Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations

2015 ◽  
Vol 85 (299) ◽  
pp. 1225-1261 ◽  
Author(s):  
Xiong Meng ◽  
Chi-Wang Shu ◽  
Boying Wu
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Hyperbolic Equations ◽  
Optimal Error Estimates ◽  
Galerkin Methods ◽  
Optimal Error ◽  
Linear Hyperbolic Equations

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.

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