2D Poisson solver

In this example, the goal is to solve the 2D Poisson problem:

with Dirichlet boundary condition

using Jacobi iteration; i.e., by discretizing the problem domain and applying the following operation to all interior points until convergence is reached:

(This example is based on the discussion of the Poisson problem in [3]).

A sequential program for this computation is straightforward. It maintains two copies of variable u, one for the current iteration (uk) and one for the next iteration (ukp1). At each iteration, it computes the values of ukp1 based on the values of uk. Observe that the boundary points are handled differently -- they maintain a constant value. Every NCHECK steps, the maximum of is computed to check for convergence.

An equivalent parallel program using the mesh archetype is not much more complicated:

• The declarations of the dimensions of the grid (NX and NY) are moved to INCLUDE file mesh_uparms.h. Parameter NGHOST is 1 for this application, since ukp1 depends on values of uk at most one step away. Parameters XPROCS and YPROCS are chosen by the user.
• Grid-based variables uk and ukp1 are distributed among grid processes. Observe the use of archetype-supplied constants to declare their dimensions.
• The whole grid is initialized in the host process and then copied to the grid processes (routines mesh_HtoG_host and mesh_HtoG_grid). (It could also be initialized directly in the grid processes; this approach was chosen for simplicity.)
• At each iteration, mesh_update_bdry is called to update the ghost boundaries before they are used in the grid computation. The special handling for the (global) boundary points is provided by using archetype library routines xintersect and yintersect to determine which points in the local section are in the interior of the global grid. Computing a global maximum for the convergence test is accomplished by computing a local maximum in each grid process and then calling archetype library routine mesh_merge_real_maxabs to find the global maximum. (After calling mesh_merge_real_maxabs, every process, including the host process, has the value of the global maximum.)
• When convergence is reached (or MAXSTEPS iterations have been performed), grid values are copied back to the host process for printing (routines mesh_GtoH_host and mesh_GtoH_grid).

Observe that the code executed by the host and grid processes has the same high-level structure -- both execute the main loop, for example, including the convergence test. This ensures that proper synchronization is maintained.

• Sequential program
• Parallel program (user-supplied code)