static const char help[] = "Time-dependent advection-reaction PDE in 1d, demonstrates IMEX methods.\n"; /* u_t + a1*u_x = -k1*u + k2*v + s1 v_t + a2*v_x = k1*u - k2*v + s2 0 < x < 1; a1 = 1, k1 = 10^6, s1 = 0, a2 = 0, k2 = 2*k1, s2 = 1 Initial conditions: u(x,0) = 1 + s2*x v(x,0) = k0/k1*u(x,0) + s1/k1 Upstream boundary conditions: u(0,t) = 1-sin(12*t)^4 Useful command-line parameters: -ts_arkimex_fully_implicit - treats advection implicitly, only relevant with -ts_type arkimex (default) -ts_type rosw - use Rosenbrock-W method (linearly implicit IMEX, amortizes assembly and preconditioner setup) */ #include #include #include typedef PetscScalar Field[2]; typedef struct _User *User; struct _User { PetscReal a[2]; /* Advection speeds */ PetscReal k[2]; /* Reaction coefficients */ PetscReal s[2]; /* Source terms */ }; static PetscErrorCode FormRHSFunction(TS,PetscReal,Vec,Vec,void*); static PetscErrorCode FormIFunction(TS,PetscReal,Vec,Vec,Vec,void*); static PetscErrorCode FormIJacobian(TS,PetscReal,Vec,Vec,PetscReal,Mat,Mat,void*); static PetscErrorCode FormInitialSolution(TS,Vec,void*); int main(int argc,char **argv) { TS ts; /* time integrator */ SNES snes; /* nonlinear solver */ SNESLineSearch linesearch; /* line search */ Vec X; /* solution, residual vectors */ Mat J; /* Jacobian matrix */ PetscInt steps,mx; PetscErrorCode ierr; DM da; PetscReal ftime,dt; struct _User user; /* user-defined work context */ TSConvergedReason reason; ierr = PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr; /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create distributed array (DMDA) to manage parallel grid and vectors - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,11,2,2,NULL,&da);CHKERRQ(ierr); ierr = DMSetFromOptions(da);CHKERRQ(ierr); ierr = DMSetUp(da);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Extract global vectors from DMDA; - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = DMCreateGlobalVector(da,&X);CHKERRQ(ierr); /* Initialize user application context */ ierr = PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Advection-reaction options",""); { user.a[0] = 1; ierr = PetscOptionsReal("-a0","Advection rate 0","",user.a[0],&user.a[0],NULL);CHKERRQ(ierr); user.a[1] = 0; ierr = PetscOptionsReal("-a1","Advection rate 1","",user.a[1],&user.a[1],NULL);CHKERRQ(ierr); user.k[0] = 1e6; ierr = PetscOptionsReal("-k0","Reaction rate 0","",user.k[0],&user.k[0],NULL);CHKERRQ(ierr); user.k[1] = 2*user.k[0]; ierr = PetscOptionsReal("-k1","Reaction rate 1","",user.k[1],&user.k[1],NULL);CHKERRQ(ierr); user.s[0] = 0; ierr = PetscOptionsReal("-s0","Source 0","",user.s[0],&user.s[0],NULL);CHKERRQ(ierr); user.s[1] = 1; ierr = PetscOptionsReal("-s1","Source 1","",user.s[1],&user.s[1],NULL);CHKERRQ(ierr); } ierr = PetscOptionsEnd();CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create timestepping solver context - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr); ierr = TSSetDM(ts,da);CHKERRQ(ierr); ierr = TSSetType(ts,TSARKIMEX);CHKERRQ(ierr); ierr = TSSetRHSFunction(ts,NULL,FormRHSFunction,&user);CHKERRQ(ierr); ierr = TSSetIFunction(ts,NULL,FormIFunction,&user);CHKERRQ(ierr); ierr = DMSetMatType(da,MATAIJ);CHKERRQ(ierr); ierr = DMCreateMatrix(da,&J);CHKERRQ(ierr); ierr = TSSetIJacobian(ts,J,J,FormIJacobian,&user);CHKERRQ(ierr); /* A line search in the nonlinear solve can fail due to ill-conditioning unless an absolute tolerance is set. Since * this problem is linear, we deactivate the line search. For a linear problem, it is usually recommended to also use * SNESSetType(snes,SNESKSPONLY). */ ierr = TSGetSNES(ts,&snes);CHKERRQ(ierr); ierr = SNESGetLineSearch(snes,&linesearch);CHKERRQ(ierr); ierr = SNESLineSearchSetType(linesearch,SNESLINESEARCHBASIC);CHKERRQ(ierr); ftime = .1; ierr = TSSetMaxTime(ts,ftime);CHKERRQ(ierr); ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set initial conditions - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = FormInitialSolution(ts,X,&user);CHKERRQ(ierr); ierr = TSSetSolution(ts,X);CHKERRQ(ierr); ierr = VecGetSize(X,&mx);CHKERRQ(ierr); dt = .1 * PetscMax(user.a[0],user.a[1]) / mx; /* Advective CFL, I don't know why it needs so much safety factor. */ ierr = TSSetTimeStep(ts,dt);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 = TSGetStepNumber(ts,&steps);CHKERRQ(ierr); ierr = TSGetConvergedReason(ts,&reason);CHKERRQ(ierr); ierr = PetscPrintf(PETSC_COMM_WORLD,"%s at time %g after %D steps\n",TSConvergedReasons[reason],(double)ftime,steps);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Free work space. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = MatDestroy(&J);CHKERRQ(ierr); ierr = VecDestroy(&X);CHKERRQ(ierr); ierr = TSDestroy(&ts);CHKERRQ(ierr); ierr = DMDestroy(&da);CHKERRQ(ierr); ierr = PetscFinalize(); return ierr; } static PetscErrorCode FormIFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ptr) { User user = (User)ptr; DM da; DMDALocalInfo info; PetscInt i; Field *f; const Field *x,*xdot; PetscErrorCode ierr; PetscFunctionBeginUser; ierr = TSGetDM(ts,&da);CHKERRQ(ierr); ierr = DMDAGetLocalInfo(da,&info);CHKERRQ(ierr); /* Get pointers to vector data */ ierr = DMDAVecGetArrayRead(da,X,(void*)&x);CHKERRQ(ierr); ierr = DMDAVecGetArrayRead(da,Xdot,(void*)&xdot);CHKERRQ(ierr); ierr = DMDAVecGetArray(da,F,&f);CHKERRQ(ierr); /* Compute function over the locally owned part of the grid */ for (i=info.xs; ik[0]*x[i][0] - user->k[1]*x[i][1] - user->s[0]; f[i][1] = xdot[i][1] - user->k[0]*x[i][0] + user->k[1]*x[i][1] - user->s[1]; } /* Restore vectors */ ierr = DMDAVecRestoreArrayRead(da,X,(void*)&x);CHKERRQ(ierr); ierr = DMDAVecRestoreArrayRead(da,Xdot,(void*)&xdot);CHKERRQ(ierr); ierr = DMDAVecRestoreArray(da,F,&f);CHKERRQ(ierr); PetscFunctionReturn(0); } static PetscErrorCode FormRHSFunction(TS ts,PetscReal t,Vec X,Vec F,void *ptr) { User user = (User)ptr; DM da; Vec Xloc; DMDALocalInfo info; PetscInt i,j; PetscReal hx; Field *f; const Field *x; PetscErrorCode ierr; PetscFunctionBeginUser; ierr = TSGetDM(ts,&da);CHKERRQ(ierr); ierr = DMDAGetLocalInfo(da,&info);CHKERRQ(ierr); hx = 1.0/(PetscReal)info.mx; /* Scatter ghost points to local vector,using the 2-step process DMGlobalToLocalBegin(),DMGlobalToLocalEnd(). By placing code between these two statements, computations can be done while messages are in transition. */ ierr = DMGetLocalVector(da,&Xloc);CHKERRQ(ierr); ierr = DMGlobalToLocalBegin(da,X,INSERT_VALUES,Xloc);CHKERRQ(ierr); ierr = DMGlobalToLocalEnd(da,X,INSERT_VALUES,Xloc);CHKERRQ(ierr); /* Get pointers to vector data */ ierr = DMDAVecGetArrayRead(da,Xloc,(void*)&x);CHKERRQ(ierr); ierr = DMDAVecGetArray(da,F,&f);CHKERRQ(ierr); /* Compute function over the locally owned part of the grid */ for (i=info.xs; ia; PetscReal u0t[2]; u0t[0] = 1.0 - PetscPowRealInt(PetscSinReal(12*t),4); u0t[1] = 0.0; for (j=0; j<2; j++) { if (i == 0) f[i][j] = a[j]/hx*(1./3*u0t[j] + 0.5*x[i][j] - x[i+1][j] + 1./6*x[i+2][j]); else if (i == 1) f[i][j] = a[j]/hx*(-1./12*u0t[j] + 2./3*x[i-1][j] - 2./3*x[i+1][j] + 1./12*x[i+2][j]); else if (i == info.mx-2) f[i][j] = a[j]/hx*(-1./6*x[i-2][j] + x[i-1][j] - 0.5*x[i][j] - 1./3*x[i+1][j]); else if (i == info.mx-1) f[i][j] = a[j]/hx*(-x[i][j] + x[i-1][j]); else f[i][j] = a[j]/hx*(-1./12*x[i-2][j] + 2./3*x[i-1][j] - 2./3*x[i+1][j] + 1./12*x[i+2][j]); } } /* Restore vectors */ ierr = DMDAVecRestoreArrayRead(da,Xloc,(void*)&x);CHKERRQ(ierr); ierr = DMDAVecRestoreArray(da,F,&f);CHKERRQ(ierr); ierr = DMRestoreLocalVector(da,&Xloc);CHKERRQ(ierr); PetscFunctionReturn(0); } /* --------------------------------------------------------------------- */ /* IJacobian - Compute IJacobian = dF/dU + a dF/dUdot */ PetscErrorCode FormIJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat J,Mat Jpre,void *ptr) { User user = (User)ptr; PetscErrorCode ierr; DMDALocalInfo info; PetscInt i; DM da; const Field *x,*xdot; PetscFunctionBeginUser; ierr = TSGetDM(ts,&da);CHKERRQ(ierr); ierr = DMDAGetLocalInfo(da,&info);CHKERRQ(ierr); /* Get pointers to vector data */ ierr = DMDAVecGetArrayRead(da,X,(void*)&x);CHKERRQ(ierr); ierr = DMDAVecGetArrayRead(da,Xdot,(void*)&xdot);CHKERRQ(ierr); /* Compute function over the locally owned part of the grid */ for (i=info.xs; ik; PetscScalar v[2][2]; v[0][0] = a + k[0]; v[0][1] = -k[1]; v[1][0] = -k[0]; v[1][1] = a+k[1]; ierr = MatSetValuesBlocked(Jpre,1,&i,1,&i,&v[0][0],INSERT_VALUES);CHKERRQ(ierr); } /* Restore vectors */ ierr = DMDAVecRestoreArrayRead(da,X,(void*)&x);CHKERRQ(ierr); ierr = DMDAVecRestoreArrayRead(da,Xdot,(void*)&xdot);CHKERRQ(ierr); ierr = MatAssemblyBegin(Jpre,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); ierr = MatAssemblyEnd(Jpre,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); if (J != Jpre) { ierr = MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); ierr = MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); } PetscFunctionReturn(0); } PetscErrorCode FormInitialSolution(TS ts,Vec X,void *ctx) { User user = (User)ctx; DM da; PetscInt i; DMDALocalInfo info; Field *x; PetscReal hx; PetscErrorCode ierr; PetscFunctionBeginUser; ierr = TSGetDM(ts,&da);CHKERRQ(ierr); ierr = DMDAGetLocalInfo(da,&info);CHKERRQ(ierr); hx = 1.0/(PetscReal)info.mx; /* Get pointers to vector data */ ierr = DMDAVecGetArray(da,X,&x);CHKERRQ(ierr); /* Compute function over the locally owned part of the grid */ for (i=info.xs; ik[1] != 0.0 ? 1.0/user->k[1] : 1.0; x[i][0] = 1 + user->s[1]*r; x[i][1] = user->k[0]*ik*x[i][0] + user->s[1]*ik; } ierr = DMDAVecRestoreArray(da,X,&x);CHKERRQ(ierr); PetscFunctionReturn(0); } /*TEST test: args: -nox -da_grid_x 200 -ts_monitor_draw_solution -ts_arkimex_type 4 -ts_adapt_type none -ts_dt .005 -ts_max_time .1 requires: !single test: suffix: 2 args: -nox -da_grid_x 200 -ts_monitor_draw_solution -ts_type rosw -ts_dt 1e-3 -ts_adapt_type none -ts_dt .005 -ts_max_time .1 nsize: 2 test: suffix: 3 args: -nox -da_grid_x 200 -ts_monitor_draw_solution -ts_type rosw -ts_rosw_type ra34pw2 -ts_dt 5e-3 -ts_max_time .1 -ts_adapt_type none nsize: 2 test: suffix: 4 args: -ts_type eimex -da_grid_x 200 -ts_eimex_order_adapt true -ts_dt 0.001 -ts_monitor -ts_max_steps 100 filter: sed "s/ITS/TIME/g" nsize: 2 TEST*/