Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws

TL;DR

This paper introduces a new collocation Runge-Kutta based technique for solving high-order well-balanced methods for systems of balance laws, improving the accuracy and stability of numerical solutions in complex physical models.

Contribution

It develops a novel collocation RK approach to efficiently solve local non-linear problems in high-order well-balanced schemes, enhancing their applicability to various physical systems.

Findings

Effective in preserving stationary solutions in test cases

Improves accuracy of shallow water and gas dynamics simulations

Handles resonant problems with a new general technique

Abstract

In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations -- with and without Manning friction -- or Euler equations of gas dynamics with gravity effects.