Implicit method for the solution of supersonic and hypersonic 3D flow problems with Lower-Upper Symmetric-Gauss-Seidel preconditioner on multiple graphics processing units

Abstract

The paper describes a numerical method for the solution of stationary gas dynamics 3D spatial equations on unstructured grids that is designed for multiple graphics processing unit (GPU) computational architecture. Discretization of governing equations is done using first and second order TVD schemes. The Newton's method with simple pseudo time-step homotopy is used to solve the problem. Each iteration step involves solution of the linear system originated from the linearization of gas dynamics equations. Krylov subspace iterative methods are used to solve the linear system. The main aim of the paper is to describe a preconditioning Lower-Upper Symmetric-Gauss-Seidel (LU-SGS) method and its adaptation on multiple GPU computational systems. It is shown that deliberately reordered matrices with rearranged solution process of arising lower and upper triangular linear systems allow one to obtain close algebraic properties to the original single threaded LU-SGS. The method is benchmarked against published results. The analysis of computational efficiency and acceleration is presented for different flows with Mach number ranging from 1.2 up to 25.