Implicit and multigrid procedures for steady-state computations with upwind algorithms

A computational study for the convergence acceleration of Euler and Navier–Stokes computations with upwind schemes has been conducted in a unified framework. It involves the flux-vector splitting algorithms due to Steger–Warming and Van Leer, the flux-difference splitting algorithms due to Roe and Osher and the hybrid algorithms, AUSM (Advection Upstream Splitting Method) and HUS (Hybrid Upwind Splitting). Implicit time integration with line Gauss–Seidel relaxation and multigrid are among the procedures which have been systematically investigated on an individual as well as cumulative basis. The upwind schemes have been tested in various implicit–explicit operator combinations such that the optimal among them can be determined based on extensive computations for two-dimensional flows in subsonic, transonic, supersonic and hypersonic flow regimes. In this study, the performance of these implicit time-integration procedures has been systematically compared with those corresponding to a multigrid accelerated explicit Runge–Kutta method. It has been demonstrated that a multigrid method employed in conjunction with an implicit time-integration scheme yields distinctly superior convergence as compared to those associated with either of the acceleration procedures provided that effective smoothers, which have been identified in this investigation, are prescribed in the implicit operator.