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arXiv · 1612.00175

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

Sondre Tesdal Galtung
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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.

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Sondre Tesdal Galtung. 2016-12-01. A Convergent Crank-Nicolson Galerkin Scheme for the Benjamin-Ono Equation. https://doi.org/10.3934/dcds.2018051

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