The Degasperis–Procesi equation is split into a system of a hyperbolic equation and an elliptic equation. For the hyperbolic equation, we use the high order finite
difference WENO-Z scheme to approximate the nonlinear flux. For the elliptic equation, the wavelet collocation method is employed to discretize the high order derivative.
Due to the combination of the WENO-Z reconstruction and the wavelet collocation, the
splitting method shows an excellent ability in capturing the formation and propagation of shockpeakon solutions. The numerical simulations for different solutions of the
Degasperis–Procesi equation are conducted to illustrate high accuracy and capability of
the proposed method.