Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws
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 modi...
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MDPI AG
2021-07-01
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| author | Irene Gómez-Bueno Manuel Jesús Castro Díaz Carlos Parés Giovanni Russo |
| author_facet | Irene Gómez-Bueno Manuel Jesús Castro Díaz Carlos Parés Giovanni Russo |
| author_sort | Irene Gómez-Bueno |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-10T09:12:06Z |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T09:12:06Z |
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| spelling | doaj.art-78ad0acb9c374c82984fa58f894b15bd2023-11-22T05:56:52ZengMDPI AGMathematics2227-73902021-07-01915179910.3390/math9151799Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance LawsIrene Gómez-Bueno0Manuel Jesús Castro Díaz1Carlos Parés2Giovanni Russo3Departamento de Análisis, Estadística e I.O. y Matemática Aplicada, University of Málaga, Avda. Cervantes, 2, 29071 Málaga, SpainDepartamento de Análisis, Estadística e I.O. y Matemática Aplicada, University of Málaga, Avda. Cervantes, 2, 29071 Málaga, SpainDepartamento de Análisis, Estadística e I.O. y Matemática Aplicada, University of Málaga, Avda. Cervantes, 2, 29071 Málaga, SpainDipartimento di Matematica ed Informatica, University of Catania, Viale Andrea Doria, 6, 95125 Catania, ItalyIn 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.https://www.mdpi.com/2227-7390/9/15/1799systems of balance lawswell-balanced methodsfinite volume methodshigh order methodsreconstruction operatorscollocation methods |
| spellingShingle | Irene Gómez-Bueno Manuel Jesús Castro Díaz Carlos Parés Giovanni Russo Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws Mathematics systems of balance laws well-balanced methods finite volume methods high order methods reconstruction operators collocation methods |
| title | Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws |
| title_full | Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws |
| title_fullStr | Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws |
| title_full_unstemmed | Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws |
| title_short | Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws |
| title_sort | collocation methods for high order well balanced methods for systems of balance laws |
| topic | systems of balance laws well-balanced methods finite volume methods high order methods reconstruction operators collocation methods |
| url | https://www.mdpi.com/2227-7390/9/15/1799 |
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