Abstract 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.

中文翻译：

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Degasperis-Procesi方程的优化紧凑重构加权基本非振荡方案

摘要 本文提出了一种两步迭代法，利用u-P公式研究Degasperis-Procesi (DP)方程，将DP方程分解为非线性对流方程和亥姆霍兹方程。平流方程中的一阶导数项由优化紧凑重构加权基本非振荡 (OCRWENO) 方案近似，该方案可减少色散和耗散误差并抑制不连续附近的振荡。此外，高阶辛Runge-Kutta方案用于时间推进。为 OCRWENO 方案提供了对色散和耗散误差的严格分析。研究单光滑孤子解以检查所提出方法的准确性和阶数。峰峰，峰峰-峰峰，然后预测DP方程的peakon-antipeakon和shockpeakon解。最后，解决了 DP 方程中平滑初始条件的破波现象。此外，计算了与 DP 方程的双哈密尔顿形式相关的两个保守量，以证明本方法的能力。