static char help[] ="Solves a time-dependent nonlinear PDE. Uses implicit\n\ timestepping. Runtime options include:\n\ -M , where = number of grid points\n\ -debug : Activate debugging printouts\n\ -nox : Deactivate x-window graphics\n\n"; /* Concepts: TS^time-dependent nonlinear problems Processors: n */ /* ------------------------------------------------------------------------ This program solves the PDE u * u_xx u_t = --------- 2*(t+1)^2 on the domain 0 <= x <= 1, with boundary conditions u(t,0) = t + 1, u(t,1) = 2*t + 2, and initial condition u(0,x) = 1 + x*x. The exact solution is: u(t,x) = (1 + x*x) * (1 + t) Note that since the solution is linear in time and quadratic in x, the finite difference scheme actually computes the "exact" solution. We use by default the backward Euler method. ------------------------------------------------------------------------- */ /* Include "petscts.h" to use the PETSc timestepping routines. Note that this file automatically includes "petscsys.h" and other lower-level PETSc include files. Include the "petscdmda.h" to allow us to use the distributed array data structures to manage the parallel grid. */ #include #include #include #include /* User-defined application context - contains data needed by the application-provided callback routines. */ typedef struct { MPI_Comm comm; /* communicator */ DM da; /* distributed array data structure */ Vec localwork; /* local ghosted work vector */ Vec u_local; /* local ghosted approximate solution vector */ Vec solution; /* global exact solution vector */ PetscInt m; /* total number of grid points */ PetscReal h; /* mesh width: h = 1/(m-1) */ PetscBool debug; /* flag (1 indicates activation of debugging printouts) */ } AppCtx; /* User-defined routines, provided below. */ extern PetscErrorCode InitialConditions(Vec,AppCtx*); extern PetscErrorCode RHSFunction(TS,PetscReal,Vec,Vec,void*); extern PetscErrorCode RHSJacobian(TS,PetscReal,Vec,Mat,Mat,void*); extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*); extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*); int main(int argc,char **argv) { AppCtx appctx; /* user-defined application context */ TS ts; /* timestepping context */ Mat A; /* Jacobian matrix data structure */ Vec u; /* approximate solution vector */ PetscInt time_steps_max = 100; /* default max timesteps */ PetscErrorCode ierr; PetscReal dt; PetscReal time_total_max = 100.0; /* default max total time */ PetscBool mymonitor = PETSC_FALSE; PetscReal bounds[] = {1.0, 3.3}; /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Initialize program and set problem parameters - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr; ierr = PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),1,bounds);CHKERRQ(ierr); appctx.comm = PETSC_COMM_WORLD; appctx.m = 60; ierr = PetscOptionsGetInt(NULL,NULL,"-M",&appctx.m,NULL);CHKERRQ(ierr); ierr = PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);CHKERRQ(ierr); ierr = PetscOptionsHasName(NULL,NULL,"-mymonitor",&mymonitor);CHKERRQ(ierr); appctx.h = 1.0/(appctx.m-1.0); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create vector data structures - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ /* Create distributed array (DMDA) to manage parallel grid and vectors and to set up the ghost point communication pattern. There are M total grid values spread equally among all the processors. */ ierr = DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,appctx.m,1,1,NULL,&appctx.da);CHKERRQ(ierr); ierr = DMSetFromOptions(appctx.da);CHKERRQ(ierr); ierr = DMSetUp(appctx.da);CHKERRQ(ierr); /* Extract global and local vectors from DMDA; we use these to store the approximate solution. Then duplicate these for remaining vectors that have the same types. */ ierr = DMCreateGlobalVector(appctx.da,&u);CHKERRQ(ierr); ierr = DMCreateLocalVector(appctx.da,&appctx.u_local);CHKERRQ(ierr); /* Create local work vector for use in evaluating right-hand-side function; create global work vector for storing exact solution. */ ierr = VecDuplicate(appctx.u_local,&appctx.localwork);CHKERRQ(ierr); ierr = VecDuplicate(u,&appctx.solution);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create timestepping solver context; set callback routine for right-hand-side function evaluation. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr); ierr = TSSetProblemType(ts,TS_NONLINEAR);CHKERRQ(ierr); ierr = TSSetRHSFunction(ts,NULL,RHSFunction,&appctx);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set optional user-defined monitoring routine - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ if (mymonitor) { ierr = TSMonitorSet(ts,Monitor,&appctx,NULL);CHKERRQ(ierr); } /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - For nonlinear problems, the user can provide a Jacobian evaluation routine (or use a finite differencing approximation). Create matrix data structure; set Jacobian evaluation routine. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = MatCreate(PETSC_COMM_WORLD,&A);CHKERRQ(ierr); ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,appctx.m,appctx.m);CHKERRQ(ierr); ierr = MatSetFromOptions(A);CHKERRQ(ierr); ierr = MatSetUp(A);CHKERRQ(ierr); ierr = TSSetRHSJacobian(ts,A,A,RHSJacobian,&appctx);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set solution vector and initial timestep - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ dt = appctx.h/2.0; ierr = TSSetTimeStep(ts,dt);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Customize timestepping solver: - Set the solution method to be the Backward Euler method. - Set timestepping duration info Then set runtime options, which can override these defaults. For example, -ts_max_steps -ts_final_time to override the defaults set by TSSetMaxSteps()/TSSetMaxTime(). - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSSetType(ts,TSBEULER);CHKERRQ(ierr); ierr = TSSetMaxSteps(ts,time_steps_max);CHKERRQ(ierr); ierr = TSSetMaxTime(ts,time_total_max);CHKERRQ(ierr); ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);CHKERRQ(ierr); ierr = TSSetFromOptions(ts);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Solve the problem - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ /* Evaluate initial conditions */ ierr = InitialConditions(u,&appctx);CHKERRQ(ierr); /* Run the timestepping solver */ ierr = TSSolve(ts,u);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Free work space. All PETSc objects should be destroyed when they are no longer needed. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSDestroy(&ts);CHKERRQ(ierr); ierr = VecDestroy(&u);CHKERRQ(ierr); ierr = MatDestroy(&A);CHKERRQ(ierr); ierr = DMDestroy(&appctx.da);CHKERRQ(ierr); ierr = VecDestroy(&appctx.localwork);CHKERRQ(ierr); ierr = VecDestroy(&appctx.solution);CHKERRQ(ierr); ierr = VecDestroy(&appctx.u_local);CHKERRQ(ierr); /* Always call PetscFinalize() before exiting a program. This routine - finalizes the PETSc libraries as well as MPI - provides summary and diagnostic information if certain runtime options are chosen (e.g., -log_view). */ ierr = PetscFinalize(); return ierr; } /* --------------------------------------------------------------------- */ /* InitialConditions - Computes the solution at the initial time. Input Parameters: u - uninitialized solution vector (global) appctx - user-defined application context Output Parameter: u - vector with solution at initial time (global) */ PetscErrorCode InitialConditions(Vec u,AppCtx *appctx) { PetscScalar *u_localptr,h = appctx->h,x; PetscInt i,mybase,myend; PetscErrorCode ierr; /* Determine starting point of each processor's range of grid values. */ ierr = VecGetOwnershipRange(u,&mybase,&myend);CHKERRQ(ierr); /* Get a pointer to vector data. - For default PETSc vectors, VecGetArray() returns a pointer to the data array. Otherwise, the routine is implementation dependent. - You MUST call VecRestoreArray() when you no longer need access to the array. - Note that the Fortran interface to VecGetArray() differs from the C version. See the users manual for details. */ ierr = VecGetArray(u,&u_localptr);CHKERRQ(ierr); /* We initialize the solution array by simply writing the solution directly into the array locations. Alternatively, we could use VecSetValues() or VecSetValuesLocal(). */ for (i=mybase; idebug) { ierr = PetscPrintf(appctx->comm,"initial guess vector\n");CHKERRQ(ierr); ierr = VecView(u,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); } return 0; } /* --------------------------------------------------------------------- */ /* ExactSolution - Computes the exact solution at a given time. Input Parameters: t - current time solution - vector in which exact solution will be computed appctx - user-defined application context Output Parameter: solution - vector with the newly computed exact solution */ PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx) { PetscScalar *s_localptr,h = appctx->h,x; PetscInt i,mybase,myend; PetscErrorCode ierr; /* Determine starting and ending points of each processor's range of grid values */ ierr = VecGetOwnershipRange(solution,&mybase,&myend);CHKERRQ(ierr); /* Get a pointer to vector data. */ ierr = VecGetArray(solution,&s_localptr);CHKERRQ(ierr); /* Simply write the solution directly into the array locations. Alternatively, we could use VecSetValues() or VecSetValuesLocal(). */ for (i=mybase; icomm) that is associated with the current communicator. This saves the effort of calling PetscViewerDrawOpen() to create the window. Note that if we wished to plot several items in separate windows we would create each viewer with PetscViewerDrawOpen() and store them in the application context, appctx. PetscReal buffering makes graphics look better. */ ierr = PetscViewerDrawGetDraw(PETSC_VIEWER_DRAW_(appctx->comm),0,&draw);CHKERRQ(ierr); ierr = PetscDrawSetDoubleBuffer(draw);CHKERRQ(ierr); ierr = VecView(u,PETSC_VIEWER_DRAW_(appctx->comm));CHKERRQ(ierr); /* Compute the exact solution at this timestep */ ierr = ExactSolution(time,appctx->solution,appctx);CHKERRQ(ierr); /* Print debugging information if desired */ if (appctx->debug) { ierr = PetscPrintf(appctx->comm,"Computed solution vector\n");CHKERRQ(ierr); ierr = VecView(u,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); ierr = PetscPrintf(appctx->comm,"Exact solution vector\n");CHKERRQ(ierr); ierr = VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); } /* Compute the 2-norm and max-norm of the error */ ierr = VecAXPY(appctx->solution,-1.0,u);CHKERRQ(ierr); ierr = VecNorm(appctx->solution,NORM_2,&en2);CHKERRQ(ierr); en2s = PetscSqrtReal(appctx->h)*en2; /* scale the 2-norm by the grid spacing */ ierr = VecNorm(appctx->solution,NORM_MAX,&enmax);CHKERRQ(ierr); /* PetscPrintf() causes only the first processor in this communicator to print the timestep information. */ ierr = PetscPrintf(appctx->comm,"Timestep %D: time = %g 2-norm error = %g max norm error = %g\n",step,(double)time,(double)en2s,(double)enmax);CHKERRQ(ierr); /* Print debugging information if desired */ if (appctx->debug) { ierr = PetscPrintf(appctx->comm,"Error vector\n");CHKERRQ(ierr); ierr = VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); } return 0; } /* --------------------------------------------------------------------- */ /* RHSFunction - User-provided routine that evalues the right-hand-side function of the ODE. This routine is set in the main program by calling TSSetRHSFunction(). We compute: global_out = F(global_in) Input Parameters: ts - timesteping context t - current time global_in - vector containing the current iterate ctx - (optional) user-provided context for function evaluation. In this case we use the appctx defined above. Output Parameter: global_out - vector containing the newly evaluated function */ PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec global_in,Vec global_out,void *ctx) { AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */ DM da = appctx->da; /* distributed array */ Vec local_in = appctx->u_local; /* local ghosted input vector */ Vec localwork = appctx->localwork; /* local ghosted work vector */ PetscErrorCode ierr; PetscInt i,localsize; PetscMPIInt rank,size; PetscScalar *copyptr,sc; const PetscScalar *localptr; /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Get ready for local function computations - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ /* 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,global_in,INSERT_VALUES,local_in);CHKERRQ(ierr); ierr = DMGlobalToLocalEnd(da,global_in,INSERT_VALUES,local_in);CHKERRQ(ierr); /* Access directly the values in our local INPUT work array */ ierr = VecGetArrayRead(local_in,&localptr);CHKERRQ(ierr); /* Access directly the values in our local OUTPUT work array */ ierr = VecGetArray(localwork,©ptr);CHKERRQ(ierr); sc = 1.0/(appctx->h*appctx->h*2.0*(1.0+t)*(1.0+t)); /* Evaluate our function on the nodes owned by this processor */ ierr = VecGetLocalSize(local_in,&localsize);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Compute entries for the locally owned part - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ /* Handle boundary conditions: This is done by using the boundary condition u(t,boundary) = g(t,boundary) for some function g. Now take the derivative with respect to t to obtain u_{t}(t,boundary) = g_{t}(t,boundary) In our case, u(t,0) = t + 1, so that u_{t}(t,0) = 1 and u(t,1) = 2t+ 2, so that u_{t}(t,1) = 2 */ ierr = MPI_Comm_rank(appctx->comm,&rank);CHKERRQ(ierr); ierr = MPI_Comm_size(appctx->comm,&size);CHKERRQ(ierr); if (!rank) copyptr[0] = 1.0; if (rank == size-1) copyptr[localsize-1] = 2.0; /* Handle the interior nodes where the PDE is replace by finite difference operators. */ for (i=1; idebug) { ierr = PetscPrintf(appctx->comm,"RHS function vector\n");CHKERRQ(ierr); ierr = VecView(global_out,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); } return 0; } /* --------------------------------------------------------------------- */ /* RHSJacobian - User-provided routine to compute the Jacobian of the nonlinear right-hand-side function of the ODE. Input Parameters: ts - the TS context t - current time global_in - global input vector dummy - optional user-defined context, as set by TSetRHSJacobian() Output Parameters: AA - Jacobian matrix BB - optionally different preconditioning matrix str - flag indicating matrix structure Notes: RHSJacobian computes entries for the locally owned part of the Jacobian. - Currently, all PETSc parallel matrix formats are partitioned by contiguous chunks of rows across the processors. - Each processor needs to insert only elements that it owns locally (but any non-local elements will be sent to the appropriate processor during matrix assembly). - Always specify global row and columns of matrix entries when using MatSetValues(). - Here, we set all entries for a particular row at once. - Note that MatSetValues() uses 0-based row and column numbers in Fortran as well as in C. */ PetscErrorCode RHSJacobian(TS ts,PetscReal t,Vec global_in,Mat AA,Mat BB,void *ctx) { AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */ Vec local_in = appctx->u_local; /* local ghosted input vector */ DM da = appctx->da; /* distributed array */ PetscScalar v[3],sc; const PetscScalar *localptr; PetscErrorCode ierr; PetscInt i,mstart,mend,mstarts,mends,idx[3],is; /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Get ready for local Jacobian computations - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ /* 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,global_in,INSERT_VALUES,local_in);CHKERRQ(ierr); ierr = DMGlobalToLocalEnd(da,global_in,INSERT_VALUES,local_in);CHKERRQ(ierr); /* Get pointer to vector data */ ierr = VecGetArrayRead(local_in,&localptr);CHKERRQ(ierr); /* Get starting and ending locally owned rows of the matrix */ ierr = MatGetOwnershipRange(BB,&mstarts,&mends);CHKERRQ(ierr); mstart = mstarts; mend = mends; /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Compute entries for the locally owned part of the Jacobian. - Currently, all PETSc parallel matrix formats are partitioned by contiguous chunks of rows across the processors. - Each processor needs to insert only elements that it owns locally (but any non-local elements will be sent to the appropriate processor during matrix assembly). - Here, we set all entries for a particular row at once. - We can set matrix entries either using either MatSetValuesLocal() or MatSetValues(). - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ /* Set matrix rows corresponding to boundary data */ if (mstart == 0) { v[0] = 0.0; ierr = MatSetValues(BB,1,&mstart,1,&mstart,v,INSERT_VALUES);CHKERRQ(ierr); mstart++; } if (mend == appctx->m) { mend--; v[0] = 0.0; ierr = MatSetValues(BB,1,&mend,1,&mend,v,INSERT_VALUES);CHKERRQ(ierr); } /* Set matrix rows corresponding to interior data. We construct the matrix one row at a time. */ sc = 1.0/(appctx->h*appctx->h*2.0*(1.0+t)*(1.0+t)); for (i=mstart; i