Stable Spectral-Volume Methods

TL;DR

This paper introduces a stabilized spectral-volume method based on entropy principles, adapting techniques from discontinuous-Galerkin methods to ensure entropy compliance and improve numerical stability for fluid dynamics equations.

Contribution

It presents a novel stabilization approach for spectral-volume methods using Dafermos' entropy rate criterion, including a new discrete filtering technique that guarantees entropy inequality satisfaction.

Findings

Method satisfies the entropy inequality.

Effective stabilization demonstrated on Burgers' and Euler equations.

Improved numerical stability for spectral-volume schemes.

Abstract

A novel approach for the stabilization of the Spectral-Volume (SV) method based on Dafermos' entropy rate criterion is presented. The method is an adaption of an already existing approach for the stabilization of the Discontinuous-Galerkin (DG) method. It employs the same estimates for the maximal possible entropy dissipation rate as the DG version. However, a new way to compute the discrete conservative filter had to be derived due to the differences of the underlying schemes. The resulting modified SV scheme even satisfies the entropy inequality. Tests are carried out for Burgers' equation and for the Euler equations of gas dynamics.