On the Galerkin / finite-element method for the Serre equations

Abstract

A highly accurate numerical scheme is presented for the Serre system of partial differential equations, which models the propagation of dispersive shallow water waves in the fully-nonlinear regime. The fully-discrete scheme utilizes the Galerkin / finite-element method based on smooth periodic splines in space, and an explicit fourth-order Runge-Kutta method in time. Computations compared with exact solitary and cnoidal wave solutions show that the scheme achieves the optimal orders of accuracy in space and time. These computations also show that the stability of this scheme does not impose restrictive conditions on the temporal step size. In addition, solitary, cnoidal, and dispersive shock waves are studied in detail using this numerical scheme for the Serre system and compared with the 'classical' Boussinesq system for small-amplitude shallow water waves. The results show that the interaction of solitary waves in the Serre system is more inelastic. The efficacy of the numerical scheme for modeling dispersive shocks is shown by comparison with asymptotic results. These results have application to the modeling of shallow water waves of intermediate or large amplitude, such as occurs in the nearshore zone.

34 pages, 20 figures, 10 tables, 42 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/

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Cited by in corpus (14)

• Dispersive shock waves and modulation theory
• Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics
• Hamiltonian formulation of the extended Green-Naghdi equations
• A modified Galerkin/finite element method for the numerical solution of the Serre-Green-Naghdi system
• On the nonlinear dynamics of the traveling-wave solutions of the Serre system
• Solitary wave solutions and their interactions for fully nonlinear water waves with surface tension in the generalized Serre equations
• Mechanical balance laws for fully nonlinear and weakly dispersive water waves
• Symmetries, conservation laws, invariant solutions and difference schemes of the one-dimensional Green-Naghdi equations
• Extended Lagrangian approach for the numerical study of multidimensional dispersive waves: applications to the Serre-Green-Naghdi equations
• On weakly singular and fully nonlinear travelling shallow capillary-gravity waves in the critical regime
• On the multi-symplectic structure of the Serre-Green-Naghdi equations
• Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart
• Hamiltonian regularisation of shallow water equations with uneven bottom
• On time relaxed schemes and formulations for dispersive wave equations