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

2020 ◽  
pp. 69-75 ◽  
Author(s):  
Manuel J. Castro ◽  
Carlos Parés
Keyword(s):  
High Order ◽  
Balance Laws ◽  
High Order Methods ◽  
Systems Of 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 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 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.

High order boundary treatment for high order methods in computational fluid dynamics

2016 ◽  
Author(s):  
André Ribeiro de Barros Aguiar ◽  
Carlos Breviglieri ◽  
Fábio Mallaco Moreira ◽  
Eduardo Jourdan ◽  
João Luiz F. Azevedo
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
High Order ◽  
High Order Methods ◽  
Boundary Treatment

High-Order Methods For Wave Propagation

2008 ◽  
Author(s):  
Miguel R. Visbal ◽  
Scott E. Sherer ◽  
Michael D. White
Keyword(s):  
Wave Propagation ◽  
High Order ◽  
High Order Methods

Numerical simulation of quench propagation at early phase by high-order methods

Cryogenics ◽  
2006 ◽  
Vol 46 (7-8) ◽  
pp. 589-596
Author(s):  
Shaolin Mao ◽  
Cesar A. Luongo ◽  
David A. Kopriva
Keyword(s):  
Numerical Simulation ◽  
Early Phase ◽  
High Order ◽  
High Order Methods ◽  
Quench Propagation
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