A class of high-order discontinuous-Galerkin methods satisfying infinitely many entropy conditions with provable error estimates and strong convergence for general nonlinear conservation laws

Yuanzhe Wei; Chi-Wang Shu

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arXiv · 2609.04687

A class of high-order discontinuous-Galerkin methods satisfying infinitely many entropy conditions with provable error estimates and strong convergence for general nonlinear conservation laws

Yuanzhe Wei·Chi-Wang Shu

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Abstract

We propose a novel framework for deriving semi-discrete discontinuous-Galerkin (DG) methods using operator semigroups for scalar conservation laws, then apply it to construct a class of high-order OFDG-type schemes [13] satisfying infinitely many local entropy inequalities with general E-fluxes on non-uniform meshes. Such schemes are further generalized to systems of conservation laws in any number of space dimensions by using entropy stable numerical fluxes in the sense of [1]. Finally, we prove optimal error estimates for smooth solutions to nonlinear scalar conservation laws, and prove strong convergence for discontinuous solutions to strictly convex conservation laws via compensated compactness.

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Yuanzhe Wei, Chi-Wang Shu. 2026-09-04. A class of high-order discontinuous-Galerkin methods satisfying infinitely many entropy conditions with provable error estimates and strong convergence for general nonlinear conservation laws. https://arxiv.org/abs/2609.04687

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