An asymptotically stable compact upwind-biased finite-difference scheme for hyperbolic systems

Asymptotic stability of high-order finite-difference schemes for linear hyperbolic systems is investigated using the Nyquist criterion of linear-system theory. This criterion leads to a sufficient stability condition which is evaluated numerically. A fifth-order compact upwind-biased finite-difference scheme is developed which is asymptotically stable, according to the Nyquist criterion, for linear 2 × 2 systems. Moreover, this scheme is optimised with respect to its dispersion properties. The suitability of the scheme for discretisation of the compressible Navier-Stokes equations is demonstrated by computing inviscid and viscous eigensolutions of compressible Couette flow.