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Analysis of local discontinuous Galerkin methods with generalized numerical fluxes for linearized KdV equations


Authors: Jia Li, Dazhi Zhang, Xiong Meng and Boying Wu
Journal: Math. Comp. 89 (2020), 2085-2111
MSC (2010): Primary 65M12, 65M15, 65M60
DOI: https://doi.org/10.1090/mcom/3550
Published electronically: May 5, 2020
MathSciNet review: 4109561
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Abstract: In this paper, we consider the local discontinuous Galerkin (LDG) method using generalized numerical fluxes for linearized Korteweg-de Vries equations. In particular, since the dispersion term dominates, we are able to choose a downwind-biased flux in possession of the anti-dissipation property for the convection term to compensate the numerical dissipation of the dispersion term. This is beneficial to obtain a lower growth of the error and to accurately capture the exact solution without phase errors for long time simulations, when compared with traditional upwind and alternating fluxes. By establishing relations of three different numerical viscosity coefficients, we first show a uniform stability for the auxiliary variables and the prime variable as well as its time derivative. Moreover, the numerical initial condition is suitably chosen, which is the LDG approximation with the same fluxes to a steady-state equation. Finally, optimal error estimates are obtained by virtue of generalized Gauss-Radau projections. Numerical experiments are given to verify the theoretical results.


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Additional Information

Jia Li
Affiliation: School of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, People’s Republic of China
Email: jli@hit.edu.cn

Dazhi Zhang
Affiliation: School of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, People’s Republic of China
Email: zhangdazhi@hit.edu.cn

Xiong Meng
Affiliation: School of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, People’s Republic of China
Email: xiongmeng@hit.edu.cn

Boying Wu
Affiliation: School of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, People’s Repbulic of China
Email: mathwby@hit.edu.cn

DOI: https://doi.org/10.1090/mcom/3550
Keywords: Linearized KdV equations, local discontinuous Galerkin method, generalized numerical fluxes, numerical initial condition, anti-dissipation
Received by editor(s): October 5, 2018
Received by editor(s) in revised form: November 3, 2019
Published electronically: May 5, 2020
Additional Notes: The research of the first author was supported by NSFC grants 11971132 and 11501149.
The research of the second author was supported by National Key Research and Development Program of China with grant number 2017YFB1401801.
The research of the third author was supported by NSFC grants 11971132 and 11501149.
The third author is the corresponding author.
Article copyright: © Copyright 2020 American Mathematical Society