1. Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws
- Author
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Giovanni Russo, Irene Gómez-Bueno, Carlos Parés, and Manuel Jesús Castro Díaz
- Subjects
finite volume methods ,Computer science ,General Mathematics ,systems of balance laws ,reconstruction operators ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Operator (computer programming) ,QA1-939 ,Computer Science (miscellaneous) ,0101 mathematics ,Engineering (miscellaneous) ,Shallow water equations ,Collocation ,shallow water equations ,Basis (linear algebra) ,Numerical analysis ,Euler equations ,high order methods ,Quadrature (mathematics) ,Burgers' equation ,010101 applied mathematics ,Law ,collocation methods ,symbols ,well-balanced methods ,Mathematics - 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.
- Published
- 2021
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