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

TL;DR

The paper tackles the challenge of producing high-order numerical methods for systems of balance laws that exactly preserve stationary solutions (well-balanced). It develops a collocation Runge-Kutta framework to solve the local nonlinear problems required to construct well-balanced reconstructions, employing Gauss-Legendre collocation to achieve high-order accuracy while controlling quadrature to maintain discrete equilibria. A general strategy for handling resonant (sonic) states is provided, ensuring solvability and selecting admissible solutions when flux Jacobians become singular. The methods are demonstrated on Burgers-type problems, shallow-water systems with gravity and Manning friction, and compressible Euler equations with gravity, showing improved accuracy and efficiency relative to prior control-based approaches and broad applicability to 1D balance laws.

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.