Stability analysis of numerical boundary conditions and implicit difference approximations for hyperbolic equations

Implicit noniterative finite-difference schemes have recently been developed by several authors for multidimensional systems of nonlinear hyperbolic partial differential equations. When applied to linear model equations with periodic boundary conditions those schemes are unconditionally stable (A-stable). As applied in practice the algorithms often face a severe time-step restriction. A major source of the difficulty is the treatment of the numerical boundary conditions. One conjecture has been that unconditional stability requires implicit numerical boundary conditions. An apparent counterexample was the space-time extrapolation considered by Gustafsson, Kreiss, and Sundstrom. In this paper we examine space (implicit) and space-time (explicit) extrapolation using normal mode analysis for a finite and infinite number of spatial mesh intervals. The results indicate that for unconditional stability with a finite number of spatial mesh intervals the numerical boundary conditions must be implicit.