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

Irene Gómez-Bueno, Manuel Jesús Castro Díaz, Carlos Parés and Giovanni Russo
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Irene Gómez-Bueno: Departamento de Análisis, Estadística e I.O. y Matemática Aplicada, University of Málaga, Avda. Cervantes, 2, 29071 Málaga, Spain
Manuel Jesús Castro Díaz: Departamento de Análisis, Estadística e I.O. y Matemática Aplicada, University of Málaga, Avda. Cervantes, 2, 29071 Málaga, Spain
Carlos Parés: Departamento de Análisis, Estadística e I.O. y Matemática Aplicada, University of Málaga, Avda. Cervantes, 2, 29071 Málaga, Spain
Giovanni Russo: Dipartimento di Matematica ed Informatica, University of Catania, Viale Andrea Doria, 6, 95125 Catania, Italy

Mathematics, 2021, vol. 9, issue 15, 1-40

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—without and with Manning friction—or Euler equations of gas dynamics with gravity effects.

Keywords: systems of balance laws; well-balanced methods; finite volume methods; high order methods; reconstruction operators; collocation methods; shallow water equations; Euler equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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