On the Nonlinearity of Modern Shock-Capturing Schemes

Abstract

In this paper we review the development of shock-capturing methods, paying special attention to the increasing nonlinearity in the design of the numerical schemes. We study the nature of the nonlinearity and examine its relation to upwind differencing. This nonlinearity is essential in the sense that analysis by local linearization is not justified and may even lead to incorrect conclusions; examples to demonstrate this point are given.