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Analysis of the local discontinuous Galerkin method with generalized fluxes for one-dimensional nonlinear convection-diffusion systems

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Abstract

In this paper, we present optimal error estimates of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear convection-diffusion systems. The upwind-biased flux with the adjustable numerical viscosity for the convective term is chosen based on the local characteristic decomposition, which is helpful in resolving discontinuities of degenerate parabolic equations without enforcing any limiting procedure. For the diffusive term, a pair of generalized alternating fluxes are considered. By constructing and analyzing generalized Gauss-Radau projections with respect to different convective or diffusive terms, we derive optimal error estimates for nonlinear convection-diffusion systems with the symmetrizable flux Jacobian and fully nonlinear diffusive problems. Numerical experiments including long time simulations, different boundary conditions and degenerate equations with discontinuous initial data are provided to demonstrate the sharpness of theoretical results.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11971132 and 11971131), Natural Science Foundation of Heilongjiang Province (Grant No. YQ2021A002) and Guangdong Basic and Applied Basic Research Foundation (Grant No. 2020B1515310006). The authors thank the referees for their helpful suggestions that result in the improvement of the paper.

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  1. School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Hongjuan Zhang, Boying Wu & Xiong Meng

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  1. Hongjuan Zhang
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  2. Boying Wu
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  3. Xiong Meng
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Correspondence to Xiong Meng.

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Zhang, H., Wu, B. & Meng, X. Analysis of the local discontinuous Galerkin method with generalized fluxes for one-dimensional nonlinear convection-diffusion systems. Sci. China Math. 66, 2641–2664 (2023). https://doi.org/10.1007/s11425-022-2035-y

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