Explicit Richardson extrapolation methods and their analyses for solving two-dimensional nonlinear wave equation with delays

Dingwen Deng; Jingliang Chen; Dingwen Deng; Jingliang Chen

doi:10.3934/nhm.2023017

Networks and Heterogeneous Media

2023, Volume 18, Issue 1: 412-443. doi: 10.3934/nhm.2023017

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Explicit Richardson extrapolation methods and their analyses for solving two-dimensional nonlinear wave equation with delays

Dingwen Deng ^,,
Jingliang Chen

College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang 330063, China

Received: 07 October 2022 Revised: 02 December 2022 Accepted: 21 December 2022 Published: 09 January 2023

In this study, we construct two explicit finite difference methods (EFDMs) for nonlinear wave equation with delay. The first EFDM is developed by modifying the standard second-order EFDM used to solve linear second-order wave equations, of which stable requirement is accepted. The second EFDM is devised for nonlinear wave equation with delay by extending the famous Du Fort-Frankel scheme initially applied to solve linear parabolic equation. The error estimations of these two EFDMs are given by applying the discrete energy methods. Besides, Richardson extrapolation methods (REMs), which are used along with them, are established to improve the convergent rates of the numerical solutions. Finally, numerical results confirm the accuracies of the algorithms and the correctness of theoretical findings. There are few studies on numerical solutions of wave equations with delay by Du Fort-Frankel-type scheme. Therefore, a main contribution of this study is that Du Fort-Frankel scheme and a corresponding new REM are constructed to solve nonlinear wave equation with delay, efficiently.
- Explicit finite difference methods,
- Du Fort-Frankel scheme,
- richardson extrapolation methods,
- convergence analyses,
- wave equation with delay
Citation: Dingwen Deng, Jingliang Chen. Explicit Richardson extrapolation methods and their analyses for solving two-dimensional nonlinear wave equation with delays[J]. Networks and Heterogeneous Media, 2023, 18(1): 412-443. doi: 10.3934/nhm.2023017

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Abstract

In this study, we construct two explicit finite difference methods (EFDMs) for nonlinear wave equation with delay. The first EFDM is developed by modifying the standard second-order EFDM used to solve linear second-order wave equations, of which stable requirement is accepted. The second EFDM is devised for nonlinear wave equation with delay by extending the famous Du Fort-Frankel scheme initially applied to solve linear parabolic equation. The error estimations of these two EFDMs are given by applying the discrete energy methods. Besides, Richardson extrapolation methods (REMs), which are used along with them, are established to improve the convergent rates of the numerical solutions. Finally, numerical results confirm the accuracies of the algorithms and the correctness of theoretical findings. There are few studies on numerical solutions of wave equations with delay by Du Fort-Frankel-type scheme. Therefore, a main contribution of this study is that Du Fort-Frankel scheme and a corresponding new REM are constructed to solve nonlinear wave equation with delay, efficiently.

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