ABSTRACT:

The equal width (EW) equation is solved by a fourth-order compact method, based on the fourth-order padé compact scheme in space and the fourth-order Runge-Kutta method in time integration. The propagation of a single solitary wave is studied to assess the properties of the method. The development of the EW undular bore is investigated by proposed scheme, and comparison is made with other recently published methods.

INTRODUCTION

A bore is a transition between different uniform flows of water, which is most easily studied in uniform, rectangular, open channels. In its most common form, a bore is a turbulent, breaking zone of water whose length is a few times the depth of water. However, if the bore is weak in that the change in water level is much less than the depth of water, the bore consists of a train of many waves whose wavelength is several times the depth of water. Experimental evidence indicates that when the ratio of the change in level with respect to the initial depth of water is less than 0.28 the bore is undular, otherwise at least one undulation is breaking (if the above ratio is greater than 0.28 but less than 0.75). Peregrine (1966) was the first to derive the regularized long wave (RLW) equation to model the development of an undular bore. The equal width (EW) wave equation, which is less well-known and proposed by Morrison et al (1984), is an alternative description of nonlinear dispersive waves to the more usual Korteweg-de Vries equation and RLW equation (Benjamin, Bona and Mahony, 1972). It has been shown to have solitary wave solutions and to govern a large number of important physical phenomena such as the nonlinear transverse waves in shallow water, ion-acoustic and magnetohydrodynamic waves in plasma and phonon packets in nonlinear crystals.

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