1D heat equation

In this example, the goal is to solve the 1D heat diffusion equation:

using the approximation:

A sequential program for this computation is straightforward. It maintains two copies of variable U, one for the current time step (uk) and one for the next time step (ukp1). At each time step, it computes the values of ukp1 based on the values of uk. Observe that the two boundary points are handled differently -- they maintain a constant value.

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

• The declaration of the size of the grid (NX) is 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. Parameter XPROCS is chosen by the user.
• Grid-based variables uk and ukp1 are distributed among grid processes. Observe the use of archetype-supplied constants IXLO and IXHI to declare the size of the local sections -- these constants give the proper dimensions of the local section, including ghost boundaries, while allowing the indices of the local section proper (exclusive of ghost boundaries) to range from 1 to NXlsize.
• The whole array 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 time step, 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 routine xintersect to determine which points in the local section are in the interior of the global array (global indices 2 through NX - 1). The grid values are then 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 time-step loop, for example. This ensures that proper synchronization is maintained.

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