# Boundary-value problems and shoaling analysis for the BBM equation

## Bergen Open Research Archive

 dc.contributor.author Senthilkumar, Amutha dc.date.accessioned 2012-04-24T07:57:19Z dc.date.available 2012-04-24T07:57:19Z dc.date.issued 2012-02-20 dc.date.submitted 2012-02-20 en dc.identifier.uri http://hdl.handle.net/1956/5772 dc.description.abstract In this thesis we study the BBM equation en u_t+u_x+ \frac{3}{2}uu_x - \frac{1}{6} u_{xxt}= 0 which describes approximately the two-dimensional propagation of surface waves in a uniform horizontal channel containing an incompressible and inviscid fluid which in its undisturbed state has depth $h$. Here $u(x,t)$ represents the deviation of the water surface from its undisturbed position, and the flow is assumed to be irrotational. The BBM equation features a bounded dispersion relation (Benjamin, Bona and Mahony ). We utilize this boundedness to prove existence, uniqueness and regularity results for solutions of the BBM equation supplemented with an initial condition and various types of boundary conditions. We also treat the water-wave problem over an uneven bottom. In particular, we consider two different models for the propagation of long waves in channels of decreasing depth and we provide both analytical and numerical results for these models. For the numerical simulation we use a spectral discretization coupled with a four-stage Runge-Kutta time integration scheme. After verifying numerically that the algorithm is fourth-order accurate in time, we run the solitary wave with uneven bottom and examine how solitary waves respond to this non-uniform depth. Our numerical simulations are compared with previous numerical and experimental results of Madsen and Mei and Peregrine. dc.format.extent 747345 bytes en dc.format.mimetype application/pdf en dc.language.iso eng en dc.publisher The University of Bergen en dc.subject BBM equation en dc.title Boundary-value problems and shoaling analysis for the BBM equation en dc.type Master thesis eng dc.type.degree Master i Anvendt og beregningsorientert matematikk en dc.type.course MAB399 en dc.subject.nsi VDP::Mathematics and natural science: 400::Mathematics: 410::Analysis: 411 en dc.subject.archivecode Mastergrad en dc.subject.nus 753109 en dc.type.program MAMN-MAB en
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