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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Irene Gómez-Bueno & Manuel Jesús Castro Díaz & Carlos Parés & Giovanni Russo, 2021. "Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws," Mathematics, MDPI, vol. 9(15), pages 1-40, July.
  • Handle: RePEc:gam:jmathe:v:9:y:2021:i:15:p:1799-:d:604202
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    Cited by:

    1. Caballero-Cárdenas, C. & Castro, M.J. & Morales de Luna, T. & Muñoz-Ruiz, M.L., 2023. "Implicit and implicit-explicit Lagrange-projection finite volume schemes exactly well-balanced for 1D shallow water system," Applied Mathematics and Computation, Elsevier, vol. 443(C).

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