A semi-discrete scheme derived from variational principles for global conservative solutions of a Camassa-Holm system
Abstract
We define a kinetic and a potential energy such that the principle of stationary action from Lagrangian mechanics yields a Camassa--Holm system (2CH) as the governing equations. After discretizing these energies, we use the same variational principle to derive a semi-discrete system of equations as an approximation of the 2CH system. The discretizaton is only available in Lagrangian coordinates and requires the inversion of a discrete Sturm--Liouville operator with time-varying coefficients. We show the existence of fundamental solutions for this operator at initial time with appropriate decay. By propagating the fundamental solutions in time, we define an equivalent semi-discrete system for which we prove that there exists unique global solutions. Finally, we show how the solutions of the semi-discrete system can be used to construct a sequence of functions converging to the conservative solution of the 2CH system.
Keywords
Cite
@article{arxiv.2003.03114,
title = {A semi-discrete scheme derived from variational principles for global conservative solutions of a Camassa-Holm system},
author = {Sondre Tesdal Galtung and Xavier Raynaud},
journal= {arXiv preprint arXiv:2003.03114},
year = {2021}
}
Comments
54 pages, 3 figures. Previously titled 'Well-posedness and convergence of a variational discretization of the Camassa-Holm equation'. Reference conversions for 'A numerical study of variational discretizations of the Camassa-Holm equation (arxiv:2006:15562): eq. (4.42) -> eq. (4.35), Lem. 3.3 -> Lem. 3.1, eq. (3.23) -> eq. (3.11), Lem. 4.1 -> Lem. 4.1, Section 5.2 -> Section 7