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Superconvergence of Direct Discontinuous Galerkin Methods:Eigen-Structure Analysis Based on Fourier Approach

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This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes.Based on the eigen-structure analysis,we carry out error estimates of the DDG solutions,which can be decomposed into three parts:(ⅰ)dissipation errors of the physically relevant eigenvalue,which grow linearly with the time and are of order 2k for Pk(k=2,3)approximations;(ⅱ )projection error from a special projection of the exact solution,which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue;(ⅲ)dissipative errors of non-physically relevant eigenvalues,which decay exponentially with respect to the spatial mesh size Δx.We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree P2 approximations,but are not for odd degree P3 approximations.Numerical experiments are provided to verify the theoretical results.

Direct discontinuous Galerkin(DDG)method with interface correctionSymmetric DDG methodSuperconvergenceFourier analysisEigen-structure

Xuechun Liu、Haijin Wang、Jue Yan、Xinghui Zhong

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School of Mathematical Sciences,Zhejiang University,Hangzhou 310027,Zhejiang,China

School of Science,Nanjing University of Posts and Telecommunications,Nanjing 210023,Jiangsu,China

Department of Mathematics,Iowa State University,Ames 50011,IA,USA

国家自然科学基金国家自然科学基金Natural Science Foundation for Colleges Universities of Jiangsu Province of China国家自然科学基金Simons Foundation国家自然科学基金国家自然科学基金

118714281207121420KJB110011DMS-16203356377161187142812272347

2024

10.1007/s42967-022-00246-6

应用数学与计算数学学报
上海大学

应用数学与计算数学学报

影响因子：0.165
ISSN：1006-6330
年,卷(期)：2024.6(1)
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