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Journal of Computational Mathematics ›› 2023, Vol. 41 ›› Issue (3): 370-394.DOI: 10.4208/jcm.2107-m2021-0051

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EXPONENTIAL TIME DIFFERENCING-PADé FINITE ELEMENT METHOD FOR NONLINEAR CONVECTION-DIFFUSION-REACTION EQUATIONS WITH TIME CONSTANT DELAY

Haishen Dai1, Qiumei Huang1, Cheng Wang2   

  1. 1. School of Mathematics, Faculty of Science, Beijing University of Technology, Beijing 100124, China;
    2. Department of Mathematics, University of Massachusetts, North Dartmouth, MA 02747, USA
  • Received:2020-09-14 Revised:2021-07-13 Published:2023-03-14
  • Contact: Qiumei Huang, Email:qmhuang@bjut.edu.cn
  • Supported by:
    This work is supported in part by NSFC 11971047 (Q. Huang) and NSF DMS-2012669 (C. Wang).
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Haishen Dai, Qiumei Huang, Cheng Wang. EXPONENTIAL TIME DIFFERENCING-PADé FINITE ELEMENT METHOD FOR NONLINEAR CONVECTION-DIFFUSION-REACTION EQUATIONS WITH TIME CONSTANT DELAY[J]. Journal of Computational Mathematics, 2023, 41(3): 370-394.

In this paper, ETD3-Padé and ETD4-Padé Galerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions. An ETD-based RK is used for time integration of the corresponding equation. To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator, the Padé approach is used for such an exponential operator approximation, which in turn leads to the corresponding ETD-Padé schemes. An unconditional L2 numerical stability is proved for the proposed numerical schemes, under a global Lipshitz continuity assumption. In addition, optimal rate error estimates are provided, which gives the convergence order of O(k3 + hr) (ETD3- Padé) or O(k4 + hr) (ETD4-Padé) in the L2 norm, respectively. Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.
Key words: Nonlinear delayed convection diffusion reaction equations, ETD-Padé scheme, Lipshitz continuity, L2 stability analysis, Convergence analysis and error estimate

CLC Number: 

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