Two-dimensional parallel solver for the solution of Navier-Stokes equations with constant and variable coefficients using ADI on cells

The paper proposed a new algorithm for the parallel solution of two-dimensional Navier - Stokes type equation with constant and non-constant coefficients which is mapped onto cell topology. This paper is a further development in the application of the local Fourier methods to the solutions of PDE's in multidomain regions. The extension of the above solution to problems with non-constant coefficients is suggested via spectral multidomain preconditioner. This approach is efficient when we have good local approximations in each subdomain. By dividing the computational domain into a large enough number of subdomains we can guarantee it. The new achievement here is that we are able to handle decomposition of the domain into cells that is the decomposed in both directions, x and y. An appropriate alternate direction implicit (ADI) scheme was developed. It enables the reduction of a 2-D problem to a collection of uncoupled 1-D ODE's. In effect, the 1-D solver becomes the basic routine to solve a 2-D problem using splitting of the differential operators by ADI. Detailed performance analysis is given where the issue of the communication among the domains (processors) is examined. We show that by using the Richardson method only local communication is required. The algorithm was implemented on IBM SP2, network of ALPHA workstations, and MOSIX [A. Barak, S. Guday, R. Wheeler, The MOSIX Distributed Operating System, Load Balancing for UNIX, Lecture Notes in Computer Science, Vol. 672, Springer-Verlag, 1993; A. Barak, O. Laden, Z. Yarom, The NOW MOSIX and its Preemptive Process Migration Scheme, IEEE TCOS 7 (2) (1995) 5-11] which is a network of i586. All are implemented using the PVM software package and the same ADI program was running on these different multiprocessor configurations. It achieved efficiency of 55-70% depending on the multiprocessor.