Collocation Methods for High-Order Well-Balanced Methods for Systems of 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.

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Cited by in corpus (8)

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• Approximately well-balanced Discontinuous Galerkin methods using bases enriched with Physics-Informed Neural Networks
• A high-order, fully well-balanced, unconditionally positivity-preserving finite volume framework for flood simulations
• Well-balanced POD-based reduced-order models for finite volume approximation of hyperbolic balance laws
• A fully well-balanced hydrodynamic reconstruction
• Towards a fully well-balanced and entropy-stable scheme for the Euler equations with gravity: General equations of state