Volume 37, Issue 4
The Structure-Preserving Methods for the Degasperis-Procesi Equation

Yuze Zhang, Yushun Wang & Yanhong Yang

J. Comp. Math., 37 (2019), pp. 475-487.

Published online: 2019-02

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  • Abstract

This paper gives several structure-preserving schemes for the Degasperis-Procesi equation which has bi-Hamiltonian structures consisted of both complex and non-local Hamiltonian differential operators. For this sake, few structure-preserving schemes have been proposed so far. In our work, based on one of the bi-Hamiltonian structures, a symplectic scheme and two new energy-preserving schemes are constructed. The symplecticity comes straightly from the application of the implicit midpoint method on the semi-discrete system which is proved to remain Hamiltonian, while the energy conservation is derived by the combination of the averaged vector field method of second and fourth order, respectively. Some numerical results are presented to show that the three schemes do have the advantages in numerical stability, accuracy in long time computing and ability to preserve the invariants of the DP equation.

  • Keywords

Degasperis-Procesi equation, bi-Hamiltonian structure, Structure-preserving scheme, Fourier pseudospectral method.

  • AMS Subject Headings

65D15, 65L05, 65L70, 65P10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

16903152r@connect.polyu.hk (Yuze Zhang)

wangyushun@njnu.edu.cn (Yushun Wang)

yzynj@163.com (Yanhong Yang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-37-475, author = {Zhang , Yuze and Wang , Yushun and Yang , Yanhong }, title = {The Structure-Preserving Methods for the Degasperis-Procesi Equation}, journal = {Journal of Computational Mathematics}, year = {2019}, volume = {37}, number = {4}, pages = {475--487}, abstract = {

This paper gives several structure-preserving schemes for the Degasperis-Procesi equation which has bi-Hamiltonian structures consisted of both complex and non-local Hamiltonian differential operators. For this sake, few structure-preserving schemes have been proposed so far. In our work, based on one of the bi-Hamiltonian structures, a symplectic scheme and two new energy-preserving schemes are constructed. The symplecticity comes straightly from the application of the implicit midpoint method on the semi-discrete system which is proved to remain Hamiltonian, while the energy conservation is derived by the combination of the averaged vector field method of second and fourth order, respectively. Some numerical results are presented to show that the three schemes do have the advantages in numerical stability, accuracy in long time computing and ability to preserve the invariants of the DP equation.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1805-m2017-0184}, url = {http://global-sci.org/intro/article_detail/jcm/13003.html} }
TY - JOUR T1 - The Structure-Preserving Methods for the Degasperis-Procesi Equation AU - Zhang , Yuze AU - Wang , Yushun AU - Yang , Yanhong JO - Journal of Computational Mathematics VL - 4 SP - 475 EP - 487 PY - 2019 DA - 2019/02 SN - 37 DO - http://dor.org/10.4208/jcm.1805-m2017-0184 UR - https://global-sci.org/intro/article_detail/jcm/13003.html KW - Degasperis-Procesi equation, bi-Hamiltonian structure, Structure-preserving scheme, Fourier pseudospectral method. AB -

This paper gives several structure-preserving schemes for the Degasperis-Procesi equation which has bi-Hamiltonian structures consisted of both complex and non-local Hamiltonian differential operators. For this sake, few structure-preserving schemes have been proposed so far. In our work, based on one of the bi-Hamiltonian structures, a symplectic scheme and two new energy-preserving schemes are constructed. The symplecticity comes straightly from the application of the implicit midpoint method on the semi-discrete system which is proved to remain Hamiltonian, while the energy conservation is derived by the combination of the averaged vector field method of second and fourth order, respectively. Some numerical results are presented to show that the three schemes do have the advantages in numerical stability, accuracy in long time computing and ability to preserve the invariants of the DP equation.

Yuze Zhang, Yushun Wang & Yanhong Yang. (2019). The Structure-Preserving Methods for the Degasperis-Procesi Equation. Journal of Computational Mathematics. 37 (4). 475-487. doi:10.4208/jcm.1805-m2017-0184
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