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.