Stability and convergence of a conservative finite difference scheme for the modified Hunter--Saxton equation
Provides a reliable numerical method for a complex PDE with mixed derivatives, addressing a gap in numerical analysis for this specific equation.
The paper develops a conservative finite difference scheme for the modified Hunter-Saxton equation, proving its stability in the uniform norm and uniform convergence to smooth solutions.
The modified Hunter--Saxton equation models the propagation of short capillary-gravity waves. As it 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, to develop a reliable numerical method for this problem, we derive a conservative finite difference scheme. Then, we rigorously prove not only its stability in the sense of the uniform norm but also its uniform convergence to sufficiently smooth exact solutions. Discrete conservation laws are used to overcome the difficulty due to the mixed derivative.