High-order well-balanced methods for systems of balance laws based on collocation RK ODE solvers

Abstract

The aim of this work is to develop high-order well-balanced schemes for 1d systems of balance laws. A general methodology for developing such numerical methods was proposed by two of the authors that requires the computation, at every cell and at every time step, of the stationary solution whose cell average is equal to the numerical approximation already obtained. Since solving these problems can be difficult and expensive, our goal is to introduce a general procedure that can be applied to any one-dimensional system of balance laws based on the application of collocation RK methods to approximate the stationary solution with given cell-average.