An inviscid regularization of hyperbolic conservation laws

Abstract

This article examines the utilization of a spatial averaging technique to the nonlinear terms of the partial differential equations as an inviscid shock-regularization of hyperbolic conservation laws. A central motivation is to promote the idea of applying filtering techniques such as the observable divergence method, rather than viscous regularization, as an alternative to the simulation of shocks and turbulence in inviscid flows while, on the other hand, generalizing and unifying previous mathematical and numerical analysis of the method applied to the one-dimensional Burgers’ and Euler equations. This article primarily concerns the mathematical analysis of the technique and examines two fundamental issues. The first is on the global existence and uniqueness of classical solutions for the regularization under the more general setting of quasilinear, symmetric hyperbolic systems in higher dimensions. The second issue examines one-dimensional scalar conservation laws and shows that the inviscid regularization method captures the unique entropy or physically relevant solution of the original, non-averaged problem as filtering vanishes.