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

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. (C) 2021 Elsevier Ltd. All rights reserved.

Automated summary

The study introduces a general framework for constructing well-balanced finite volume methods for hyperbolic balance laws. The proposed method can be applied to follow any solution of any system of hyperbolic balance laws in multiple spatial dimensions. By modifying the standard finite volume approach, the well-balancing property is achieved and maintained even with high order accuracy.