Factorized Implicit Upwind Methods Applied to Inviscid Flows at High Mach Number

The performance of several approximate factorization methods, coupled with a e nite volume spatial discretization using Roe’ s approximate Riemann solver, are compared by means of numerical tests on a two-dimensional steady inviscid e ow past a blunt body at Mach numbers ranging from 5 to 20. The comparisons are carried out evaluating, by numerical experiments, the optimal Courant number of each method. The alternating direction implicit and the lower ‐upper symmetric Gauss ‐Seidel methods result in the most efe cient factorizations, in terms of CPU time. The former behaves smoothly with increasing Mach number, and its performance is not affected by the grid size. The latter may achieve higher efe ciency but is strongly dependent on the number of relaxation steps performed, requiring optimization in terms of Mach number and grid size. I. Introduction A N IMPLICIT time integration is usually required to improve convergence toward steady state in time-marching, shockcapturing methods. This requirement is particularly true for high Mach number real gas e ows where the practical stability limit of explicit methods is particularly severe. The large block banded linearized operators, resulting from the application of multistep implicitmethodsto multidimensionalstructuredgrids,aresolvedmost efe ciently by resorting to approximate factorization (AF). The implicit operator may be split along the grid directions, as in the early alternating direction implicit (ADI) schemes of Briley and McDonald 1 andBeamand Warming, 2 orintotwolower ‐upper(LU) factors, as in the original LU schemes of Steger and Warming 3 and Jameson and Turkel. 4 The ADI scheme, without the addition of an appropriate amount of numerical damping, may, however, become unstable in three dimensions when formulated in delta form. Therefore, the LU factorization has become increasingly popular, and many improvements to this technique have been proposed in recent years. 5 Yoon and Jameson, 6,7 combined the LU factoriza