Convergence Rates of a Fully Discrete Galerkin Scheme for the Benjamin-Ono Equation

TL;DR

This paper analyzes the convergence rates of a fully discrete Galerkin scheme for the Benjamin-Ono equation, providing theoretical results under regular initial data and supporting them with numerical examples.

Contribution

It establishes theoretical convergence rates for the Galerkin scheme applied to the Benjamin-Ono equation under regular initial data.

Findings

Convergence rates are proven for the scheme with regular initial data.

Numerical examples confirm the theoretical convergence rates.

The scheme is shown to be locally convergent in finite time.

Abstract

We consider a recently proposed fully discrete Galerkin scheme for the Benjamin-Ono equation which has been found to be locally convergent in finite time for initial data in $L^2(\mathbb{R})$. By assuming that the initial data is sufficiently regular we obtain theoretical convergence rates for the scheme both in the full line and periodic versions of the associated initial value problem. These rates are illustrated with some numerical examples.