A Convergent Crank-Nicolson Galerkin Scheme for the Benjamin-Ono Equation

TL;DR

This paper proves the convergence of a Crank-Nicolson Galerkin finite element scheme for the Benjamin-Ono equation, establishing strong convergence to weak solutions and demonstrating the method with numerical examples.

Contribution

It introduces a convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation, extending techniques from Korteweg-de Vries equations and utilizing local smoothing effects.

Findings

Scheme converges strongly in L^2 for initial data in L^2.

Convergence proof uses local smoothing effect and recent KdV results.

Numerical examples illustrate the effectiveness of the scheme.

Abstract

In this paper we prove the convergence of a Crank-Nicolson type Galerkin finite element scheme for the initial value problem associated to the Benjamin-Ono equation. The proof is based on a recent result for a similar discrete scheme for the Korteweg-de Vries equation and utilizes a local smoothing effect to bound the $H^{1/2}$-norm of the approximations locally. This enables us to show that the scheme converges strongly in $L^{2}(0,T;L^{2}_{\text{loc}}(\mathbb{R}))$ to a weak solution of the equation for initial data in $L^{2}(\mathbb{R})$ and some $T > 0$. Finally we illustrate the method with some numerical examples.