Parametric Interpolation Framework for 1-D Scalar Conservation Laws with Non-Convex Flux Functions

TL;DR

This paper introduces a high-order numerical framework for 1-D scalar conservation laws with non-convex flux functions, utilizing a generalized equal-area principle and area-preserving parametric interpolation to accurately locate shocks and satisfy entropy conditions.

Contribution

It presents a novel parametric interpolation method that ensures correct shock selection and high-order accuracy for non-convex flux functions, improving upon standard approaches.

Findings

Locates shock positions to fifth order in space

Conserves area exactly in numerical solutions

Numerically satisfies Oleinik entropy condition

Abstract

In this paper we present a novel framework for obtaining high order numerical methods for 1-D scalar conservation laws with non-convex flux functions. When solving Riemann problems, the Oleinik entropy condition, [16], is satisfied when the resulting shocks and rarefactions correspond to correct portions of the appropriate (upper or lower) convex envelope of the flux function. We show that the standard equal-area principle fails to select these solutions in general, and therefore we introduce a generalized equal-area principle which always selects the weak solution corresponding to the correct convex envelope. The resulting numerical scheme presented here relies on the area-preserving parametric interpolation framework introduced in [14] and locates shock position to fifth order in space, conserves area exactly and admits weak solutions which satisfy the Oleinik entropy condition numerically regardless of the initial states.