Optimal error estimates of discontinuous Galerkin methods with generalized fluxes for wave equations on unstructured meshes

2020 ◽  
pp. 1
Author(s):  
Zheng Sun ◽  
Yulong Xing
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Wave Equations ◽  
Discontinuous Galerkin Methods ◽  
Unstructured Meshes ◽  
Optimal Error Estimates ◽  
Galerkin Methods ◽  
Optimal Error

Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using Pk elements

2020 ◽  
Vol 54 (2) ◽  
pp. 705-726
Author(s):  
Yong Liu ◽  
Chi-Wang Shu ◽  
Mengping Zhang
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Hyperbolic Equations ◽  
Optimal Error Estimates ◽  
Galerkin Methods ◽  
Optimal Error ◽  
Piecewise Polynomials ◽  
Cartesian Meshes ◽  
Theoretical Results

In this paper, we study the optimal error estimates of the classical discontinuous Galerkin method for time-dependent 2-D hyperbolic equations using Pk elements on uniform Cartesian meshes, and prove that the error in the L2 norm achieves optimal (k + 1)th order convergence when upwind fluxes are used. For the linear constant coefficient case, the results hold true for arbitrary piecewise polynomials of degree k ≥ 0. For variable coefficient and nonlinear cases, we give the proof for piecewise polynomials of degree k = 0, 1, 2, 3 and k = 2, 3, respectively, under the condition that the wind direction does not change. The theoretical results are verified by numerical examples.

scholarly journals DISCONTINUOUS GALERKIN METHODS FOR THE MULTI-DIMENSIONAL VLASOV–POISSON PROBLEM

2012 ◽  
Vol 22 (12) ◽  
pp. 1250042 ◽  
Author(s):  
BLANCA AYUSO DE DIOS ◽  
JOSÉ A. CARRILLO ◽  
CHI-WANG SHU
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Mixed Finite Element ◽  
Galerkin Approximation ◽  
Mixed Finite Element Method ◽  
Optimal Error Estimates ◽  
Galerkin Methods ◽  
Poisson System ◽  
Optimal Error ◽  
Poisson Problem

We introduce and analyze two new semi-discrete numerical methods for the multi-dimensional Vlasov–Poisson system. The schemes are constructed by combining a discontinuous Galerkin approximation to the Vlasov equation together with a mixed finite element method for the Poisson problem. We show optimal error estimates in the case of smooth compactly supported initial data. We propose a scheme that preserves the total energy of the system.

scholarly journals Central discontinuous Galerkin methods on overlapping meshes for wave equations

2020 ◽  
Author(s):  
Yong Liu ◽  
Jianfang Lu ◽  
Chi-Wang Shu ◽  
Mengping Zhang
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Wave Equations ◽  
Dispersion Analysis ◽  
Second Order ◽  
Optimal Order ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Optimal Order Of Convergence ◽  
Central Discontinuous Galerkin

In this paper, we study the central discontinuous Galerkin (DG) method on overlapping meshes for second order wave equations. We consider the first order hyperbolic system, which is equivalent to the second order scalar equation, and construct the corresponding central DG scheme. We then provide the stability analysis and the optimal error estimates for the proposed central DG scheme for one- and multi-dimensional cases with piecewise ${P}^k$ elements. The optimal error estimates are valid for uniform Cartesian meshes and polynomials of arbitrary degree $k\geq 0$. In particular, we adopt the techniques in \cite{liu2018central, liu2020pk} and obtain the local projection that is crucial in deriving the optimal order of convergence. The construction of the projection here is more challenging since the unknowns are highly coupled in the proposed scheme. Dispersion analysis is performed on the proposed scheme for one dimensional problems, indicating that the numerical solution with $P^1$ elements reaches its minimum with a suitable parameter in the dissipation term. Several numerical examples including accuracy tests and long time simulation are presented to validate the theoretical results.

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