An integral equation method for the numerical analysis of gravity waves in a channel with free boundary

We consider the numerical solution of an evolution equation of second order on a boundary in R^2 that arises in the theory of gravity waves in fluids. First, by Laguerre transformation with respect to time we reduce the non-stationary problem to a sequence of operator equations. Then we handle the cross-section and longitudinal section cases of the channel in different ways. The potential theory for Laplace equation in a bounded domain with corners and the Green's function technique in a semi-infinite domain lead to systems of boundary integral equations of the second kind. For a full discretization we use a Nystrom method. In the case of cross-section the trigonometrical quadrature rules in combination with special mesh grading transformation for the accounting of densities singularities in corners are applied. In the longitudinal section case the quadratures are based on sinc-approximation. The results of numerical experiments are presented.

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