static char help[] = "Time-dependent PDE in 2d. Modified from ex13.c for illustrating how to solve DAEs. \n"; /* u_t = uxx + uyy 0 < x < 1, 0 < y < 1; At t=0: u(x,y) = exp(c*r*r*r), if r=PetscSqrtReal((x-.5)*(x-.5) + (y-.5)*(y-.5)) < .125 u(x,y) = 0.0 if r >= .125 Boundary conditions: Drichlet BC: At x=0, x=1, y=0, y=1: u = 0.0 Neumann BC: At x=0, x=1: du(x,y,t)/dx = 0 At y=0, y=1: du(x,y,t)/dy = 0 mpiexec -n 2 ./ex15 -da_grid_x 40 -da_grid_y 40 -ts_max_steps 2 -snes_monitor -ksp_monitor ./ex15 -da_grid_x 40 -da_grid_y 40 -draw_pause .1 -boundary 1 -ts_monitor_draw_solution ./ex15 -da_grid_x 40 -da_grid_y 40 -draw_pause .1 -boundary 1 -Jtype 2 -nstencilpts 9 */ #include #include #include /* User-defined data structures and routines */ /* AppCtx: used by FormIFunction() and FormIJacobian() */ typedef struct { DM da; PetscInt nstencilpts; /* number of stencil points: 5 or 9 */ PetscReal c; PetscInt boundary; /* Type of boundary condition */ PetscBool viewJacobian; } AppCtx; extern PetscErrorCode FormIFunction(TS, PetscReal, Vec, Vec, Vec, void *); extern PetscErrorCode FormIJacobian(TS, PetscReal, Vec, Vec, PetscReal, Mat, Mat, void *); extern PetscErrorCode FormInitialSolution(Vec, void *); int main(int argc, char **argv) { TS ts; /* nonlinear solver */ Vec u, r; /* solution, residual vectors */ Mat J, Jmf = NULL; /* Jacobian matrices */ DM da; PetscReal dt; AppCtx user; /* user-defined work context */ SNES snes; PetscInt Jtype; /* Jacobian type 0: user provide Jacobian; 1: slow finite difference; 2: fd with coloring; */ PetscFunctionBeginUser; PetscCall(PetscInitialize(&argc, &argv, NULL, help)); /* Initialize user application context */ user.da = NULL; user.nstencilpts = 5; user.c = -30.0; user.boundary = 0; /* 0: Drichlet BC; 1: Neumann BC */ user.viewJacobian = PETSC_FALSE; PetscCall(PetscOptionsGetInt(NULL, NULL, "-nstencilpts", &user.nstencilpts, NULL)); PetscCall(PetscOptionsGetInt(NULL, NULL, "-boundary", &user.boundary, NULL)); PetscCall(PetscOptionsHasName(NULL, NULL, "-viewJacobian", &user.viewJacobian)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create distributed array (DMDA) to manage parallel grid and vectors - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ if (user.nstencilpts == 5) { PetscCall(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DMDA_STENCIL_STAR, 11, 11, PETSC_DECIDE, PETSC_DECIDE, 1, 1, NULL, NULL, &da)); } else if (user.nstencilpts == 9) { PetscCall(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DMDA_STENCIL_BOX, 11, 11, PETSC_DECIDE, PETSC_DECIDE, 1, 1, NULL, NULL, &da)); } else SETERRQ(PETSC_COMM_WORLD, PETSC_ERR_SUP, "nstencilpts %" PetscInt_FMT " is not supported", user.nstencilpts); PetscCall(DMSetFromOptions(da)); PetscCall(DMSetUp(da)); user.da = da; /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Extract global vectors from DMDA; - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(DMCreateGlobalVector(da, &u)); PetscCall(VecDuplicate(u, &r)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create timestepping solver context - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(TSCreate(PETSC_COMM_WORLD, &ts)); PetscCall(TSSetProblemType(ts, TS_NONLINEAR)); PetscCall(TSSetType(ts, TSBEULER)); PetscCall(TSSetDM(ts, da)); PetscCall(TSSetIFunction(ts, r, FormIFunction, &user)); PetscCall(TSSetMaxTime(ts, 1.0)); PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set initial conditions - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(FormInitialSolution(u, &user)); PetscCall(TSSetSolution(ts, u)); dt = .01; PetscCall(TSSetTimeStep(ts, dt)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set Jacobian evaluation routine - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(DMSetMatType(da, MATAIJ)); PetscCall(DMCreateMatrix(da, &J)); Jtype = 0; PetscCall(PetscOptionsGetInt(NULL, NULL, "-Jtype", &Jtype, NULL)); if (Jtype == 0) { /* use user provided Jacobian evaluation routine */ PetscCheck(user.nstencilpts == 5, PETSC_COMM_WORLD, PETSC_ERR_SUP, "user Jacobian routine FormIJacobian() does not support nstencilpts=%" PetscInt_FMT, user.nstencilpts); PetscCall(TSSetIJacobian(ts, J, J, FormIJacobian, &user)); } else { /* use finite difference Jacobian J as preconditioner and '-snes_mf_operator' for Mat*vec */ PetscCall(TSGetSNES(ts, &snes)); PetscCall(MatCreateSNESMF(snes, &Jmf)); if (Jtype == 1) { /* slow finite difference J; */ PetscCall(SNESSetJacobian(snes, Jmf, J, SNESComputeJacobianDefault, NULL)); } else if (Jtype == 2) { /* Use coloring to compute finite difference J efficiently */ PetscCall(SNESSetJacobian(snes, Jmf, J, SNESComputeJacobianDefaultColor, 0)); } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Jtype is not supported"); } /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Sets various TS parameters from user options - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(TSSetFromOptions(ts)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Solve nonlinear system - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(TSSolve(ts, u)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Free work space. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(MatDestroy(&J)); PetscCall(MatDestroy(&Jmf)); PetscCall(VecDestroy(&u)); PetscCall(VecDestroy(&r)); PetscCall(TSDestroy(&ts)); PetscCall(DMDestroy(&da)); PetscCall(PetscFinalize()); return 0; } /* --------------------------------------------------------------------- */ /* FormIFunction = Udot - RHSFunction */ PetscErrorCode FormIFunction(TS ts, PetscReal t, Vec U, Vec Udot, Vec F, void *ctx) { AppCtx *user = (AppCtx *)ctx; DM da = user->da; PetscInt i, j, Mx, My, xs, ys, xm, ym; PetscReal hx, hy, sx, sy; PetscScalar u, uxx, uyy, **uarray, **f, **udot; Vec localU; PetscFunctionBeginUser; PetscCall(DMGetLocalVector(da, &localU)); PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, &My, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE)); hx = 1.0 / (PetscReal)(Mx - 1); sx = 1.0 / (hx * hx); hy = 1.0 / (PetscReal)(My - 1); sy = 1.0 / (hy * hy); PetscCheck(user->nstencilpts != 9 || hx == hy, PETSC_COMM_WORLD, PETSC_ERR_SUP, "hx must equal hy when nstencilpts = 9 for this example"); /* 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. */ PetscCall(DMGlobalToLocalBegin(da, U, INSERT_VALUES, localU)); PetscCall(DMGlobalToLocalEnd(da, U, INSERT_VALUES, localU)); /* Get pointers to vector data */ PetscCall(DMDAVecGetArrayRead(da, localU, &uarray)); PetscCall(DMDAVecGetArray(da, F, &f)); PetscCall(DMDAVecGetArray(da, Udot, &udot)); /* Get local grid boundaries */ PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL)); /* Compute function over the locally owned part of the grid */ for (j = ys; j < ys + ym; j++) { for (i = xs; i < xs + xm; i++) { /* Boundary conditions */ if (i == 0 || j == 0 || i == Mx - 1 || j == My - 1) { if (user->boundary == 0) { /* Drichlet BC */ f[j][i] = uarray[j][i]; /* F = U */ } else { /* Neumann BC */ if (i == 0 && j == 0) { /* SW corner */ f[j][i] = uarray[j][i] - uarray[j + 1][i + 1]; } else if (i == Mx - 1 && j == 0) { /* SE corner */ f[j][i] = uarray[j][i] - uarray[j + 1][i - 1]; } else if (i == 0 && j == My - 1) { /* NW corner */ f[j][i] = uarray[j][i] - uarray[j - 1][i + 1]; } else if (i == Mx - 1 && j == My - 1) { /* NE corner */ f[j][i] = uarray[j][i] - uarray[j - 1][i - 1]; } else if (i == 0) { /* Left */ f[j][i] = uarray[j][i] - uarray[j][i + 1]; } else if (i == Mx - 1) { /* Right */ f[j][i] = uarray[j][i] - uarray[j][i - 1]; } else if (j == 0) { /* Bottom */ f[j][i] = uarray[j][i] - uarray[j + 1][i]; } else if (j == My - 1) { /* Top */ f[j][i] = uarray[j][i] - uarray[j - 1][i]; } } } else { /* Interior */ u = uarray[j][i]; /* 5-point stencil */ uxx = (-2.0 * u + uarray[j][i - 1] + uarray[j][i + 1]); uyy = (-2.0 * u + uarray[j - 1][i] + uarray[j + 1][i]); if (user->nstencilpts == 9) { /* 9-point stencil: assume hx=hy */ uxx = 2.0 * uxx / 3.0 + (0.5 * (uarray[j - 1][i - 1] + uarray[j - 1][i + 1] + uarray[j + 1][i - 1] + uarray[j + 1][i + 1]) - 2.0 * u) / 6.0; uyy = 2.0 * uyy / 3.0 + (0.5 * (uarray[j - 1][i - 1] + uarray[j - 1][i + 1] + uarray[j + 1][i - 1] + uarray[j + 1][i + 1]) - 2.0 * u) / 6.0; } f[j][i] = udot[j][i] - (uxx * sx + uyy * sy); } } } /* Restore vectors */ PetscCall(DMDAVecRestoreArrayRead(da, localU, &uarray)); PetscCall(DMDAVecRestoreArray(da, F, &f)); PetscCall(DMDAVecRestoreArray(da, Udot, &udot)); PetscCall(DMRestoreLocalVector(da, &localU)); PetscCall(PetscLogFlops(11.0 * ym * xm)); PetscFunctionReturn(PETSC_SUCCESS); } /* --------------------------------------------------------------------- */ /* FormIJacobian() - Compute IJacobian = dF/dU + a dF/dUdot This routine is not used with option '-use_coloring' */ PetscErrorCode FormIJacobian(TS ts, PetscReal t, Vec U, Vec Udot, PetscReal a, Mat J, Mat Jpre, void *ctx) { PetscInt i, j, Mx, My, xs, ys, xm, ym, nc; AppCtx *user = (AppCtx *)ctx; DM da = user->da; MatStencil col[5], row; PetscScalar vals[5], hx, hy, sx, sy; PetscFunctionBeginUser; PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, &My, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE)); PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL)); hx = 1.0 / (PetscReal)(Mx - 1); sx = 1.0 / (hx * hx); hy = 1.0 / (PetscReal)(My - 1); sy = 1.0 / (hy * hy); for (j = ys; j < ys + ym; j++) { for (i = xs; i < xs + xm; i++) { nc = 0; row.j = j; row.i = i; if (user->boundary == 0 && (i == 0 || i == Mx - 1 || j == 0 || j == My - 1)) { col[nc].j = j; col[nc].i = i; vals[nc++] = 1.0; } else if (user->boundary > 0 && i == 0) { /* Left Neumann */ col[nc].j = j; col[nc].i = i; vals[nc++] = 1.0; col[nc].j = j; col[nc].i = i + 1; vals[nc++] = -1.0; } else if (user->boundary > 0 && i == Mx - 1) { /* Right Neumann */ col[nc].j = j; col[nc].i = i; vals[nc++] = 1.0; col[nc].j = j; col[nc].i = i - 1; vals[nc++] = -1.0; } else if (user->boundary > 0 && j == 0) { /* Bottom Neumann */ col[nc].j = j; col[nc].i = i; vals[nc++] = 1.0; col[nc].j = j + 1; col[nc].i = i; vals[nc++] = -1.0; } else if (user->boundary > 0 && j == My - 1) { /* Top Neumann */ col[nc].j = j; col[nc].i = i; vals[nc++] = 1.0; col[nc].j = j - 1; col[nc].i = i; vals[nc++] = -1.0; } else { /* Interior */ col[nc].j = j - 1; col[nc].i = i; vals[nc++] = -sy; col[nc].j = j; col[nc].i = i - 1; vals[nc++] = -sx; col[nc].j = j; col[nc].i = i; vals[nc++] = 2.0 * (sx + sy) + a; col[nc].j = j; col[nc].i = i + 1; vals[nc++] = -sx; col[nc].j = j + 1; col[nc].i = i; vals[nc++] = -sy; } PetscCall(MatSetValuesStencil(Jpre, 1, &row, nc, col, vals, INSERT_VALUES)); } } PetscCall(MatAssemblyBegin(Jpre, MAT_FINAL_ASSEMBLY)); PetscCall(MatAssemblyEnd(Jpre, MAT_FINAL_ASSEMBLY)); if (J != Jpre) { PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY)); PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY)); } if (user->viewJacobian) { PetscCall(PetscPrintf(PetscObjectComm((PetscObject)Jpre), "Jpre:\n")); PetscCall(MatView(Jpre, PETSC_VIEWER_STDOUT_WORLD)); } PetscFunctionReturn(PETSC_SUCCESS); } /* ------------------------------------------------------------------- */ PetscErrorCode FormInitialSolution(Vec U, void *ptr) { AppCtx *user = (AppCtx *)ptr; DM da = user->da; PetscReal c = user->c; PetscInt i, j, xs, ys, xm, ym, Mx, My; PetscScalar **u; PetscReal hx, hy, x, y, r; PetscFunctionBeginUser; PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, &My, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE)); hx = 1.0 / (PetscReal)(Mx - 1); hy = 1.0 / (PetscReal)(My - 1); /* Get pointers to vector data */ PetscCall(DMDAVecGetArray(da, U, &u)); /* Get local grid boundaries */ PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL)); /* Compute function over the locally owned part of the grid */ for (j = ys; j < ys + ym; j++) { y = j * hy; for (i = xs; i < xs + xm; i++) { x = i * hx; r = PetscSqrtReal((x - .5) * (x - .5) + (y - .5) * (y - .5)); if (r < .125) u[j][i] = PetscExpReal(c * r * r * r); else u[j][i] = 0.0; } } /* Restore vectors */ PetscCall(DMDAVecRestoreArray(da, U, &u)); PetscFunctionReturn(PETSC_SUCCESS); } /*TEST test: args: -da_grid_x 20 -da_grid_y 20 -boundary 0 -ts_max_steps 10 -ts_monitor test: suffix: 2 args: -da_grid_x 20 -da_grid_y 20 -boundary 0 -ts_max_steps 10 -Jtype 2 -ts_monitor test: suffix: 3 requires: !single args: -da_grid_x 20 -da_grid_y 20 -boundary 1 -ts_max_steps 10 -ts_monitor test: suffix: 4 requires: !single nsize: 2 args: -da_grid_x 20 -da_grid_y 20 -boundary 1 -ts_max_steps 10 -ts_monitor test: suffix: 5 nsize: 1 args: -da_grid_x 20 -da_grid_y 20 -boundary 0 -ts_max_steps 10 -Jtype 1 -ts_monitor TEST*/