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

Galtung, Sondre Tesdal
Master thesis
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URI
http://hdl.handle.net/11250/2395092
Date
2016
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  • Institutt for matematiske fag [1434]
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.
Publisher
NTNU

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