Well-Balanced High-Order Finite Volume Methods for Systems of Balance Laws

2020 ◽  
Vol 82 (2) ◽  
Author(s):  
Manuel J. Castro ◽  
Carlos Parés
Keyword(s):  
Finite Volume ◽  
Finite Volume Methods ◽  
High Order ◽  
Balance Laws ◽  
Systems Of Balance Laws

scholarly journals High order well-balanced finite volume methods for multi-dimensional systems of hyperbolic balance laws

2021 ◽  
Vol 219 ◽  
pp. 104858
Author(s):  
Jonas P. Berberich ◽  
Praveen Chandrashekar ◽  
Christian Klingenberg
Keyword(s):  
Finite Volume ◽  
Finite Volume Methods ◽  
High Order ◽  
Balance Laws ◽  
Hyperbolic Balance Laws

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

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1799
Author(s):  
Irene Gómez-Bueno ◽  
Manuel Jesús Castro Díaz ◽  
Carlos Parés ◽  
Giovanni Russo
Keyword(s):  
High Order ◽  
Collocation Methods ◽  
Quadrature Formulas ◽  
Balance Laws ◽  
Source Terms ◽  
Time Step ◽  
Systems Of Balance Laws ◽  
Reconstruction Operator ◽  
Non Linear ◽  
Linear Problems

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.

High Order Finite Volume Methods Applied to Sediment Transport and Submarine Avalanches

2008 ◽  
pp. 247-258 ◽  
Author(s):  
D. Bresch ◽  
M. J. C. Díaz ◽  
E. D. Fernández-Nieto ◽  
A. M. Ferreiro ◽  
A. Mangeney
Keyword(s):  
Sediment Transport ◽  
Finite Volume ◽  
Finite Volume Methods ◽  
High Order
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