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Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems

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Abstract

In this paper, we discuss the local discontinuous Galerkin methods coupled with two specific explicit-implicit-null time discretizations for solving one-dimensional nonlinear diffusion problems Ut = (a(U)Ux)x. The basic idea is to add and subtract two equal terms a0Uxx on the right-hand side of the partial differential equation, then to treat the term a0Uxx implicitly and the other terms (a(U)Ux)x − a0Uxx explicitly. We give stability analysis for the method on a simplified model by the aid of energy analysis, which gives a guidance for the choice of a0, i.e., a0 ≽ max{a(u)}/2 to ensure the unconditional stability of the first order and second order schemes. The optimal error estimate is also derived for the simplified model, and numerical experiments are given to demonstrate the stability, accuracy and performance of the schemes for nonlinear diffusion equations.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11601241, 11671199, 11571290 and 11672082), Natural Science Foundation of Jiangsu Province (Grant No. BK20160877), ARO (Grant No. W911NF-15-1-0226) and National Science Foundation of USA (Grant No. DMS-1719410).

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Authors and Affiliations

  1. School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

    Haijin Wang

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Qiang Zhang

  3. College of Shipbuilding Engineering, Harbin Engineering University, Harbin, 150001, China

    Shiping Wang

  4. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Chi-Wang Shu

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  1. Haijin Wang
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  2. Qiang Zhang
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  3. Shiping Wang
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  4. Chi-Wang Shu
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Correspondence to Chi-Wang Shu.

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Wang, H., Zhang, Q., Wang, S. et al. Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems. Sci. China Math. 63, 183–204 (2020). https://doi.org/10.1007/s11425-018-9524-x

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  • DOI: https://doi.org/10.1007/s11425-018-9524-x

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