Development of a Combined Compact Difference Scheme to Simulate Soliton Collision in a Shallow Water Equation (Communications in Computational Physics, 2018)
In this paper a three-step scheme is applied to solve the Camassa-Holm(CH) shallow water equation. The differential order of the CH equation has beenreduced in order to facilitate development of numerical scheme in a comparativelysmaller grid stencil. Here a three-point seventh-order spatially accurate upwindingcombined compact difference (CCD) scheme is proposed to approximate the first-orderderivative term. We conduct modified equation analysis on the CCD scheme andeliminate the leading discretization error terms for accurately predicting unidirectionalwave propagation. The Fourier analysis is carried out as well on the proposed numericalscheme to minimize the dispersive error. For preserving Hamiltonians in Camassa-Holmequation, a symplecticity conserving time integrator has been employed. Theother main emphasis of the present study is the use of u−P−α formulation to get nondissipativeCH solution for peakon-antipeakon and soliton-anticuspon head-on wavecollision problems.