Spectral methods on triangles and other domains
Journal of Scientific Computing
A spectral element basin model for the shallow water equations
Journal of Computational Physics
The spectral element method for the shallow water equations on the sphere
Journal of Computational Physics
Journal of Computational Physics
A Taylor-Galerkin method for simulating nonlinear dispersive water waves
Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations
Journal of Computational Physics
Journal of Scientific Computing
A Discontinuous Spectral Element Model for Boussinesq-Type Equations
Journal of Scientific Computing
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Two-dimensional shallow water systems are frequently used in engineering practice to model environmental flows. The benefit of these systems are that, by integration over the water depth, a two-dimensional system is obtained which approximates the full three-dimensional problem. Nevertheless, for most applications the need to propagate waves over many wavelengths means that the numerical solution of these equations remains particularly challenging. The requirement for an accurate discretization in geometrically complex domains makes the use of spectral/hp elements attractive. However, to allow for the possibility of discontinuous solutions the most natural formulation of the system is within a discontinuous Galerkin (DG) framework. In this paper we consider the unstructured spectral/hp DG formulation of (i) weakly nonlinear dispersive Boussinesq equations and (ii) nonlinear shallow water equations (a subset of the Boussinesq equations). Discretization of the Boussinesq equations involves resolving third order mixed derivatives. To efficiently handle these high order terms a new scalar formulation based on the divergence of the momentum equations is presented. Numerical computations illustrate the exponential convergence with regard to expansion order and finally, we simulate solitary wave solutions.