static const char help[] = "Time-dependent PDE in 1d. Simplified from ex15.c for illustrating how to solve DAEs. \n"; /* u_t = uxx 0 < x < 1; At t=0: u(x) = exp(c*r*r*r), if r=PetscSqrtReal((x-.5)*(x-.5)) < .125 u(x) = 0.0 if r >= .125 Boundary conditions: Dirichlet BC: At x=0, x=1, u = 0.0 Neumann BC: At x=0, x=1: du(x,t)/dx = 0 mpiexec -n 2 ./ex17 -da_grid_x 40 -ts_max_steps 2 -snes_monitor -ksp_monitor ./ex17 -da_grid_x 40 -monitor_solution ./ex17 -da_grid_x 100 -ts_type theta -ts_theta_theta 0.5 # Midpoint is not L-stable ./ex17 -jac_type fd_coloring -da_grid_x 500 -boundary 1 ./ex17 -da_grid_x 100 -ts_type gl -ts_adapt_type none -ts_max_steps 2 */ #include #include #include typedef enum {JACOBIAN_ANALYTIC,JACOBIAN_FD_COLORING,JACOBIAN_FD_FULL} JacobianType; static const char *const JacobianTypes[] = {"analytic","fd_coloring","fd_full","JacobianType","fd_",0}; /* User-defined data structures and routines */ typedef struct { PetscReal c; PetscInt boundary; /* Type of boundary condition */ PetscBool viewJacobian; } AppCtx; 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; /* nonlinear solver */ Vec u; /* solution, residual vectors */ Mat J; /* Jacobian matrix */ PetscInt nsteps; PetscReal vmin,vmax,norm; PetscErrorCode ierr; DM da; PetscReal ftime,dt; AppCtx user; /* user-defined work context */ JacobianType jacType; 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,11,1,1,NULL,&da);CHKERRQ(ierr); ierr = DMSetFromOptions(da);CHKERRQ(ierr); ierr = DMSetUp(da);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Extract global vectors from DMDA; - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = DMCreateGlobalVector(da,&u);CHKERRQ(ierr); /* Initialize user application context */ user.c = -30.0; user.boundary = 0; /* 0: Dirichlet BC; 1: Neumann BC */ user.viewJacobian = PETSC_FALSE; ierr = PetscOptionsGetInt(NULL,NULL,"-boundary",&user.boundary,NULL);CHKERRQ(ierr); ierr = PetscOptionsHasName(NULL,NULL,"-viewJacobian",&user.viewJacobian);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create timestepping solver context - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr); ierr = TSSetProblemType(ts,TS_NONLINEAR);CHKERRQ(ierr); ierr = TSSetType(ts,TSTHETA);CHKERRQ(ierr); ierr = TSThetaSetTheta(ts,1.0);CHKERRQ(ierr); /* Make the Theta method behave like backward Euler */ ierr = TSSetIFunction(ts,NULL,FormIFunction,&user);CHKERRQ(ierr); ierr = DMSetMatType(da,MATAIJ);CHKERRQ(ierr); ierr = DMCreateMatrix(da,&J);CHKERRQ(ierr); jacType = JACOBIAN_ANALYTIC; /* use user-provide Jacobian */ ierr = TSSetDM(ts,da);CHKERRQ(ierr); /* Use TSGetDM() to access. Setting here allows easy use of geometric multigrid. */ ftime = 1.0; ierr = TSSetMaxTime(ts,ftime);CHKERRQ(ierr); ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set initial conditions - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = FormInitialSolution(ts,u,&user);CHKERRQ(ierr); ierr = TSSetSolution(ts,u);CHKERRQ(ierr); dt = .01; ierr = TSSetTimeStep(ts,dt);CHKERRQ(ierr); /* Use slow fd Jacobian or fast fd Jacobian with colorings. Note: this requirs snes which is not created until TSSetUp()/TSSetFromOptions() is called */ ierr = PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Options for Jacobian evaluation",NULL);CHKERRQ(ierr); ierr = PetscOptionsEnum("-jac_type","Type of Jacobian","",JacobianTypes,(PetscEnum)jacType,(PetscEnum*)&jacType,0);CHKERRQ(ierr); ierr = PetscOptionsEnd();CHKERRQ(ierr); if (jacType == JACOBIAN_ANALYTIC) { ierr = TSSetIJacobian(ts,J,J,FormIJacobian,&user);CHKERRQ(ierr); } else if (jacType == JACOBIAN_FD_COLORING) { SNES snes; ierr = TSGetSNES(ts,&snes);CHKERRQ(ierr); ierr = SNESSetJacobian(snes,J,J,SNESComputeJacobianDefaultColor,0);CHKERRQ(ierr); } else if (jacType == JACOBIAN_FD_FULL) { SNES snes; ierr = TSGetSNES(ts,&snes);CHKERRQ(ierr); ierr = SNESSetJacobian(snes,J,J,SNESComputeJacobianDefault,&user);CHKERRQ(ierr); } /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set runtime options - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSSetFromOptions(ts);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Integrate ODE system - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSSolve(ts,u);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Compute diagnostics of the solution - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = VecNorm(u,NORM_1,&norm);CHKERRQ(ierr); ierr = VecMax(u,NULL,&vmax);CHKERRQ(ierr); ierr = VecMin(u,NULL,&vmin);CHKERRQ(ierr); ierr = TSGetStepNumber(ts,&nsteps);CHKERRQ(ierr); ierr = TSGetTime(ts,&ftime);CHKERRQ(ierr); ierr = PetscPrintf(PETSC_COMM_WORLD,"timestep %D: time %g, solution norm %g, max %g, min %g\n",nsteps,(double)ftime,(double)norm,(double)vmax,(double)vmin);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Free work space. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = MatDestroy(&J);CHKERRQ(ierr); ierr = VecDestroy(&u);CHKERRQ(ierr); ierr = TSDestroy(&ts);CHKERRQ(ierr); ierr = DMDestroy(&da);CHKERRQ(ierr); ierr = PetscFinalize(); return ierr; } /* ------------------------------------------------------------------- */ static PetscErrorCode FormIFunction(TS ts,PetscReal ftime,Vec U,Vec Udot,Vec F,void *ptr) { AppCtx *user=(AppCtx*)ptr; DM da; PetscErrorCode ierr; PetscInt i,Mx,xs,xm; PetscReal hx,sx; PetscScalar *u,*udot,*f; Vec localU; PetscFunctionBeginUser; ierr = TSGetDM(ts,&da);CHKERRQ(ierr); ierr = DMGetLocalVector(da,&localU);CHKERRQ(ierr); ierr = DMDAGetInfo(da,PETSC_IGNORE,&Mx,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);CHKERRQ(ierr); hx = 1.0/(PetscReal)(Mx-1); sx = 1.0/(hx*hx); /* 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 = DMGlobalToLocalBegin(da,U,INSERT_VALUES,localU);CHKERRQ(ierr); ierr = DMGlobalToLocalEnd(da,U,INSERT_VALUES,localU);CHKERRQ(ierr); /* Get pointers to vector data */ ierr = DMDAVecGetArrayRead(da,localU,&u);CHKERRQ(ierr); ierr = DMDAVecGetArrayRead(da,Udot,&udot);CHKERRQ(ierr); ierr = DMDAVecGetArray(da,F,&f);CHKERRQ(ierr); /* Get local grid boundaries */ ierr = DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);CHKERRQ(ierr); /* Compute function over the locally owned part of the grid */ for (i=xs; iboundary == 0) { /* Dirichlet BC */ if (i == 0 || i == Mx-1) f[i] = u[i]; /* F = U */ else f[i] = udot[i] + (2.*u[i] - u[i-1] - u[i+1])*sx; } else { /* Neumann BC */ if (i == 0) f[i] = u[0] - u[1]; else if (i == Mx-1) f[i] = u[i] - u[i-1]; else f[i] = udot[i] + (2.*u[i] - u[i-1] - u[i+1])*sx; } } /* Restore vectors */ ierr = DMDAVecRestoreArrayRead(da,localU,&u);CHKERRQ(ierr); ierr = DMDAVecRestoreArrayRead(da,Udot,&udot);CHKERRQ(ierr); ierr = DMDAVecRestoreArray(da,F,&f);CHKERRQ(ierr); ierr = DMRestoreLocalVector(da,&localU);CHKERRQ(ierr); PetscFunctionReturn(0); } /* --------------------------------------------------------------------- */ /* IJacobian - Compute IJacobian = dF/dU + a dF/dUdot */ PetscErrorCode FormIJacobian(TS ts,PetscReal t,Vec U,Vec Udot,PetscReal a,Mat J,Mat Jpre,void *ctx) { PetscErrorCode ierr; PetscInt i,rstart,rend,Mx; PetscReal hx,sx; AppCtx *user = (AppCtx*)ctx; DM da; MatStencil col[3],row; PetscInt nc; PetscScalar vals[3]; PetscFunctionBeginUser; ierr = TSGetDM(ts,&da);CHKERRQ(ierr); ierr = MatGetOwnershipRange(Jpre,&rstart,&rend);CHKERRQ(ierr); ierr = DMDAGetInfo(da,PETSC_IGNORE,&Mx,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);CHKERRQ(ierr); hx = 1.0/(PetscReal)(Mx-1); sx = 1.0/(hx*hx); for (i=rstart; iboundary == 0 && (i == 0 || i == Mx-1)) { col[nc].i = i; vals[nc++] = 1.0; } else if (user->boundary > 0 && i == 0) { /* Left Neumann */ col[nc].i = i; vals[nc++] = 1.0; col[nc].i = i+1; vals[nc++] = -1.0; } else if (user->boundary > 0 && i == Mx-1) { /* Right Neumann */ col[nc].i = i-1; vals[nc++] = -1.0; col[nc].i = i; vals[nc++] = 1.0; } else { /* Interior */ col[nc].i = i-1; vals[nc++] = -1.0*sx; col[nc].i = i; vals[nc++] = 2.0*sx + a; col[nc].i = i+1; vals[nc++] = -1.0*sx; } ierr = MatSetValuesStencil(Jpre,1,&row,nc,col,vals,INSERT_VALUES);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); } if (user->viewJacobian) { ierr = PetscPrintf(PETSC_COMM_WORLD,"Jpre:\n");CHKERRQ(ierr); ierr = MatView(Jpre,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); } PetscFunctionReturn(0); } /* ------------------------------------------------------------------- */ PetscErrorCode FormInitialSolution(TS ts,Vec U,void *ptr) { AppCtx *user=(AppCtx*)ptr; PetscReal c =user->c; DM da; PetscErrorCode ierr; PetscInt i,xs,xm,Mx; PetscScalar *u; PetscReal hx,x,r; PetscFunctionBeginUser; ierr = TSGetDM(ts,&da);CHKERRQ(ierr); ierr = DMDAGetInfo(da,PETSC_IGNORE,&Mx,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);CHKERRQ(ierr); hx = 1.0/(PetscReal)(Mx-1); /* Get pointers to vector data */ ierr = DMDAVecGetArray(da,U,&u);CHKERRQ(ierr); /* Get local grid boundaries */ ierr = DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);CHKERRQ(ierr); /* Compute function over the locally owned part of the grid */ for (i=xs; i