2018/02/10 by Shunichi Sato, Sato, Shun
Mathematics · Physics and Astronomy · #65M06 #65M12 #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1802.03539
openalex publication_date 2018/02/10 · openalex created_date 2023/02/19 · openalex updated_date 2026/07/28
The modified Hunter--Saxton equation models the propagation of short\ncapillary-gravity waves. As it involves a mixed derivative, its initial value\nproblem on the periodic domain is much more complicated than the standard\nevolutionary equations. Although its local well-posedness has recently been\nproved, the behavior of its solution is yet to be investigated. In this paper,\nto develop a reliable numerical method for this problem, we derive a\nconservative finite difference scheme. Then, we rigorously prove not only its\nstability in the sense of the uniform norm but also its uniform convergence to\nsufficiently smooth exact solutions. Discrete conservation laws are used to\novercome the difficulty due to the mixed derivative.\n