Optimal error estimate of two linear and momentum-preserving Fourier pseudo-spectral schemes for the RLW equation

TL;DR

This paper introduces two new linear-implicit, momentum-preserving Fourier pseudo-spectral schemes for the RLW equation, providing optimal error estimates and demonstrating their effectiveness through numerical tests.

Contribution

The paper develops and analyzes two novel linear-implicit, momentum-preserving Fourier pseudo-spectral schemes with proven stability and optimal error bounds for the RLW equation.

Findings

Both schemes conserve discrete momentum.

Error estimates of order $ au^2 + N^{-r}$ are established.

Numerical tests confirm theoretical accuracy and efficiency.

Abstract

In this paper, two novel linear-implicit and momentum-preserving Fourier pseudo-spectral schemes are proposed and analyzed for the regularized long-wave equation. The numerical methods are based on the blend of the Fourier pseudo-spectral method in space and the linear-implicit Crank-Nicolson method or the leap-frog scheme in time. The two fully discrete linear schemes are shown to possess the discrete momentum conservation law, and the linear systems resulting from the schemes are proved uniquely solvable. Due to the momentum conservative property of the proposed schemes, the Fourier pseudo-spectral solution is proved to be bounded in the discrete $L^{\infty}$ norm. Then by using the standard energy method, both the linear-implicit Crank-Nicolson momentum-preserving scheme and the linear-implicit leap-frog momentum-preserving scheme are shown to have the accuracy of $\mathcal{O}(\tau^2+N^{-r})$ in the discrete $L^{\infty}$ norm without any restrictions on the grid ratio, where $N$ is the number of nodes and $\tau$ is the time step size. Numerical examples are carried out to verify the correction of the theory analysis and the efficiency of the proposed schemes.