Stability Analysis of Inverse Lax-Wendroff Procedure for High order Compact Finite Difference Schemes

doi:10.1007/s42967-022-00228-8

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (1): 142-189.doi: 10.1007/s42967-022-00228-8

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Stability Analysis of Inverse Lax-Wendroff Procedure for High order Compact Finite Difference Schemes

Tingting Li¹, Jianfang Lu², Pengde Wang³

1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, Henan, China;
2. School of Mathematics, South China University of Technology, Guangzhou, 510641, Guangdong, China;
3. College of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450046, Henan, China

Received:2022-07-21 Revised:2022-10-04 Published:2024-04-16
Contact: Pengde Wang,E-mail:pengde_wang@yeah.net E-mail:pengde_wang@yeah.net
Supported by:
P. Wang’s research is partially supported by the NSFC grant 11801140, 12171177, the Young Elite Scientists Sponsorship Program by Henan Association for Science and Technology of China grant 2022HYTP0009, and the Program for Young Key Teacher of Henan Province of China grant 2021GGJS067.

Abstract

Abstract: This paper considers the finite difference (FD) approximations of diffusion operators and the boundary treatments for different boundary conditions. The proposed schemes have the compact form and could achieve arbitrary even order of accuracy. The main idea is to make use of the lower order compact schemes recursively, so as to obtain the high order compact schemes formally. Moreover, the schemes can be implemented efficiently by solving a series of tridiagonal systems recursively or the fast Fourier transform (FFT). With mathematical induction, the eigenvalues of the proposed differencing operators are shown to be bounded away from zero, which indicates the positive definiteness of the operators. To obtain numerical boundary conditions for the high order schemes, the simplified inverse Lax-Wendroff (SILW) procedure is adopted and the stability analysis is performed by the Godunov-Ryabenkii method and the eigenvalue spectrum visualization method. Various numerical experiments are provided to demonstrate the effectiveness and robustness of our algorithms.

Key words: Compact scheme, Diffusion operators, Inverse Lax-Wendroff (ILW), Fourier analysis, Eigenvalue analysis

Tingting Li, Jianfang Lu, Pengde Wang. Stability Analysis of Inverse Lax-Wendroff Procedure for High order Compact Finite Difference Schemes[J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 142-189.

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Stability Analysis of Inverse Lax-Wendroff Procedure for High order Compact Finite Difference Schemes

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