High order finite difference WENO methods with unequal-sized sub-stencils for the Degasperis-Procesi type equations

TL;DR

This paper introduces two novel high-order finite difference WENO schemes with unequal-sized sub-stencils for solving the nonlinear high-order Degasperis-Procesi equations, improving accuracy and flexibility in handling peakon solutions and shock waves.

Contribution

The paper develops two new WENO schemes with unequal-sized sub-stencils for DP equations, offering simpler stencil choice and maintaining high polynomial degree accuracy.

Findings

Schemes achieve high order accuracy in numerical tests.

Methods effectively handle peakon solutions and shocks.

Schemes demonstrate non-oscillatory behavior.

Abstract

In this paper, we develop two finite difference weighted essentially non-oscillatory (WENO) schemes with unequal-sized sub-stencils for solving the Degasperis-Procesi (DP) and $\mu$-Degasperis-Procesi ($\mu$DP) equations, which contain nonlinear high order derivatives, and possibly peakon solutions or shock waves. By introducing auxiliary variable(s), we rewrite the DP equation as a hyperbolic-elliptic system, and the \mdp equation as a first order system. Then we choose a linear finite difference scheme with suitable order of accuracy for the auxiliary variable(s), and two finite difference WENO schemes with unequal-sized sub-stencils for the primal variable. One WENO scheme uses one large stencil and several smaller stencils, and the other WENO scheme is based on the multi-resolution framework which uses a series of unequal-sized hierarchical central stencils. Comparing with the classical WENO scheme which uses several small stencils of the same size to make up a big stencil, both WENO schemes with unequal-sized sub-stencils are simple in the choice of the stencil and enjoy the freedom of arbitrary positive linear weights. Another advantage is that the final reconstructed polynomial on the target cell is a polynomial of the same degree as the polynomial over the big stencil, while the classical finite difference WENO reconstruction can only be obtained for specific points inside the target interval. Numerical tests are provided to demonstrate the high order accuracy and non-oscillatory properties of the proposed schemes.