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

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 H1/2 -norm of the approximations locally. This enables us to show that the scheme converges strongly in L2 (0, T; L2 loc(R)) to a weak solution of the equation for initial data in L2 (R) and some T > 0. Finally we illustrate the method with some numerical examples