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.

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.

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.

A Reconstructed Discontinuous Galerkin Method for the Euler Equations on Arbitrary Grids

2012 ◽  
Vol 12 (5) ◽  
pp. 1495-1519 ◽  
Author(s):  
Hong Luo ◽  
Luqing Luo ◽  
Robert Nourgaliev
Keyword(s):  
Galerkin Method ◽  
Euler Equations ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Least Squares Method ◽  
Polynomial Solution ◽  
Third Order ◽  
Von Neumann ◽  
Flow Problems ◽  
Dg Method

AbstractA reconstruction-based discontinuous Galerkin (RDG(P1P2)) method, a variant of P1P2 method, is presented for the solution of the compressible Euler equations on arbitrary grids. In this method, an in-cell reconstruction, designed to enhance the accuracy of the discontinuous Galerkin method, is used to obtain a quadratic polynomial solution (P2) from the underlying linear polynomial (P1) discontinuous Galerkin solution using a least-squares method. The stencils used in the reconstruction involve only the von Neumann neighborhood (face-neighboring cells) and are compact and consistent with the underlying DG method. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG(P1P2) method is third-order accurate, and outperforms the third-order DG method (DG(P2)) in terms of both computing costs and storage requirements.

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