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. The phrase well-balancing is used in a wider sense, since the method can be applied to exactly follow any solution of any system of hyperbolic balance laws in multiple spatial dimensions. 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. The potentially high order of accuracy of the method is maintained under the modification. We show 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. Different grid geometries and reconstruction methods are used. We demonstrate high order convergence numerically.