Volume 19, Issue 1
Geometric Numerical Integration for Peakon b-Family Equations

Wenjun Cai, Yajuan Sun & Yushun Wang

Commun. Comput. Phys., 19 (2016), pp. 24-52.

Published online: 2018-04

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  • Abstract

In this paper, we study the Camassa-Holm equation and the Degasperis-Procesi equation. The two equations are in the family of integrable peakon equations, and both have very rich geometric properties. Based on these geometric structures, we construct the geometric numerical integrators for simulating their soliton solutions. The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties, however, they also have the significant differences, for example, there exist the shock wave solutions for the Degasperis-Procesi equation. By using the symplectic Fourier pseudo-spectral integrator, we simulate the peakon solutions of the two equations. To illustrate the smooth solitons and shock wave solutions of the DP equation, we use the splitting technique and combine the composition methods. In the numerical experiments, comparisons of these two kinds of methods are presented in terms of accuracy, computational cost and invariants preservation.

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@Article{CiCP-19-24, author = {}, title = {Geometric Numerical Integration for Peakon b-Family Equations}, journal = {Communications in Computational Physics}, year = {2018}, volume = {19}, number = {1}, pages = {24--52}, abstract = {

In this paper, we study the Camassa-Holm equation and the Degasperis-Procesi equation. The two equations are in the family of integrable peakon equations, and both have very rich geometric properties. Based on these geometric structures, we construct the geometric numerical integrators for simulating their soliton solutions. The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties, however, they also have the significant differences, for example, there exist the shock wave solutions for the Degasperis-Procesi equation. By using the symplectic Fourier pseudo-spectral integrator, we simulate the peakon solutions of the two equations. To illustrate the smooth solitons and shock wave solutions of the DP equation, we use the splitting technique and combine the composition methods. In the numerical experiments, comparisons of these two kinds of methods are presented in terms of accuracy, computational cost and invariants preservation.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.171114.140715a}, url = {http://global-sci.org/intro/article_detail/cicp/11079.html} }
TY - JOUR T1 - Geometric Numerical Integration for Peakon b-Family Equations JO - Communications in Computational Physics VL - 1 SP - 24 EP - 52 PY - 2018 DA - 2018/04 SN - 19 DO - http://doi.org/10.4208/cicp.171114.140715a UR - https://global-sci.org/intro/article_detail/cicp/11079.html KW - AB -

In this paper, we study the Camassa-Holm equation and the Degasperis-Procesi equation. The two equations are in the family of integrable peakon equations, and both have very rich geometric properties. Based on these geometric structures, we construct the geometric numerical integrators for simulating their soliton solutions. The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties, however, they also have the significant differences, for example, there exist the shock wave solutions for the Degasperis-Procesi equation. By using the symplectic Fourier pseudo-spectral integrator, we simulate the peakon solutions of the two equations. To illustrate the smooth solitons and shock wave solutions of the DP equation, we use the splitting technique and combine the composition methods. In the numerical experiments, comparisons of these two kinds of methods are presented in terms of accuracy, computational cost and invariants preservation.

Wenjun Cai, Yajuan Sun & Yushun Wang. (2020). Geometric Numerical Integration for Peakon b-Family Equations. Communications in Computational Physics. 19 (1). 24-52. doi:10.4208/cicp.171114.140715a
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