Year: 2022
Author: Qi Hong, Chunwu Wang, Yue Chen, Yuezheng Gong, Qi Hong, Chunwu Wang
Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 3 : pp. 768–792
Abstract
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2021-0172
Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 3 : pp. 768–792
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 25
Keywords: Quadratic auxiliary variable approach symplectic Runge-Kutta scheme energy-preserving algorithm Fourier pseudo-spectral method.
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