Convergent difference schemes for the Hunter–Saxton equation

Abstract:

We propose and analyze several finite difference schemes for the Hunter–Saxton equation \begin{equation} u_t+u u_x= \frac 12 \int _0^x (u_x)^2 dx, \qquad x>0, t>0. \tag {HS} \end{equation} This equation has been suggested as a simple model for nematic liquid crystals. We prove that the numerical approximations converge to the unique dissipative solution of (HS), as identified by Zhang and Zheng. A main aspect of the analysis, in addition to the derivation of several a priori estimates that yield some basic convergence results, is to prove strong convergence of the discrete spatial derivative of the numerical approximations of $u$, which is achieved by analyzing various renormalizations (in the sense of DiPerna and Lions) of the numerical schemes. Finally, we demonstrate through several numerical examples the proposed schemes as well as some other schemes for which we have no rigorous convergence results.