Hybrid Explicit-Implicit, Unconditionally Stable Scheme for Unsteady Compressible Flows.

A variable explicit-implicit scheme was proposed by Collins et al. [(CCG) scheme] (Collins, J. E, Colella, P., and Glaz, H. M., "An Implicit-Explicit Eulerian Godunov Scheme for Compressible Flows" Journal of Computational Physics, Vol. 116, 1995, pp. 195-211) for one-dimensional hyperbolic systems. By suitable blending of explicit and implicit schemes, this approach aims to ensure a maximum norm diminishing (MND) property at all Courant numbers. Although proven for linear equations, however, the CCG scheme, fails to maintain this property unconditionally for non-linear equations and requires a specific time-step restriction. The remedy for this limitation of the CCG scheme is shown, and a new general method for the design of unconditionally MND hybrid schemes for nonlinear hyperbolic equations is proposed. An extension to the compressible Navier-Stokes equations is also discussed. The method is tested and verified by calculation of the Burgers equation on a highly stiff nonuniform grid. It is also applied to simulate the reflection of a normal shock wave from the end wall of a two-dimensional channel, which is attended with unsteady, viscous interaction processes including shock bifurcation. [ABSTRACT FROM AUTHOR]