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

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. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1645–1664, 2015