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

TL;DR

This paper introduces a second-order well-balanced central scheme for 2D hyperbolic conservation laws, combining the deviation method with the KT scheme to achieve accuracy, stability, and non-oscillatory solutions.

Contribution

It presents a novel combination of well-balanced deviation and KT schemes for 2D hyperbolic systems, ensuring accuracy and stability without Riemann solvers.

Findings

Scheme is non-oscillatory in numerical experiments.

Achieves second-order accuracy and well-balanced property.

Demonstrates effectiveness on Euler equations with gravitational source term.

Abstract

We develop a second-order accurate central scheme for the two-dimensional hyperbolic system of in-homogeneous conservation laws. The main idea behind the scheme is that we combine the well-balanced deviation method with the Kurganov-Tadmor (KT) scheme. The approach satisfies the well-balanced property and retains the advantages of KT scheme: Riemann-solver-free and the avoidance of oversampling on the regions between Riemann-fans. The scheme is implemented and applied to a number of numerical experiments for the Euler equations with gravitational source term and the results are non-oscillatory. Based on the same idea, we construct a semi-discrete scheme where we combine the above two methods and illustrate the maximum principle.