Parallel implicit Euler solver on homogeneous and heterogeneous computing environments

Abstract

In this paper, we present a domain decomposition approach for designing a parallel implicit Euler solver on workstation clusters. First, the computing domain is tessellated by an unstructured mesh based on a background quadtree. Using the Hilbert-Peano space-filling curve to traverse the quadtree and defining a sequential order, the background quadtree is also used to guide the partitioning of the unstructured mesh. This allows us to partition unstructured meshes for both homogeneous and heterogeneous computing environments. Then, we propose a parallel implicit Euler solver based on a parallel symmetric approximate LU factorization iterative algorithm (ALU). We show that the number of substitution steps for the lower sweep or the upper sweep of the parallel ALU algorithm is four. Experimental studies for the NACA0012 airfoil, the NASA EET wing and artillery shell within shock tube are reported. VER the pact decade, a lot of efforts were dedicated to the field of computational fluid dynamics. Especially, the implementation of unstructured meshes, which are suitable for the complex geometries, is popularly accepted in solving the Euler/Navier-Stokes equation. In this paper, we focus in implementing the unstructured Euler/NavierStokes solver on distributed memory parallel computers (DMPCs). We propose a novel partitioning strategy for homogeneous and heterogeneous computing, and propose parallel algorithms to accelerate the computation. We then present experimental studies on a workstation cluster. Parallel Euler/Navier-Stokes solvers for DMPCs were proposed based on different capabilities, such as using Runge-Kutta scheme for solving the partial differential equations,1 dealing with structured meshes,2'3 and others. However, for huge computational problems, the implicit or the multigrid methods are necessary to propagate the information across the computing domain and to accelerate the convergence rates. A complete survey of using various implicit methods for solving sparse linear systems, which are arising from Euler/Navier-Stokes equations, in parallel environments can be seen elsewhere.4 Since the scale of the sparse linear system is often very large, it is not practical to use any direct method to solve it, as many non-zero entries may fill in the matrix, which results in an unacceptable computational complexity and the requirement of huge memory space. Therefore, solving the sparse linear system by an iterative implicit method is more suitable for the computation. To implement the iterative *This work was partially supported by the NSC under Grants NSC