# High order well-balanced finite volume methods for multi-dimensional systems of hyperbolic balance laws

## TL;DR

This paper presents a flexible finite volume framework that ensures exact preservation of known solutions for multi-dimensional hyperbolic balance laws, maintaining high order accuracy and applicable to various physical systems.

## Contribution

A novel general framework for constructing well-balanced finite volume methods that exactly follow known solutions in multiple dimensions, applicable to diverse hyperbolic systems.

## Key findings

- Successfully applied to compressible Euler equations with gravity
- Maintains high order accuracy while being well-balanced
- Effective for magnetohydrodynamics equations

## Abstract

We introduce a general framework for the construction of well-balanced finite volume methods for hyperbolic balance laws. We use the phrase well-balancing in a broader sense, since our proposed method can be applied to exactly follow any solution of any system of hyperbolic balance laws in multiple spatial dimensions and not only time independent solutions. The solution has to be known a priori, either as an analytical expression or as discrete data. The proposed framework modifies the standard finite volume approach such that the well-balancing property is obtained and in case the method is high order accurate, this is maintained under our modification. We present numerical tests for the compressible Euler equations with and without gravity source term and with different equations of state, and for the equations of compressible ideal magnetohydrodynamics.