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Volume 18, Issue 3
Stability of High Order Finite Difference Schemes with Implicit-Explicit Time-Marching for Convection-Diffusion and Convection-Dispersion Equations

Meiqi Tan, Juan Cheng & Chi-Wang Shu

Int. J. Numer. Anal. Mod., 18 (2021), pp. 362-383.

Published online: 2021-03

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  • Abstract

The main purpose of this paper is to analyze the stability of the implicit-explicit (IMEX) time-marching methods coupled with high order finite difference spatial discretization for solving the linear convection-diffusion and convection-dispersion equations in one dimension. Both Runge-Kutta and multistep IMEX methods are considered. Stability analysis is performed on the above mentioned schemes with uniform meshes and periodic boundary condition by the aid of the Fourier method. For the convection-diffusion equations, the result shows that the high order IMEX finite difference schemes are subject to the time step restriction $∆t ≤$ max{$τ_0, c∆x$}, where $τ_0$ is a positive constant proportional to the diffusion coefficient and c is the Courant number. For the convection-dispersion equations, we show that the IMEX finite difference schemes are stable under the standard CFL condition $∆t ≤ c∆x$. Numerical experiments are also given to verify the main results.

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@Article{IJNAM-18-362, author = {Tan , MeiqiCheng , Juan and Shu , Chi-Wang}, title = {Stability of High Order Finite Difference Schemes with Implicit-Explicit Time-Marching for Convection-Diffusion and Convection-Dispersion Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2021}, volume = {18}, number = {3}, pages = {362--383}, abstract = {

The main purpose of this paper is to analyze the stability of the implicit-explicit (IMEX) time-marching methods coupled with high order finite difference spatial discretization for solving the linear convection-diffusion and convection-dispersion equations in one dimension. Both Runge-Kutta and multistep IMEX methods are considered. Stability analysis is performed on the above mentioned schemes with uniform meshes and periodic boundary condition by the aid of the Fourier method. For the convection-diffusion equations, the result shows that the high order IMEX finite difference schemes are subject to the time step restriction $∆t ≤$ max{$τ_0, c∆x$}, where $τ_0$ is a positive constant proportional to the diffusion coefficient and c is the Courant number. For the convection-dispersion equations, we show that the IMEX finite difference schemes are stable under the standard CFL condition $∆t ≤ c∆x$. Numerical experiments are also given to verify the main results.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/18730.html} }
TY - JOUR T1 - Stability of High Order Finite Difference Schemes with Implicit-Explicit Time-Marching for Convection-Diffusion and Convection-Dispersion Equations AU - Tan , Meiqi AU - Cheng , Juan AU - Shu , Chi-Wang JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 362 EP - 383 PY - 2021 DA - 2021/03 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/18730.html KW - Convection-diffusion equation, convection-dispersion equation, stability, IMEX, finite difference, Fourier method. AB -

The main purpose of this paper is to analyze the stability of the implicit-explicit (IMEX) time-marching methods coupled with high order finite difference spatial discretization for solving the linear convection-diffusion and convection-dispersion equations in one dimension. Both Runge-Kutta and multistep IMEX methods are considered. Stability analysis is performed on the above mentioned schemes with uniform meshes and periodic boundary condition by the aid of the Fourier method. For the convection-diffusion equations, the result shows that the high order IMEX finite difference schemes are subject to the time step restriction $∆t ≤$ max{$τ_0, c∆x$}, where $τ_0$ is a positive constant proportional to the diffusion coefficient and c is the Courant number. For the convection-dispersion equations, we show that the IMEX finite difference schemes are stable under the standard CFL condition $∆t ≤ c∆x$. Numerical experiments are also given to verify the main results.

Meiqi Tan, Juan Cheng & Chi-Wang Shu. (2021). Stability of High Order Finite Difference Schemes with Implicit-Explicit Time-Marching for Convection-Diffusion and Convection-Dispersion Equations. International Journal of Numerical Analysis and Modeling. 18 (3). 362-383. doi:
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