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Volume 15, Issue 3
A Novel Class of Energy-Preserving Runge-Kutta Methods for the Korteweg-de Vries Equation

Yue Chen, Yuezheng Gong, Qi Hong & Chunwu Wang

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 768-792.

Published online: 2022-07

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  • 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.

  • AMS Subject Headings

65M06, 65M70

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COPYRIGHT: © Global Science Press

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@Article{NMTMA-15-768, author = {Chen , YueGong , YuezhengHong , Qi and Wang , Chunwu}, title = {A Novel Class of Energy-Preserving Runge-Kutta Methods for the Korteweg-de Vries Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {3}, pages = {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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2021-0172}, url = {http://global-sci.org/intro/article_detail/nmtma/20815.html} }
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 -

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.

Yue Chen, Yuezheng Gong, Qi Hong & Chunwu Wang. (2022). A Novel Class of Energy-Preserving Runge-Kutta Methods for the Korteweg-de Vries Equation. Numerical Mathematics: Theory, Methods and Applications. 15 (3). 768-792. doi:10.4208/nmtma.OA-2021-0172
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