In this paper we propose parallel algorithm for the solution of partial differential equations over a rectangular domain using the Crank-Nicholson method by cooperation with the DuFort-Frankel method and apply it on a model problem, namely, the heat conduction equation. One of the well known parallel techniques in solving partial differential equations in cluster computing environment is the domain decomposition technique. Using this technique, the whole domain is decomposed into subdomains, each of them has its own boundaries that are called the interface points. Parallelization is realized by approximating interface values using the unconditionally stable DuFort-Frankel explicit scheme, and these values serve as Neumann boundary conditions for the Crank-Nicholson implicit scheme in the subdomains. The numerical results show that our algorithm is more accurate than the algorithm based on the forward explicit method to approximate the values of the interface points, especially, when we use a small number of time steps. Moreover, these numerical results show that increasing the number of processors which are used in the cluster, yields an increase in the algorithm speedup. © 2007.
Mahmoud, F. A., & Al-Towaiq, M. H. (2008). Parallel algorithm for the solutions of PDEs in linux clustered workstations. Applied Mathematics and Computation, 200(1), 178–188. https://doi.org/10.1016/j.amc.2007.11.013