Finite-Difference Hermite WENO Scheme for Degasperis-Procesi Equation

Abstract

We present a fifth-order finite-difference Hermite weighted essentially non-oscillatory (HWENO) method for solving the Degasperis–Procesi (DP) equation in this paper. First, the DP equation can be rewritten as a system of equations consisting of hyperbolic equations and elliptic equations by introducing an auxiliary variable, since the equations contain nonlinear higher order derivative terms. Then, the auxiliary variable equations are solved using the Hermite interpolation, while the HWENO scheme is performed for the hyperbolic equations. Compared with the popular WENO-type scheme, the most important feature of the HWENO scheme mentioned in this paper is the compactness of its spatial reconstruction stencil, which can achieve the fifth-order accuracy of the expected design with only three points, while the WENO method requires five points. Finally, we demonstrate the effectiveness of the HWENO method in various aspects by conducting some benchmark numerical tests.