BIT Numerical Mathematics

, Volume 59, Issue 1, pp 213–241 | Cite as

Stability and convergence of a conservative finite difference scheme for the modified Hunter–Saxton equation

  • Shun SatoEmail author


The modified Hunter–Saxton equation models the propagation of short capillary-gravity waves. As the equation involves a mixed derivative, its initial value problem on the periodic domain is much more complicated than the standard evolutionary equations. Although its local well-posedness has recently been proved, the behavior of its solution is yet to be investigated. In this paper, a conservative finite difference scheme is derived as a reliable numerical method for this problem. Then, the stability of the numerical solution in the sense of the uniform norm, and the uniform convergence of the numerical solutions to sufficiently smooth exact solutions are rigorously proved. Discrete conservation laws are used to overcome the difficulty due to the mixed derivative.


Modified Hunter–Saxton equation Geometric integration Stability Convergence 

Mathematics Subject Classification

65M06 65M12 



The author is grateful to Yuto Miyatake and Takayasu Matsuo for their insightful comments. The author also appreciate the anonymous referees’ comments.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyThe University of TokyoBunkyo-kuJapan

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