In this paper, we determine the critical time, when a weak discontinuity in the shallow water equations culminates into a bore. Invariance group properties of the governing system of partial differential equations (PDEs), admitting Lie group of point transformations with commuting infinitesimal operators, are presented. Some appropriate canonical variables are characterized that transform equat...

The Euler’s equations describing the dynamics of capillarygravity water waves in two-dimensions are considered in the limits of small-amplitude and long-wavelength under appropriate boundary conditions. Using a double-series perturbation analysis, a general Boussinesq type of equation is derived involving the small-amplitude and longwavelength parameters. A recently introduced sixth-order Bouss...

Recently the shallow water magnetohydrodynamic ~SMHD! equations have been proposed for describing the dynamics of nearly incompressible conducting fluids for which the evolution is nearly two-dimensional ~2D! with magnetohydrostatic equilibrium in the third direction. In the present paper the properties of the SMHD equations as a nonlinear system of hyperbolic conservation laws are described. C...

We consider the Cacuhy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modelling of motions for shallow water with free surface in a rotating sub-domain [18]. The global existence of the solution in the space of Besov type is shown for initial data close to a constant equilibrium state away from the vacuu...

In this paper we apply the generalized Taylor–Galerkin finite element model to simulate bore wave propagation in a domain of two dimensions. For stability and accuracy reasons, we generalize the model through the introduction of four free parameters. One set of parameters is rigorously determined to obtain the high-order finite element solution. The other set of free parameters is determined fr...

The proposed work concerns the numerical approximations of the shallow-water equations with varying topography. The main objective is to introduce an easy and systematic technique to enforce the well-balance property and to make the scheme able to deal with dry areas. To access such an issue, the derived numerical method is obtained by involving the free surface instead of the water height and ...

A spherical spectral viscosity operator is proposed as an alternative to standard horizontal diffusion terms in global atmospheric models. Implementation in NCAR’s Spectral Transform Shallow Water Model and application to a suite of standard test cases demonstrates improvement in resolution and numerical conservation of invariants at no extra computational cost. The retention in the spectral vi...

The accurate numerical simulation of wave disturbance within harbours requires consideration of both nonlinear and dispersive wave processes in order to capture such physical effects as wave refraction and diffraction, and nonlinear wave interactions such as the generation of harmonic waves. The Boussinesq equations are the simplest class of mathematical model that contain all these effects in ...

We present the derivation of the discrete Euler–Lagrange equations for an inverse spectral element ocean model based on the shallow water equations. We show that the discrete Euler–Lagrange equations can be obtained from the continuous Euler–Lagrange equations by using a correct combination of the weak and the strong forms of derivatives in the Galerkin integrals, and by changing the order with...

A space-time discontinuous Galerkin (DG) discretization is presented for the (rotating) shallow water equations over varying topography. We formulate the space-time DG finite element discretization in an efficient and conservative discretization. The HLLC flux is used as numerical flux through the finite element boundaries. When discontinuities are present, we locally apply dissipation around t...