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

Abstract

In this thesis 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 numerical method for the Korteweg--de Vries equation, and utilises a commutator estimate related to a local smoothing effect to bound the $H^{\frac{1}{2}}$-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 $L^{2}$ and some $T > 0$. Finally we illustrate the convergence with some numerical examples.