TY - JOUR
T1 - A Novel Class of Energy-Preserving Runge-Kutta Methods for the Korteweg-de Vries Equation
AU - Chen , Yue
AU - Gong , Yuezheng
AU - Hong , Qi
AU - Wang , Chunwu
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 3
SP - 768
EP - 792
PY - 2022
DA - 2022/07
SN - 15
DO - http://doi.org/10.4208/nmtma.OA-2021-0172
UR - https://global-sci.org/intro/article_detail/nmtma/20815.html
KW - Quadratic auxiliary variable approach, symplectic Runge-Kutta scheme, energy-preserving algorithm, Fourier pseudo-spectral method.
AB - <p style="text-align: justify;">In this paper, we present a quadratic auxiliary variable approach to develop a new class of energy-preserving Runge-Kutta methods for the Korteweg-de
Vries equation. The quadratic auxiliary variable approach is first proposed to reformulate the original model into an equivalent system, which transforms the energy
conservation law of the Korteweg-de Vries equation into two quadratic invariants
of the reformulated system. Then the symplectic Runge-Kutta methods are directly
employed for the reformulated model to arrive at a new kind of time semi-discrete
schemes for the original problem. Under consistent initial conditions, the proposed
methods are rigorously proved to maintain the original energy conservation law of
the Korteweg-de Vries equation. In addition, the Fourier pseudo-spectral method is
used for spatial discretization, resulting in fully discrete energy-preserving schemes.
To implement the proposed methods effectively, we present a very efficient iterative
technique, which not only greatly saves the calculation cost, but also achieves the
purpose of practically preserving structure. Ample numerical results are addressed
to confirm the expected order of accuracy, conservative property and efficiency of
the proposed algorithms.</p>