Stability and convergence of a conservative finite difference scheme for the modified Hunter–Saxton equation
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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.
KeywordsModified Hunter–Saxton equation Geometric integration Stability Convergence
Mathematics Subject Classification65M06 65M12
The author is grateful to Yuto Miyatake and Takayasu Matsuo for their insightful comments. The author also appreciate the anonymous referees’ comments.
- 1.Ben-Israel, A., Greville, T.N.E.: Generalized Inverses, 2nd edn. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 15. Springer, New York. Theory and applications (2003)Google Scholar
- 2.Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)Google Scholar
- 5.Fu, Y., Yin, Z.: Existence and singularities of solutions to an integrable equation governing short-waves in a long-wave model. J. Math. Phys. 51(9), 093,509, 16 (2010). https://doi.org/10.1063/1.3488968
- 9.Hunter, J.K.: Numerical solutions of some nonlinear dispersive wave equations. In: Computational solution of nonlinear systems of equations (Fort Collins, CO, 1988), Lectures in Appl. Math., vol. 26, pp. 301–316. Amer. Math. Soc., Providence, RI (1990)Google Scholar
- 11.Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. Lecture Notes in Math., vol. 448, pp. 25–70 (1975)Google Scholar
- 13.Lenells, J.: Poisson structure of a modified Hunter–Saxton equation. J. Phys. A 41(28), 285,207, 9 (2008). https://doi.org/10.1088/1751-8113/41/28/285207
- 22.Ridder, J., Ruf, A.M.: A convergent finite difference scheme for the Ostrovsky–Hunter equation with Dirichlet boundary conditions (2018). arxiv:1805.07255
- 23.Sato, S.: Linear gradient structures and discrete gradient methods for conservative/dissipative differential-algebraic equations (2018). arxiv:1805.04824
- 24.Sato, S., Matsuo, T.: On spatial discretization of evolutionary differential equations on the periodic domain with a mixed derivative (2017). arxiv:1704.03645