The analysis of operator splitting methods for the Camassa-Holm equation. (English) Zbl 1393.65018
Summary: In this paper, the convergence analysis of operator splitting methods for the Camassa-Holm equation is provided. The analysis is built upon the regularity of the Camassa-Holm equation and the divided equations. It is proved that the solution of the Camassa-Holm equation satisfies the locally Lipschitz condition in \(H^1\) and \(H^2\) norm, which ensures the regularity of the numerical solution. Through the calculus of Lie derivatives, we show that the Lie-Trotter and Strang splitting converge with the expected rate under suitable assumptions. Numerical experiments are presented to illustrate the theoretical result.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 76B25 | Solitary waves for incompressible inviscid fluids |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
Keywords:
operator splitting; Lie-Trotter splitting; strang splitting; convergence; Camassa-Holm equationReferences:
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