A well-balanced scheme for Euler equations with singular sources

TL;DR

This paper introduces a novel well-balanced numerical scheme for Euler equations with singular sources, using a structure-based Riemann solver to accurately capture wave phenomena and handle singularities effectively.

Contribution

A new approximate Riemann solver based on solution structure is proposed, improving the accuracy and robustness of numerical methods for Euler equations with singular sources.

Findings

The proposed scheme accurately captures wave structures in numerical tests.

It outperforms existing splitting and well-balanced schemes in extreme cases.

Numerical results demonstrate the scheme's effectiveness in handling singular sources.

Abstract

Numerical methods for the Euler equations with a singular source are discussed in this paper. The stationary discontinuity induced by the singular source and its coupling with the convection of fluid presents challenges to numerical methods. We show that the splitting scheme is not well-balanced and leads to incorrect results; in addition, some popular well-balanced schemes also give incorrect solutions in extreme cases due to the singularity of source. To fix such difficulties, we propose a solution-structure based approximate Riemann solver, in which the structure of Riemann solution is first predicted and then its corresponding approximate solver is given. The proposed solver can be applied to the calculation of numerical fluxes in a general finite volume method, which can lead to a new well-balanced scheme. Numerical tests show that the discontinuous Galerkin method based on the present approximate Riemann solver has the ability to capture each wave accurately.