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Implicit-Explicit Parallel Asynchronous Solver of Parabolic PDEs

This paper presents a method for the solution of parabolic PDEs on parallel computers, which is a combination of implicit and explicit finite difference schemes based on a domain decomposition (DD) strategy. Moreover, this method is asynchronous (i.e., no explicit synchronization is required among processors). We determine the values at subdomains' boundaries by our new high-order asynchronous explicit schemes. Then, any known high-order implicit finite difference scheme can be applied within each subdomain. We present a technique for derivation of appropriate asynchronous-explicit schemes based on Green's functions. Synchronous versions of these schemes are obtained as special cases. The applicability of this method is also demonstrated for a family of nonlinear problems. 
Our new explicit schemes are of high order and yet stable for a large time step, as established in our analysis of their numerical properties. Moreover, these schemes provide attractive properties for parallel implementation. Being asynchronous, they allow local time stepping, thus eliminating the need for a global synchronized time step. Moreover, our asynchronous computation is time stabilizing, in the sense that the calculation implicitly prevents a growing time gap between neighboring subdomains. The locality property, due to the exponential decay of Green's functions, implies that communication is needed only between neighboring processors. 
Hence, this method which is designed to minimize the overhead associated with the synchronization of the multiple processors is specifically suitable for parallel computers having a high synchronization cost or highly varying load, even in cases in which some processors have persistent speed differences. Furthermore, the implementation of different resolution in each subdomain (e.g., irregular or unstructured grid) makes it valuable as an adaptive algorithm. 
The above schemes were implemented and tested on the shared-memory multi-user Cray J90 and Sequent Balance machines. These implementations prove high accuracy and high degree of parallelism. This work is complementary to our previous work on asynchronous schemes [Comput. Math. Appl., 24 (1992), pp. 33--53; Appl. Numer. Math., 12 (1993), pp. 27--45; Numer. Algorithms, 6 (1994), pp. 275--296; Numer. Algorithms, 12 (1996), pp. 159--192].