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Volume 42, Issue 1
Analysis of the Implicit-Explicit Ultra-Weak Discontinuous Galerkin Method for Convection-Diffusion Problems

Haijin Wang, Anping Xu & Qi Tao

DOI: 10.4208/jcm.2202-m2021-0290

J. Comp. Math., 42 (2024), pp. 1-23.

Published online: 2023-12

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  • Abstract

In this paper, we first present the optimal error estimates of the semi-discrete ultra-weak discontinuous Galerkin method for solving one-dimensional linear convection-diffusion equations. Then, coupling with a kind of Runge-Kutta type implicit-explicit time discretization which treats the convection term explicitly and the diffusion term implicitly, we analyze the stability and error estimates of the corresponding fully discrete schemes. The fully discrete schemes are proved to be stable if the time-step $\tau ≤ \tau_0,$ where $\tau_0$ is a constant independent of the mesh-size $h.$ Furthermore, by the aid of a special projection and a careful estimate for the convection term, the optimal error estimate is also obtained for the third order fully discrete scheme. Numerical experiments are displayed to verify the theoretical results.

  • Keywords

The ultra-weak discontinuous Galerkin method, Convection-diffusion, Implicit-explicit time discretization, Stability, Error estimate.

  • AMS Subject Headings

65M12, 65M15, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-1, author = {Wang , HaijinXu , Anping and Tao , Qi}, title = {Analysis of the Implicit-Explicit Ultra-Weak Discontinuous Galerkin Method for Convection-Diffusion Problems}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {42}, number = {1}, pages = {1--23}, abstract = {

In this paper, we first present the optimal error estimates of the semi-discrete ultra-weak discontinuous Galerkin method for solving one-dimensional linear convection-diffusion equations. Then, coupling with a kind of Runge-Kutta type implicit-explicit time discretization which treats the convection term explicitly and the diffusion term implicitly, we analyze the stability and error estimates of the corresponding fully discrete schemes. The fully discrete schemes are proved to be stable if the time-step $\tau ≤ \tau_0,$ where $\tau_0$ is a constant independent of the mesh-size $h.$ Furthermore, by the aid of a special projection and a careful estimate for the convection term, the optimal error estimate is also obtained for the third order fully discrete scheme. Numerical experiments are displayed to verify the theoretical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2202-m2021-0290}, url = {http://global-sci.org/intro/article_detail/jcm/22150.html} }
TY - JOUR T1 - Analysis of the Implicit-Explicit Ultra-Weak Discontinuous Galerkin Method for Convection-Diffusion Problems AU - Wang , Haijin AU - Xu , Anping AU - Tao , Qi JO - Journal of Computational Mathematics VL - 1 SP - 1 EP - 23 PY - 2023 DA - 2023/12 SN - 42 DO - http://doi.org/10.4208/jcm.2202-m2021-0290 UR - https://global-sci.org/intro/article_detail/jcm/22150.html KW - The ultra-weak discontinuous Galerkin method, Convection-diffusion, Implicit-explicit time discretization, Stability, Error estimate. AB -

In this paper, we first present the optimal error estimates of the semi-discrete ultra-weak discontinuous Galerkin method for solving one-dimensional linear convection-diffusion equations. Then, coupling with a kind of Runge-Kutta type implicit-explicit time discretization which treats the convection term explicitly and the diffusion term implicitly, we analyze the stability and error estimates of the corresponding fully discrete schemes. The fully discrete schemes are proved to be stable if the time-step $\tau ≤ \tau_0,$ where $\tau_0$ is a constant independent of the mesh-size $h.$ Furthermore, by the aid of a special projection and a careful estimate for the convection term, the optimal error estimate is also obtained for the third order fully discrete scheme. Numerical experiments are displayed to verify the theoretical results.

Wang , HaijinXu , Anping and Tao , Qi. (2023). Analysis of the Implicit-Explicit Ultra-Weak Discontinuous Galerkin Method for Convection-Diffusion Problems. Journal of Computational Mathematics. 42 (1). 1-23. doi:10.4208/jcm.2202-m2021-0290
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