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

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 15
Author(s):  
Ernesto Guerrero Fernández ◽  
Cipriano Escalante ◽  
Manuel J. Castro Díaz
Keyword(s):  
Numerical Method ◽  
Discontinuous Galerkin ◽  
Stationary Solutions ◽  
High Order ◽  
General Strategy ◽  
Balance Laws ◽  
Smooth Solutions ◽  
Systems Of Balance Laws ◽  
Spurious Oscillations ◽  
Time Marching

This work introduces a general strategy to develop well-balanced high-order Discontinuous Galerkin (DG) numerical schemes for systems of balance laws. The essence of our approach is a local projection step that guarantees the exactly well-balanced character of the resulting numerical method for smooth stationary solutions. The strategy can be adapted to some well-known different time marching DG discretisations. Particularly, in this article, Runge–Kutta DG and ADER DG methods are studied. Additionally, a limiting procedure based on a modified WENO approach is described to deal with the spurious oscillations generated in the presence of non-smooth solutions, keeping the well-balanced properties of the scheme intact. The resulting numerical method is then exactly well-balanced and high-order in space and time for smooth solutions. Finally, some numerical results are depicted using different systems of balance laws to show the performance of the introduced numerical strategy.

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 well-balanced methods for systems of balance laws: a control-based approach

2021 ◽  
Vol 394 ◽  
pp. 125820
Author(s):  
Irene Gómez-Bueno ◽  
Manuel J. Castro ◽  
Carlos Parés
Keyword(s):  
High Order ◽  
Balance Laws ◽  
Systems Of Balance Laws

scholarly journals Análisis de Von Neumann para el métodoLocal Discontinuous Galerkin en 1D

2019 ◽  
Vol 37 (2) ◽  
pp. 199-217
Author(s):  
Paul Castillo ◽  
Sergio Gómez
Keyword(s):  
Discontinuous Galerkin ◽  
Numerical Experiments ◽  
High Order ◽  
Stability Conditions ◽  
Von Neumann ◽  
Theoretical Tool ◽  
Von Neumann Analysis ◽  
Time Marching ◽  
The Stability ◽  
Theoretical Results

Using the von Neumann analysis as a theoretical tool, an analysisof the stability conditions of some explicit time marching schemes, in com-bination with the spatial discretizationLocal Discontinuous Galerkin(LDG)and high order approximations, is presented. The stabilityconstant, CFL(Courant-Friedrichs-Lewy), is studied as a function of theLDG parametersand the approximation degree. A series of numerical experiments is carriedout to validate the theoretical results.

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