Annals of Mathematical Sciences and Applications

Volume 2 (2017)

Number 2

Guest Editors: Tai-Chia Lin (National Taiwan University), Wen-Wei Lin (National Chiao Tung University), Tony Wen-Hann Sheu (National Taiwan University), Weichung Wang (National Taiwan University), Chih-wen Weng (National Chiao Tung University), and Salil Vadhan (Harvard University).

Weighted essentially non-oscillatory schemes for Degasperis–Procesi equation with discontinuous solutions

Pages: 319 – 340

DOI: https://dx.doi.org/10.4310/AMSA.2017.v2.n2.a5

Authors

Yinhua Xia (School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui Province, China)

Yan Xu (School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui Province, China)

Abstract

In this paper, we develop the high order weighted essentially nonoscillatory (WENO) schemes for solving the Degasperis–Procesi (DP) equation, including finite volume (FV) and finite difference (FD) methods. The DP equation contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The finite volume method is designed based on the total variation bounded property of the DP equation. And the finite difference method is constructed based on the $L^2$ stability of the DP equation. Due to the adoption of the WENO reconstruction, both schemes are arbitrary high order accuracy and shock capturing. The numerical simulation results for different types of solutions of the nonlinear Degasperis–Procesi equation are provided to illustrate the accuracy and capability of the methods.

Keywords

Degasperis–Procesi equation, discontinuous solution, weighted essentially non-oscillatory schemes, finite difference method, finite volume method

2010 Mathematics Subject Classification

Primary 65M06, 65M08. Secondary 35Q53.

Received 27 June 2016

Published 10 August 2017