An extension of the wave propagation algorithm first introduced by LeVeque [J. Comp. Phys. 131, 327–353 (1997)] is developed for hyperbolic systems on a general curved manifold. This extension is important in a variety of applications, including the propagation of sound waves on a curved surface, shallow water flow on the surface of the Earth, shallow water magnetohydrodynamics in the solar tac...

This article takes the form of a tutorial on the use of a particular class of mixed finite element methods, which can be thought of as the finite element extension of the C-grid staggered finite difference method. The class is often referred to as compatible finite elements, mimetic finite elements, discrete differential forms or finite element exterior calculus. We provide an elementary introd...

This paper is concerned with the global existence of dissipative solutions to the Camassa-Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as an O.D.E. in an L∞ space, containing a non-local source term which is discontinuous but has bounded directional variation along a suitable cone of directions. For a give...

This study investigates the new (3+1)-dimensional shallow water wave equation. To do so, definitions of conformable derivatives and their descriptions are given. Using Riccati equation modified Kudryashov methods, exact solutions to this problem discovered. The gathered data's contour plot surfaces related 3D 2D emphasize result's physical nature. monitor problem's activity, complete necessary....

A weakly-nonlinear analysis of the transient evolution of two-dimensional, standing waves in a rectangular basin is presented. The waves are resonated by periodic oscillation along an axis aligned with the wavenumber vector. The amplitude of oscillation is assumed to be small with respect to the basin dimensions. The effects of detuning, viscous damping, and cubic nonlinearity are all simultane...

In this paper we study a shallow water equation derivable using the Boussinesq approximation, which includes as two special cases, one equation discussed by Ablowitz et al [Stud. Appl. Math., 53 (1974) 249–315] and one by Hirota and Satsuma [J. Phys. Soc. Japan, 40 (1976) 611–612]. A catalogue of classical and nonclassical symmetry reductions, and a Painlevé analysis, are given. Of particular i...

We generalize the non-linear one-dimensional equation of a fluid layer for any depth and length as an infinite order differential equation for the steady waves. This equation can be written as a q-differential one, with its general solution written as a power series expansion with coefficients satisfying a nonlinear recurrence relation. In the limit of long and shallow water (shallow channels) ...