Development of a numerical phase optimized upwinding combined compact difference scheme for solving the Camassa-Holm equation with different initial solitary waves

Abstract

In this article, the solution of Camassa-Holm (CH) equation is solved by the proposed two-step method. In the first step, the sixth-order spatially accurate upwinding combined compact difference scheme with minimized phase error is developed in a stencil of four points to approximate the first-order derivative term. For the purpose of retaining both of the long-term accurate Hamiltonian property and the geometric structure inherited in the CH equation, the time integrator used in this study should be able to conserve symplecticity. In the second step, the Helmholtz equation governing the pressure-like variable is approximated by the sixth-order accurate three-point centered compact difference scheme. Through the fundamental and numerical verification studies, the integrity of the proposed high-order scheme is demonstrated. Another aim of this study is to reveal the wave propagation nature for the investigated shallow water equation subject to different initial wave profiles, whose peaks take the smooth, peakon, and cuspon forms. The transport phenomena for the cases with/without inclusion of the linear first-order advection term κu x in the CH equation will be addressed.

Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit

“…( 10) have been shown to have application in the unidirectional flow of a second-grade fluid, statistical mechanics and the study of heat conduction, see [Momoniat, 2015;Polyanin, 2002] and references therein. Also, this type of PDEs can be associated with the integrable Camassa-Holm equation used to model the soliton wave interaction together with the breaking of wave, [Yu et al, 2015]. Pseudo-parabolic equations can also be regarded as a Sobolev-type equation or a Sobolev-Galpern-type equation, see discussion and other applications in [Zhou et al, 2021].…”