High-order WENO finite-difference methods for hyperbolic nonconservative systems of Partial Differential Equations

TL;DR

This paper develops high-order WENO finite-difference methods tailored for nonconservative hyperbolic PDE systems, addressing the challenge of non-uniqueness in weak solutions by introducing a generalized flux approach.

Contribution

It introduces a novel strategy to extend WENO methods to nonconservative systems using a generalized flux function based on path families, enabling high-order accurate solutions.

Findings

Successfully applied to coupled Burgers system.

Achieved high-order accuracy in two-layer shallow water equations.

Preserves water-at-rest steady states.

Abstract

This work aims to extend the well-known high-order WENO finite-difference methods for systems of conservation laws to nonconservative hyperbolic systems. The main difficulty of these systems both from the theoretical and the numerical points of view comes from the fact that the definition of weak solution is not unique: according to the theory developed by Dal Maso, LeFloch, and Murat in 1995, it depends on the choice of a family of paths. A new strategy is introduced here that allows non-conservative products to be written as the derivative of a generalized flux function that is defined locally on the basis of the selected family of paths. WENO reconstructions are then applied to this generalized flux. Moreover, if a Roe linearization is available, the generalized flux function can be evaluated through matrix vector operations instead of path-integrals. Two different known techniques are used to extend the methods to problems with source terms and the well-balanced properties of the resulting schemes are studied. These numerical schemes are applied to a coupled Burgers system and to the two-layer shallow water equations in one- and two- dimensions to obtain high-order methods that preserve water-at-rest steady states.