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Analysis of Fully Discrete Approximations for Dissipative Systems and Application to Time-Dependent Nonlocal Diffusion Problems

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Abstract

In this paper we first present stability and error analysis of the fully discrete numerical schemes for general dissipative systems, in which the implicit Runge–Kutta (IRK) method is adopted for time integration. Under suitable conditions on the IRK time stepping method that we refer as the total stability, a priori error estimates can be simultaneously obtained. Then we apply such time-marching techniques and analysis framework to one-dimensional time-dependent nonlocal diffusion problems, together with the discontinuous Galerkin method being used for spatial discretization. Unconditional stability of approximations of both primal and auxiliary variables and the priori error estimates for the corresponding fully discrete systems are proved, and the results indicate the schemes are asymptotically compatible. In addition, long time asymptotic behavior of the approximate solutions is also investigated. Various numerical experiments are finally performed to verify the theoretical results.

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Author information

Authors and Affiliations

  1. Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027, USA

    Qiang Du

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Lili Ju

  3. School of Mathematical Sciences, Ocean University of China, Qingdao, Shandong, 266100, China

    Lili Ju

  4. Applied and Computational Mathematics Division, Beijing Computational Science and Research Center, Beijing, 100193, China

    Jianfang Lu

  5. South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou, Guangdong, 510631, China

    Jianfang Lu

Authors
  1. Qiang Du
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  2. Lili Ju
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Correspondence to Jianfang Lu.

Additional information

Qiang Du’s work is partially supported by US National Science Foundation Grant DMS-1719699, US AFOSR MURI Center for Material Failure Prediction Through Peridynamics, and US Army Research Office MURI Grant W911NF-15-1-0562. Lili Ju’s research is partially supported by US National Science Foundation Grant DMS-1521965. Jianfang Lu’s work is partially supported by Postdoctoral Science Foundation of China Grant 2017M610749, and Natural Science Foundation of China Grants 11771035 and U153040.

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Du, Q., Ju, L. & Lu, J. Analysis of Fully Discrete Approximations for Dissipative Systems and Application to Time-Dependent Nonlocal Diffusion Problems. J Sci Comput 78, 1438–1466 (2019). https://doi.org/10.1007/s10915-018-0815-6

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  • DOI: https://doi.org/10.1007/s10915-018-0815-6

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