Third-order nonoscillatory schemes for the Euler equations

Abstract

Two time-level third-order finite-difference shock-capturin g schemes based on applying the characteristic flux difference splitting to a modified flux that may have high-order accuracy and either monotonicity preserving or essentially nonoscillatory (ENO) property have been developed for the Euler equations of gas dynamics. Two ways to achieve high-order accuracy are described. One is based on upstream interpolation using Lagrange's formula with van Leer's smoothness monitors. The other is based on ENO interpolation using reconstruction via primitive function approaches. For multidimensional problems, dimensional splitting is adopted for explicit schemes, and standard alternating direction implicit approximate factorization procedures are used for implicit schemes. Numerical examples to illustrate the performance of the proposed schemes are given. ERY recently, a new class of uniformly high-order accurate, essentially nonoscillatory (ENO) schemes has been developed by Harten and Osher,1 Marten,2 and Harten et al.3'4 They presented a hierarchy of uniformly high-order accurate schemes that generalize Godunov's scheme5 and its secondorder accurate MUSCL extension6'7 and total variation diminishing (TVD) schemes8'9 to an arbitrary order of accuracy. In contrast to the earlier second-order TVD schemes that drop to first-order accuracy at local extrema and maintain second-order accuracy in smooth regions, the new ENO schemes are uniformly high-order accurate throughout, even at critical points. Theoretical results for the scalar coefficient case and numerical results for the scalar conservation law and for the one-dimensional Euler equations of gas dynamics have been reported with highly accurate results. Preliminary results for two-dimensional problems were reported in Ref. 2. In this paper, following van Leer,10'11 Harten,2 and Harten et al.3'4 we describe a class of third-order, essentially nonoscillatory shock-capturing schemes for the Euler equations of gas dynamics. These schemes are obtained by applying the characteristic flux-difference splitting to an appropriately modified flux vector that may have high-order accuracy and nonoscillatory property. Third-order schemes are constructed using upstream interpolation and ENO interpolation. Both explicit and implicit schemes are derived. Implicit schemes for two-dimensional Euler equations in general coordinates are also given. We apply the resulting schemes to simulate one-dimensional and two-dimensional unsteady shock tube flows and steady two-dimensional flows involving strong shocks to illustrate the performance of the schemes. In Sec. II, characteristic properties of the Euler equations related to the numerical advections are briefly summarized. Upstream interpolation using Lagrange's formula to generate a class of two time-level, 2p + 1 space point, (2p - l)thorder accurate schemes is described in Sec. III. In particular, a third-order scheme with smoothness monitors due to van Leer10 is described. In Sec. IV, the essentially nonoscillatory interpolation of Harten2 and Harten et al.3'4 using reconstruction via primitive function approach is employed to yield a third-order nonoscillatory scheme.