On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes

TL;DR

This paper investigates how the choice of the small parameter epsilon in third order WENO and CWENO reconstructions affects accuracy and convergence on non-uniform meshes, with practical implications for adaptive schemes.

Contribution

It extends previous uniform mesh studies to non-uniform meshes, demonstrating that setting epsilon proportional to local mesh size improves error and stability in high-order reconstructions.

Findings

Choosing epsilon as a function of local mesh size improves convergence.

Epsilon proportional to h_j or h_j^2 benefits error reduction.

Numerical tests confirm improved stability and accuracy on adaptive meshes.

Abstract

Third order WENO and CWENO reconstruction are widespread high order reconstruction techniques for numerical schemes for hyperbolic conservation and balance laws. In their definition, there appears a small positive parameter, usually called $\epsilon$, that was originally introduced in order to avoid a division by zero on constant states, but whose value was later shown to affect the convergence properties of the schemes. Recently, two detailed studies of the role of this parameter, in the case of uniform meshes, were published. In this paper we extend their results to the case of finite volume schemes on non-uniform meshes, which is very important for h-adaptive schemes, showing the benefits of choosing $\epsilon$ as a function of the local mesh size $h_j$. In particular we show that choosing $\epsilon=h_j^2$ or $\epsilon=h_j$ is beneficial for the error and convergence order, studying on several non-uniform grids the effect of this choice on the reconstruction error, on fully discrete schemes for the linear transport equation, on the stability of the numerical schemes. Finally we compare the different choices for $\epsilon$ in the case of a well-balanced scheme for the Saint-Venant system for shallow water flows and in the case of an h-adaptive scheme for nonlinear systems of conservation laws and show numerical tests for a two-dimensional generalisation of the CWENO reconstruction on locally adapted meshes.