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

Authors :
Giovanni Russo
Irene Gómez-Bueno
Carlos Parés
Manuel Jesús Castro Díaz
Source :
Mathematics, Volume 9, Issue 15, Mathematics, Vol 9, Iss 1799, p 1799 (2021)
Publication Year :
2021
Publisher :
Multidisciplinary Digital Publishing Institute, 2021.

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.

Details

Language :
English
ISSN :
22277390
Database :
OpenAIRE
Journal :
Mathematics
Accession number :
edsair.doi.dedup.....26fff6493d616e4e5da81170e99bd389
Full Text :
https://doi.org/10.3390/math9151799