Date of this Version
JOURNAL OF COMPUTATIONAL PHYSICS 131, 216–232 (1997).
Two families of explicit and implicit compact high-resolution shock- capturing methods for the multidimensional compressible Euler equations for fluid dynamics are constructed. Some of these schemes can be fourth- and sixth-order accurate away from discontinuities. For the semi-discrete case their shock-capturing properties are of the total variation diminishing (TVD), total variation bounded ( TVB), total variation diminishing in the mean (TVDM), essentially nonoscillatory (ENO), or positive type of scheme for 1D scalar hyperbolic conservation laws and are positive schemes in more than one dimension. These higher-order compact schemes require the same grid stencil per spatial direction as their second-order noncompact cousins. The added terms over the second-order noncompact cousins involve extra vector additions but no added flux evaluations. Due to the construction, these schemes can be viewed as approximations to genuinely multidimensional schemes in the sense that they might produce less distortion in spherical type shocks and are more accurate in vortex type flows than schemes based purely on 1D extensions. The extension of these families of compact schemes to coupled nonlinear systems can be accomplished using the Roe approximate Riemann solver, the generalized Steger and Warming flux-vector splitting, or the van Leer type flux-vector splitting. Modification to existing high-resolution second- or third-order non-compact shock-capturing computer codes is minimal. High-resolution shock-capturing properties can also be achieved via a variant of the second-order Lax–Friedrichs numerical flux without the use of Riemann solvers for coupled nonlinear systems with comparable operations count to their classical shock-capturing counterparts. An efficient and compatible high-resolution shock-capturing filter for spatially fourth- and sixth-order classical compact and noncompact schemes is discussed. The simplest extension to viscous flows can be achieved by using the standard fourth-order compact or non-compact formula for the viscous terms.