TY - JOUR
T1 - A Linearly-Fitted Conservative (Dissipative) Scheme for Efficiently Solving Conservative (Dissipative) Nonlinear Wave PDEs
AU - Liu , Kai
AU - Wu , Xinyuan
AU - Shi , Wei
JO - Journal of Computational Mathematics
VL - 6
SP - 780
EP - 800
PY - 2017
DA - 2017/12
SN - 35
DO - http://doi.org/10.4208/jcm.1612-m2016-0604
UR - https://global-sci.org/intro/article_detail/jcm/10494.html
KW - Conservative (dissipative) wave PDEs, Structure-preserving algorithm, Linearly-fitted, Average Vector Field formula, Sine-Gordon equation.
AB - <p style="text-align: justify;">The extended discrete gradient method is an extension of traditional discrete gradient method, which is specially designed to solve oscillatory Hamiltonian systems efficiently while preserving their energy exactly. In this paper, based on the extended discrete gradient method, we present an efficient approach to devising novel schemes for numerically solving conservative (dissipative) nonlinear wave partial differential equations. The new scheme can preserve the energy exactly for conservative wave equations. With a minor remedy to the extended discrete gradient method, the new scheme is applicable to dissipative wave equations. Moreover, it can preserve the dissipation structure for the dissipative wave equation as well. Another important property of the new scheme is that it is linearly-fitted, which guarantees much fast convergence for the fixed-point iteration which is required by an energy-preserving integrator. The efficiency of the new scheme is demonstrated by some numerical examples.</p>