Conservative discontinuous Galerkin methods for the nonlinear Serre equations

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Department of Mathematical Sciences


In this paper, we develop three conservative discontinuous Galerkin (DG) schemes for the one-dimensional nonlinear dispersive Serre equations, including two conserved schemes for the equations in conservative form and a Hamiltonian conserved scheme for the equations in non-conservative form. One of the schemes owns the well-balanced property via constructing a high order approximation to the source term for the Serre equations with a non-flat bottom topography. By virtue of the Hamiltonian structure of the Serre equations, we introduce an Hamiltonian invariant and then develop a DG scheme which can preserve the discrete version of such an invariant. Numerical experiments in different cases are performed to verify the accuracy and capability of these DG schemes for solving the Serre equations.

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© 2020 Elsevier Inc. Publisher’s version of record: https://doi.org/10.1016/j.jcp.2020.109729

Publication Title

Journal of Computational Physics