Kinetic functions for nonclassical shocks, entropy stability, and discrete summation by parts

TL;DR

This paper investigates numerical schemes for hyperbolic conservation laws with non-convex flux, focusing on kinetic functions that characterize nonclassical shocks, and demonstrates that entropy stability alone does not guarantee unique or bounded solutions.

Contribution

It computes kinetic functions for various numerical schemes, showing entropy stability does not ensure solution uniqueness or boundedness, and designs entropy-dissipative schemes for measure-valued solutions.

Findings

Entropy stability does not imply solution uniqueness.

Numerical schemes can produce nonclassical shocks with entropy dissipation.

Designed entropy-dissipative schemes for measure-valued solutions with delta shocks.

Abstract

We study nonlinear hyperbolic conservation laws with non-convex flux in one space dimension and, for a broad class of numerical methods based on summation by parts operators, we compute numerically the kinetic functions associated with each scheme. As established by LeFloch and collaborators, kinetic functions (for continuous or discrete models) uniquely characterize the macro-scale dynamics of small-scale dependent, undercompressive, nonclassical shock waves. We show here that various entropy-dissipative numerical schemes can yield nonclassical solutions containing classical shocks, including Fourier methods with (super-) spectral viscosity, finite difference schemes with artificial dissipation, discontinuous Galerkin schemes with or without modal filtering, and TeCNO schemes. We demonstrate numerically that entropy stability does not imply uniqueness of the limiting numerical solutions for scalar conservation laws in one space dimension, and we compute the associated kinetic functions in order to distinguish between these schemes. In addition, we design entropy-dissipative schemes for the Keyfitz-Kranzer system whose solutions are measures with delta shocks. This system illustrates the fact that entropy stability does not imply boundedness under grid refinement.