonstrates the TS solver interface to a variant of linear problems, u_t = f(u,t), namely turning (1) into a system of first order differential equations, [ y' ] = [ z ] [ z' ] [ mu[(1-y^2)z-y] ] which then we can write as a vector equation [ u_1' ] = [ u_2 ] (2) [ u_2' ] [ mu[(1-u_1^2)u_2-u_1] ] which is now in the desired form of u_t = f(u,t). One way that we can split f(u,t) in (2) is to split by component, [ u_1' ] = [ u_2 ] + [ 0 ] [ u_2' ] [ 0 ] [ mu[(1-u_1^2)u_2-u_1] ] where [ F(u,t) ] = [ u_2 ] [ 0 ] and [ G(u',u,t) ] = [ u_1' ] - [ 0 ] [ u_2' ] [ mu[(1-u_1^2)u_2-u_1] ] Using the definition of the Jacobian of G (from the PETSc user manual), in the equation G(u',u,t) = F(u,t), dG dG J(G) = a * -- - -- du' du where d is the partial derivative. In this example, dG [ 1 ; 0 ] -- = [ ] du' [ 0 ; 1 ] dG [ 0 ; 0 ] -- = [ ] du [ -mu*(1.0 + 2.0*u_1*u_2) ; mu*(1-u_1*u_1) ] Hence, [ a ; 0 ] J(G) = [ ] [ mu*(1.0 + 2.0*u_1*u_2) ; a - mu*(1-u_1*u_1) ] ------------------------------------------------------------------------- */ #include typedef struct _n_User *User; struct _n_User { PetscReal mu; PetscBool imex; PetscReal next_output; }; /* * User-defined routines */ #undef __FUNCT__ #define __FUNCT__ "RHSFunction" static PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec X,Vec F,void *ctx) { PetscErrorCode ierr; User user = (User)ctx; PetscScalar *f; const PetscScalar *x; PetscFunctionBeginUser; ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); ierr = VecGetArray(F,&f);CHKERRQ(ierr); f[0] = (user->imex ? x[1] : 0.0); f[1] = 0.0; ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); ierr = VecRestoreArray(F,&f);CHKERRQ(ierr); PetscFunctionReturn(0); } #undef __FUNCT__ #define __FUNCT__ "IFunction" static PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx) { PetscErrorCode ierr; User user = (User)ctx; const PetscScalar *x,*xdot; PetscScalar *f; PetscFunctionBeginUser; ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); ierr = VecGetArrayRead(Xdot,&xdot);CHKERRQ(ierr); ierr = VecGetArray(F,&f);CHKERRQ(ierr); f[0] = xdot[0] - (user->imex ? 0 : x[1]); f[1] = xdot[1] - user->mu*((1.0-x[0]*x[0])*x[1] - x[0]); ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); ierr = VecRestoreArrayRead(Xdot,&xdot);CHKERRQ(ierr); ierr = VecRestoreArray(F,&f);CHKERRQ(ierr); PetscFunctionReturn(0); } #undef __FUNCT__ #define __FUNCT__ "IJacobian" static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat A,Mat B,void *ctx) { PetscErrorCode ierr; User user = (User)ctx; PetscInt rowcol[] = {0,1}; const PetscScalar *x; PetscScalar J[2][2]; PetscFunctionBeginUser; ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); J[0][0] = a; J[0][1] = (user->imex ? 0 : -1.0); J[1][0] = user->mu*(1.0 + 2.0*x[0]*x[1]); J[1][1] = a - user->mu*(1.0-x[0]*x[0]); ierr = MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);CHKERRQ(ierr); ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); if (A != B) { ierr = MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); ierr = MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); } PetscFunctionReturn(0); } #undef __FUNCT__ #define __FUNCT__ "RegisterMyARK2" /* This is an example of registering an user-provided ARKIMEX scheme */ static PetscErrorCode RegisterMyARK2(void) { PetscErrorCode ierr; PetscFunctionBeginUser; { const PetscReal A[3][3] = {{0,0,0}, {0.41421356237309504880,0,0}, {0.75,0.25,0}}, At[3][3] = {{0,0,0}, {0.12132034355964257320,0.29289321881345247560,0}, {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}}; ierr = TSARKIMEXRegister("myark2",2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,NULL,NULL,0,NULL,NULL);CHKERRQ(ierr); } PetscFunctionReturn(0); } #undef __FUNCT__ #define __FUNCT__ "Monitor" /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx) { PetscErrorCode ierr; const PetscScalar *x; PetscReal tfinal, dt; User user = (User)ctx; Vec interpolatedX; PetscFunctionBeginUser; ierr = TSGetTimeStep(ts,&dt);CHKERRQ(ierr); ierr = TSGetDuration(ts,NULL,&tfinal);CHKERRQ(ierr); while (user->next_output <= t && user->next_output <= tfinal) { ierr = VecDuplicate(X,&interpolatedX);CHKERRQ(ierr); ierr = TSInterpolate(ts,user->next_output,interpolatedX);CHKERRQ(ierr); ierr = VecGetArrayRead(interpolatedX,&x);CHKERRQ(ierr); 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); ierr = VecRestoreArrayRead(interpolatedX,&x);CHKERRQ(ierr); ierr = VecDestroy(&interpolatedX);CHKERRQ(ierr); user->next_output += 0.1; } PetscFunctionReturn(0); } #undef __FUNCT__ #define __FUNCT__ "main" int main(int argc,char **argv) { TS ts; /* nonlinear solver */ Vec x; /* solution, residual vectors */ Mat A; /* Jacobian matrix */ PetscInt steps; PetscReal ftime = 0.5; PetscBool monitor = PETSC_FALSE; PetscScalar *x_ptr; PetscMPIInt size; struct _n_User user; PetscErrorCode ierr; /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Initialize program - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscInitialize(&argc,&argv,NULL,help); ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRQ(ierr); if (size != 1) SETERRQ(PETSC_COMM_SELF,1,"This is a uniprocessor example only!"); /* Register user-specified ARKIMEX method */ ierr = RegisterMyARK2();CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set runtime options - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ user.imex = PETSC_TRUE; user.next_output = 0.0; user.mu = 1.0e6; ierr = PetscOptionsGetBool(NULL,NULL,"-imex",&user.imex,NULL);CHKERRQ(ierr); ierr = PetscOptionsGetBool(NULL,NULL,"-monitor",&monitor,NULL);CHKERRQ(ierr); ierr = PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Physical parameters",NULL); ierr = PetscOptionsReal("-mu","Stiffness parameter","<1.0e6>",user.mu,&user.mu,PETSC_NULL); ierr = PetscOptionsEnd(); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create necessary matrix and vectors, solve same ODE on every process - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = MatCreate(PETSC_COMM_WORLD,&A);CHKERRQ(ierr); ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);CHKERRQ(ierr); ierr = MatSetFromOptions(A);CHKERRQ(ierr); ierr = MatSetUp(A);CHKERRQ(ierr); ierr = MatCreateVecs(A,&x,NULL);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create timestepping solver context - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr); ierr = TSSetType(ts,TSBEULER);CHKERRQ(ierr); ierr = TSSetRHSFunction(ts,NULL,RHSFunction,&user);CHKERRQ(ierr); ierr = TSSetIFunction(ts,NULL,IFunction,&user);CHKERRQ(ierr); ierr = TSSetIJacobian(ts,A,A,IJacobian,&user);CHKERRQ(ierr); ierr = TSSetDuration(ts,PETSC_DEFAULT,ftime);CHKERRQ(ierr); ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);CHKERRQ(ierr); if (monitor) { ierr = TSMonitorSet(ts,Monitor,&user,NULL);CHKERRQ(ierr); } /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set initial conditions - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = VecGetArray(x,&x_ptr);CHKERRQ(ierr); x_ptr[0] = 2.0; x_ptr[1] = -6.666665432100101e-01; ierr = VecRestoreArray(x,&x_ptr);CHKERRQ(ierr); ierr = TSSetInitialTimeStep(ts,0.0,.001);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set runtime options - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSSetFromOptions(ts);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Solve nonlinear system - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSSolve(ts,x);CHKERRQ(ierr); ierr = TSGetSolveTime(ts,&ftime);CHKERRQ(ierr); ierr = TSGetTimeStepNumber(ts,&steps);CHKERRQ(ierr); ierr = PetscPrintf(PETSC_COMM_WORLD,"steps %D, ftime %g\n",steps,(double)ftime);CHKERRQ(ierr); ierr = VecView(x,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Free work space. All PETSc objects should be destroyed when they are no longer needed. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = MatDestroy(&A);CHKERRQ(ierr); ierr = VecDestroy(&x);CHKERRQ(ierr); ierr = TSDestroy(&ts);CHKERRQ(ierr); ierr = PetscFinalize(); PetscFunctionReturn(0); }