A comparative study of high‐resolution methods for nonlinear hyperbolic problems

Abstract

AbstractThis paper reviews some of the established high‐resolution (HR) schemes for computing solutions to hyperbolic systems by means of a finite volume (FV) framework. The objective of this research is to assess the ability of various HR flux limiting techniques to capture intense shocks and discontinuities. A crucial design factor in developing these methods is selecting a suitable flux‐limiter, so a review of different flux‐limiters for standard explicit FV frameworks has also been included. A brief discussion of the HR total variation diminishing (TVD) numerical schemes, starting from some historical introduction, going on to exploring the design principles behind the HR, second‐order, oscillation free schemes, turning towards high order nonlinear techniques and ending with an overview of some of the most significant developments and applications in the last decades has been covered. Several one‐dimensional and two‐dimensional benchmark examples with severe and challenging wave configurations have been utilized to compare the various numerical approaches. These nonlinear test cases have been selected such that the computational setups are as basic as possible.