A Well-Balanced Method for an Unstaggered Central Scheme, the one-space Dimensional Case

TL;DR

This paper introduces a new well-balanced finite volume scheme for hyperbolic balance laws, combining existing methods to improve stability and accuracy, validated through numerical experiments on Euler equations with gravity.

Contribution

A novel MUSCL scheme that integrates the Kurganov-Tadmor approach with the Deviation method, creating a well-balanced, essentially TVD scheme for hyperbolic balance laws.

Findings

The scheme is well-balanced for stationary solutions.

It is essentially TVD for scalar conservation laws.

Numerical experiments confirm effectiveness on Euler equations with gravity.

Abstract

In this paper, we propose a new MUSCL scheme by combining the ideas of the Kurganov and Tadmor scheme and the so-called Deviation method which results in a well-balanced finite volume method for the hyperbolic balance laws, by evolving the difference between the exact solution and a given stationary solution. After that, we derive a semi-discrete scheme from this new scheme and it can be shown to be essentially TVD when applied to a scalar conservation law. In the end, we apply and validate the developed methods by numerical experiments and solve classical problems featuring Euler equations with gravitational source term.