A novel linearized and momentum‐preserving Fourier pseudo‐spectral scheme for the Rosenau‐Korteweg de Vries equation. Issue 2 (23rd November 2022)

• Record Type: Journal Article
• Authors: Jiang, Chaolong; Cui, Jin; Cai, Wenjun; Wang, Yushun
• Is Part Of: Numerical methods for partial differential equations. Volume 39:Issue 2(2023)
• Page Start: 1558
• Page End: 1582
• Publication Date: 2022-11-23
• Subjects: Fourier pseudo‐spectral method -- momentum‐preserving scheme -- priori estimate -- Rosenau‐KdV equation; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22945
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract: In this article, we design a novel linearized and momentum‐preserving Fourier pseudo‐spectral scheme to solve the Rosenau‐Korteweg de Vries equation. With the aid of a new semi‐norm equivalence between the Fourier pseudo‐spectral method and the finite difference method, a prior bound of the numerical solution in discrete L ∞ $$ {L}^{\infty } $$ ‐norm is obtained from the discrete momentum conservation law. Subsequently, based on the energy method and the bound of the numerical solution, we show that, without any restriction on the mesh ratio, the scheme is convergent with order O N − s + τ 2 $$ O\left({N}^{-s}+{\tau}^2\right) $$ in discrete L ∞ $$ {L}^{\infty } $$ ‐norm, where N $$ N $$ is the number of collocation points used in the spectral method and τ $$ \tau $$ is the time step. Numerical results are addressed to confirm our theoretical analysis.