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A space-time discontinuous Galerkin method for Boussinesq-type equations. (English) Zbl 1410.76167
Summary: In this work, we propose a new high order accurate, fully implicit space-time discontinuous Galerkin (DG) method for advection-diffusion-dispersion equations, i.e., for so-called Korteweg-de-Vries-type equations. In particular, we focus on Boussinesq-type models for free surface flows, which are used for the modeling of water waves that travel in deep water, where the classical shallow water equations are not valid any more. Our method follows the ideas of the local DG method (LDG) for dispersion equations proposed by J. Yan and C.-W. Shu [SIAM J. Numer. Anal. 40, No. 2, 769–791 (2002; Zbl 1021.65050); J. Sci. Comput. 17, No. 1–4, 27–47 (2002; Zbl 1003.65115)], who used an explicit Runge-Kutta method to integrate their scheme in time. However, such explicit time integrators applied to dispersive equations imply a very severe restriction on the time step, which has to be taken proportional to the cube of the mesh spacing, and which therefore can make even one-dimensional computations prohibitively expensive on fine grids. For the scalar case and with some simplifying assumptions, the scheme presented in this paper can be proven to be unconditionally stable in \(L_{2}\) norm. Furthermore, our method is based directly on a space-time finite element formulation, which also provides a natural way to discretize third order dispersive terms that contain a mixed space-time derivative. Such terms appear frequently in the context of Boussinesq-type models for free surface flows. We show numerical convergence studies for linear scalar advection-diffusion-dispersion equations and furthermore, we also study the convergence of our method using solitary wave solutions of the nonlinear Boussinesq-type model of P. A. Madson et al. [“A new form of the Boussinesq equations with improved linear dispersion characteristics”, Coast. Eng. 15, No. 4, 371–388 (1991; doi:10.1016/0378-3839(91)90017-b)].

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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