Journal of Shanghai University(Natural Science Edition) ›› 2014, Vol. 20 ›› Issue (6): 757-768.doi: 10.3969/j.issn.1007-2861.2013.07.029

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Discontinuous Galerkin Spectral Element Methods for Nonlinear Reaction-Diffusion Equations

WU Hua, HAN Xiao-fei   

  1. College of Sciences, Shanghai University, Shanghai 200444, China
  • Received:2013-05-27 Online:2014-12-23 Published:2014-12-23

Abstract: Discontinuous Galerkin spectral element methods for nonlinear reaction-diffusion equations are considered. The schemes are basically in the Legendre-Galerkin form. The nonlinear term is interpolated through the Chebyshev-Gauss-Lobatto points. The jump term tackled by the central numerical flux in space variation. The fourth-order low-storage Runge-Kutta scheme is applied for time discrete inside each subinterval. The methods can be used to solve discontinuous initial value problems and implemented in a parallel way. Stability and the optimal rate of convergence in L2-norm for the semi-discrete scheme are shown using the approximate results of the Chebyshev-Gauss-Lobatto interpolation operator without a weight function. Numerical results for the continuous and discontinuous problems are given.

Key words: Chebyshev-Gauss-Lobatto interpolation, discontinuous Galerkin method, reaction-diffusion equation, spectral element method

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