Numerical solutions of the inviscid Burgers equation with the discontinuous Galerkin method

2021 ◽  
Vol 149 (4) ◽  
pp. A139-A139
Author(s):  
Vaughn E. Holmes ◽  
Robert J. McGough
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Inviscid Burgers Equation

An Application of Discontinuous Galerkin Method for Blast Wave Simulations

2010 ◽  
Author(s):  
Emre Alpman
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Blast Wave ◽  
Finite Volume ◽  
Discontinuous Galerkin Method ◽  
Numerical Solutions ◽  
Blast Waves ◽  
Strong Discontinuities ◽  
Runge Kutta ◽  
Volume Method

An implementation of Runge-Kutta Discontinuous Galerkin method to an in-house computational fluid dynamics code capable of simulating blast waves was performed. The resultant code was tested for two shock tube problems with moderately and extremely strong discontinuities. Numerical solutions were compared with predictions of a finite volume method and exact solutions. It was observed that when there are extreme discontinuities in the flowfield, as in the case of blast waves, the limiter adopted for solution clearly affects the overall quality of the predictions. An alternative limiting technique was proposed and tested to improve the results obtained. Blast wave predictions using Runge-Kutta Discontinuous Galerkin method with the alternative limiting technique yielded slightly stronger and faster moving shock waves compared to finite volume solutions.

scholarly journals A Discontinuous Galerkin Method for Ideal Two-Fluid Plasma Equations

2011 ◽  
Vol 9 (2) ◽  
pp. 240-268 ◽  
Author(s):  
John Loverich ◽  
Ammar Hakim ◽  
Uri Shumlak
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Numerical Solutions ◽  
Three Dimensions ◽  
Third Order ◽  
Small Gauge ◽  
Electron Acoustic ◽  
Two Fluid ◽  
Two Fluid Plasma

AbstractA discontinuous Galerkin method for the ideal 5 moment two-fluid plasma system is presented. The method uses a second or third order discontinuous Galerkin spatial discretization and a third order TVD Runge-Kutta time stepping scheme. The method is benchmarked against an analytic solution of a dispersive electron acoustic square pulse as well as the two-fluid electromagnetic shock [1] and existing numerical solutions to the GEM challenge magnetic reconnection problem [2]. The algorithm can be generalized to arbitrary geometries and three dimensions. An approach to maintaining small gauge errors based on error propagation is suggested.

A Sub-Grid Structure Enhanced Discontinuous Galerkin Method for Multiscale Diffusion and Convection-Diffusion Problems

2013 ◽  
Vol 14 (2) ◽  
pp. 370-392 ◽  
Author(s):  
Eric T. Chung ◽  
Wing Tat Leung
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Heterogeneous Media ◽  
Numerical Solutions ◽  
Solution Space ◽  
Convection Diffusion ◽  
Diffusion And Convection ◽  
Convection Dominated ◽  
Diffusion Problems

AbstractIn this paper, we present an efficient computational methodology for diffusion and convection-diffusion problems in highly heterogeneous media as well as convection-dominated diffusion problem. It is well known that the numerical computation for these problems requires a significant amount of computer memory and time. Nevertheless, the solutions to these problems typically contain a coarse component, which is usually the quantity of interest and can be represented with a small number of degrees of freedom. There are many methods that aim at the computation of the coarse component without resolving the full details of the solution. Our proposed method falls into the framework of interior penalty discontinuous Galerkin method, which is proved to be an effective and accurate class of methods for numerical solutions of partial differential equations. A distinctive feature of our method is that the solution space contains two components, namely a coarse space that gives a polynomial approximation to the coarse component in the traditional way and a multiscale space which contains sub-grid structures of the solution and is essential to the computation of the coarse component. In addition, stability of the method is proved. The numerical results indicate that the method can accurately capture the coarse behavior of the solution for problems in highly heterogeneous media as well as boundary and internal layers for convection-dominated problems.

scholarly journals Application of discontinuous Galerkin method to modeling of two-dimensional flows of a multicomponent ideal gases mixture using local adaptive mesh refinement

2019 ◽  
Vol 21 (2) ◽  
pp. 244-258
Author(s):  
Ruslan V. Zhalnin ◽  
Victor F. Masyagin ◽  
Elizaveta E. Peskova ◽  
Vladimir F. Tishkin
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Numerical Algorithm ◽  
Adaptive Mesh Refinement ◽  
Numerical Solutions ◽  
Point Problem ◽  
Gas Dynamics Equations ◽  
Ideal Gases ◽  
Accuracy Order

In this article a numerical algorithm is developed for solving of gas dynamics equations for a mixture of ideal gases on adaptive locally refined grids. The algorithm is based on discontinuous Galerkin method. To avoid the appearance of non-physical oscillations near the discontinuities, the Barth-Jespersen limiter is used. The numerical algorithm is based on the data structure and algorithms of the p4est library. In present work the numerical simulation of one problem of Richtmyer-Meshkov instability development is considered and the triple point problem is solved using the developed numerical algorithm of high accuracy order. The obtained results are in good agreement with the well-known numerical solutions. The pictures plotted basing on the solution describe in detail the dynamics of the complex flows under consideration.

SELECTION OF VISCOUS TERMS APPROXIMATION IN DISCONTINUOUS GALERKIN METHOD

2013 ◽  
Vol 44 (3) ◽  
pp. 327-354
Author(s):  
Aleksey Igorevich Troshin ◽  
Vladimir Viktorovich Vlasenko ◽  
Andrey Viktorovich Wolkov
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
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