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Volume 41, Issue 3
Exponential Time Differencing-Padé Finite Element Method for Nonlinear Convection-Diffusion-Reaction Equations with Time Constant Delay

Haishen Dai, Qiumei Huang & Cheng Wang

DOI: 10.4208/jcm.2107-m2021-0051

J. Comp. Math., 41 (2023), pp. 370-394.

Published online: 2023-02

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  • Abstract

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 $L^2$ 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 $O(k^{3}+h^{r})$ (ETD3-Padé) or $O(k^{4}+h^{r})$ (ETD4-Padé) in the $L^2$ norm, respectively. Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.

  • Keywords

Nonlinear delayed convection diffusion reaction equations, ETD-Padé scheme, Lipshitz continuity, $L^2$ stability analysis, Convergence analysis and error estimate.

  • AMS Subject Headings

65N08, 65N12, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

daihaishen@emails.bjut.edu.cn (Haishen Dai)

qmhuang@bjut.edu.cn (Qiumei Huang)

cwang1@umassd.edu (Cheng Wang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-41-370, author = {Dai , HaishenHuang , Qiumei and Wang , Cheng}, title = {Exponential Time Differencing-Padé Finite Element Method for Nonlinear Convection-Diffusion-Reaction Equations with Time Constant Delay}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {3}, pages = {370--394}, abstract = {

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 $L^2$ 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 $O(k^{3}+h^{r})$ (ETD3-Padé) or $O(k^{4}+h^{r})$ (ETD4-Padé) in the $L^2$ norm, respectively. Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2107-m2021-0051}, url = {http://global-sci.org/intro/article_detail/jcm/21389.html} }
TY - JOUR T1 - Exponential Time Differencing-Padé Finite Element Method for Nonlinear Convection-Diffusion-Reaction Equations with Time Constant Delay AU - Dai , Haishen AU - Huang , Qiumei AU - Wang , Cheng JO - Journal of Computational Mathematics VL - 3 SP - 370 EP - 394 PY - 2023 DA - 2023/02 SN - 41 DO - http://doi.org/10.4208/jcm.2107-m2021-0051 UR - https://global-sci.org/intro/article_detail/jcm/21389.html KW - Nonlinear delayed convection diffusion reaction equations, ETD-Padé scheme, Lipshitz continuity, $L^2$ stability analysis, Convergence analysis and error estimate. AB -

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 $L^2$ 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 $O(k^{3}+h^{r})$ (ETD3-Padé) or $O(k^{4}+h^{r})$ (ETD4-Padé) in the $L^2$ norm, respectively. Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.

Haishen Dai, Qiumei Huang & Cheng Wang. (2023). Exponential Time Differencing-Padé Finite Element Method for Nonlinear Convection-Diffusion-Reaction Equations with Time Constant Delay. Journal of Computational Mathematics. 41 (3). 370-394. doi:10.4208/jcm.2107-m2021-0051
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