Communications on Applied Mathematics and Computation >

2024 , Vol. 6 >Issue 1: 257 - 278

DOI: https://doi.org/10.1007/s42967-022-00246-6

ORIGINAL PAPERS

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

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  • 1. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, Zhejiang, China;
    2. School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, Jiangsu, China;
    3. Department of Mathematics, Iowa State University, Ames, 50011, IA, USA

Received date: 2022-08-01

  Revised date: 2022-12-18

  Online published: 2024-04-16

Supported by

Research work of H. Wang is partially supported by the National Natural Science Foundation of China (Grant Nos. 11871428 and 12071214), and the Natural Science Foundation for Colleges and Universities of Jiangsu Province of China (Grant No. 20KJB110011). Research work of J. Yan is supported by the National Science Foundation (Grant No. DMS-1620335) and the Simons Foundation (Grant No. 637716). Research work of X. Zhong is partially supported by the National Natural Science Foundation of China (Grant Nos. 11871428 and 12272347).

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Abstract

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 △x. 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.

Key words： Direct discontinuous Galerkin (DDG) method with interface correction; Symmetric DDG method; Superconvergence; Fourier analysis; Eigen-structure

Cite this article

Xuechun Liu, Haijin Wang, Jue Yan, Xinghui Zhong . Superconvergence of Direct Discontinuous Galerkin Methods: Eigen-structure Analysis Based on Fourier Approach[J]. Communications on Applied Mathematics and Computation, 2024 , 6(1) : 257 -278 . DOI: 10.1007/s42967-022-00246-6

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