An optimized compact reconstruction weighted essentially non-oscillatory scheme for Degasperis-Procesi equation

• Published in: Numerical heat transfer. Part B, Fundamentals Vol. 77; no. 4; pp. 328 - 347
• Main Authors: An, R. D., Yu, C. H., Wu, Zhenyu
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 02.04.2020; Taylor & Francis Ltd
• Subjects: Advection; Essentially non-oscillatory schemes; Helmholtz equations; Reconstruction; Runge-Kutta method; Solitary waves; Time marching; Wave breaking
• ISSN: 1040-7790; 1521-0626
• DOI: 10.1080/10407790.2020.1713651

This article presents a two-step iterative method that uses u - P formulation to study a Degasperis-Procesi (DP) equation, with which the DP equation is decomposed into the nonlinear advection equation and the Helmholtz equation. The first-order derivative terms in the advection equation are approximated by an Optimized Compact Reconstruction Weighted Essentially Non-Oscillatory (OCRWENO) scheme that reduces dispersion and dissipation errors and suppresses oscillation near discontinuities. Besides, high-order symplectic Runge-Kutta scheme is used for time marching. A rigorous analysis of the dispersion and dissipation errors are provided for the OCRWENO scheme. Single smooth soliton solution is investigated to check the accuracy and the order of the proposed method. Peakon, peakon-peakon, peakon-antipeakon and shockpeakon solutions of the DP equation are then predicted. Finally, wave breaking phenomena of smoothed initial condition from the DP equation are addressed. In addition, two conservative quantities associated with the bi-Hamiltonian form of the DP equation are calculated to demonstrate capabilities of the present method.