Galerkin-finite element methods for the shallow water equations with characteristic boundary conditions

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We consider the shallow water equations in the supercritical and subcritical cases in one space variable, posed in a finite spatial interval with characteristic boundary conditions at the end points, which, as is well known, are transparent, i.e., allow outgoing waves to exit without generating spurious reflected waves. Assuming that the resulting initial--boundary-value problems have smooth solutions, we approximate them in space using standard Galerkin-finite element methods and prove |L²|-error estimates for thesemidiscrete problems on quasiuniform meshes. We discretize the problems inthe temporal variable using an explicit, fourth-order-accurate Runge-Kuttascheme and check, by means of numerical experiment, that the resulting fullydiscrete schemes have excellent absorption properties.