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

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.