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 */ PetscErrorCode ierr; 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; */ ierr = PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr; /* 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; ierr = PetscOptionsGetInt(NULL,NULL,"-nstencilpts",&user.nstencilpts,NULL);CHKERRQ(ierr); ierr = PetscOptionsGetInt(NULL,NULL,"-boundary",&user.boundary,NULL);CHKERRQ(ierr); ierr = PetscOptionsHasName(NULL,NULL,"-viewJacobian",&user.viewJacobian);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create distributed array (DMDA) to manage parallel grid and vectors - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ if (user.nstencilpts == 5) { ierr = DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,11,11,PETSC_DECIDE,PETSC_DECIDE,1,1,NULL,NULL,&da);CHKERRQ(ierr); } else if (user.nstencilpts == 9) { ierr = DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE,DMDA_STENCIL_BOX,11,11,PETSC_DECIDE,PETSC_DECIDE,1,1,NULL,NULL,&da);CHKERRQ(ierr); } else SETERRQ1(PETSC_COMM_WORLD,PETSC_ERR_SUP,"nstencilpts %d is not supported",user.nstencilpts); ierr = DMSetFromOptions(da);CHKERRQ(ierr); ierr = DMSetUp(da);CHKERRQ(ierr); user.da = da; /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Extract global vectors from DMDA; - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = DMCreateGlobalVector(da,&u);CHKERRQ(ierr); ierr = VecDuplicate(u,&r);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create timestepping solver context - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr); ierr = TSSetProblemType(ts,TS_NONLINEAR);CHKERRQ(ierr); ierr = TSSetType(ts,TSBEULER);CHKERRQ(ierr); ierr = TSSetDM(ts,da);CHKERRQ(ierr); ierr = TSSetIFunction(ts,r,FormIFunction,&user);CHKERRQ(ierr); ierr = TSSetMaxTime(ts,1.0);CHKERRQ(ierr); ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set initial conditions - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = FormInitialSolution(u,&user);CHKERRQ(ierr); ierr = TSSetSolution(ts,u);CHKERRQ(ierr); dt = .01; ierr = TSSetTimeStep(ts,dt);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set Jacobian evaluation routine - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = DMSetMatType(da,MATAIJ);CHKERRQ(ierr); ierr = DMCreateMatrix(da,&J);CHKERRQ(ierr); Jtype = 0; ierr = PetscOptionsGetInt(NULL,NULL, "-Jtype",&Jtype,NULL);CHKERRQ(ierr); if (Jtype == 0) { /* use user provided Jacobian evaluation routine */ if (user.nstencilpts != 5) SETERRQ1(PETSC_COMM_WORLD,PETSC_ERR_SUP,"user Jacobian routine FormIJacobian() does not support nstencilpts=%D",user.nstencilpts); ierr = TSSetIJacobian(ts,J,J,FormIJacobian,&user);CHKERRQ(ierr); } else { /* use finite difference Jacobian J as preconditioner and '-snes_mf_operator' for Mat*vec */ ierr = TSGetSNES(ts,&snes);CHKERRQ(ierr); ierr = MatCreateSNESMF(snes,&Jmf);CHKERRQ(ierr); if (Jtype == 1) { /* slow finite difference J; */ ierr = SNESSetJacobian(snes,Jmf,J,SNESComputeJacobianDefault,NULL);CHKERRQ(ierr); } else if (Jtype == 2) { /* Use coloring to compute finite difference J efficiently */ ierr = SNESSetJacobian(snes,Jmf,J,SNESComputeJacobianDefaultColor,0);CHKERRQ(ierr); } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Jtype is not supported"); } /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Sets various TS parameters from user options - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSSetFromOptions(ts);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Solve nonlinear system - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSSolve(ts,u);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Free work space. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = MatDestroy(&J);CHKERRQ(ierr); ierr = MatDestroy(&Jmf);CHKERRQ(ierr); ierr = VecDestroy(&u);CHKERRQ(ierr); ierr = VecDestroy(&r);CHKERRQ(ierr); ierr = TSDestroy(&ts);CHKERRQ(ierr); ierr = DMDestroy(&da);CHKERRQ(ierr); ierr = PetscFinalize(); return ierr; } /* --------------------------------------------------------------------- */ /* FormIFunction = Udot - RHSFunction */ PetscErrorCode FormIFunction(TS ts,PetscReal t,Vec U,Vec Udot,Vec F,void *ctx) { PetscErrorCode ierr; AppCtx *user=(AppCtx*)ctx; DM da = (DM)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; ierr = DMGetLocalVector(da,&localU);CHKERRQ(ierr); ierr = 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);CHKERRQ(ierr); hx = 1.0/(PetscReal)(Mx-1); sx = 1.0/(hx*hx); hy = 1.0/(PetscReal)(My-1); sy = 1.0/(hy*hy); if (user->nstencilpts == 9 && hx != hy) SETERRQ(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. */ 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,&uarray);CHKERRQ(ierr); ierr = DMDAVecGetArray(da,F,&f);CHKERRQ(ierr); ierr = DMDAVecGetArray(da,Udot,&udot);CHKERRQ(ierr); /* Get local grid boundaries */ ierr = DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);CHKERRQ(ierr); /* Compute function over the locally owned part of the grid */ for (j=ys; jboundary == 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 */ ierr = DMDAVecRestoreArrayRead(da,localU,&uarray);CHKERRQ(ierr); ierr = DMDAVecRestoreArray(da,F,&f);CHKERRQ(ierr); ierr = DMDAVecRestoreArray(da,Udot,&udot);CHKERRQ(ierr); ierr = DMRestoreLocalVector(da,&localU);CHKERRQ(ierr); ierr = PetscLogFlops(11.0*ym*xm);CHKERRQ(ierr); PetscFunctionReturn(0); } /* --------------------------------------------------------------------- */ /* 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) { PetscErrorCode ierr; PetscInt i,j,Mx,My,xs,ys,xm,ym,nc; AppCtx *user = (AppCtx*)ctx; DM da = (DM)user->da; MatStencil col[5],row; PetscScalar vals[5],hx,hy,sx,sy; PetscFunctionBeginUser; ierr = 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);CHKERRQ(ierr); ierr = DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);CHKERRQ(ierr); 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; jboundary == 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; } 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(PetscObjectComm((PetscObject)Jpre),"Jpre:\n");CHKERRQ(ierr); ierr = MatView(Jpre,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); } PetscFunctionReturn(0); } /* ------------------------------------------------------------------- */ PetscErrorCode FormInitialSolution(Vec U,void *ptr) { AppCtx *user=(AppCtx*)ptr; DM da =user->da; PetscReal c =user->c; PetscErrorCode ierr; PetscInt i,j,xs,ys,xm,ym,Mx,My; PetscScalar **u; PetscReal hx,hy,x,y,r; PetscFunctionBeginUser; ierr = 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);CHKERRQ(ierr); hx = 1.0/(PetscReal)(Mx-1); hy = 1.0/(PetscReal)(My-1); /* Get pointers to vector data */ ierr = DMDAVecGetArray(da,U,&u);CHKERRQ(ierr); /* Get local grid boundaries */ ierr = DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);CHKERRQ(ierr); /* Compute function over the locally owned part of the grid */ for (j=ys; j