Convergence of finite difference schemes for the Benjamin-Ono equation

TL;DR

This paper proves the convergence of finite difference schemes for the Benjamin-Ono equation, considering both full line and periodic cases, with numerical examples demonstrating the theoretical results.

Contribution

It provides the first rigorous convergence analysis of finite difference schemes for the Benjamin-Ono equation under regular initial data.

Findings

Finite difference schemes converge to classical solutions

Convergence established for both full line and periodic cases

Numerical examples confirm theoretical predictions

Abstract

In this paper, we analyze finite difference schemes for Benjamin-Ono equation, u_t = uu_x + Hu_{xx}, where H denotes the Hilbert transform. Both the decaying case on the full line and the periodic case are considered. If the initial data are sufficiently regular, fully discrete finite difference schemes shown to converge to a classical solution. Finally, the convergence is illustrated by several examples.