Rate of convergence for numerical $\alpha$-dissipative solutions of the Hunter-Saxton equation

We prove that $\alpha$-dissipative solutions to the Cauchy problem of the Hunter-Saxton equation, where $\alpha \in W^{1,\mathrm{\infty }}(\mathbb{R},[0,1))$, can be computed numerically with order $\mathcal{O}(\mathrm{\Delta }{x}^{1\mathrm{/}8}+\mathrm{\Delta }{x}^{\beta \mathrm{/}4})$ in $L^{\mathrm{\infty }}(\mathbb{R})$, provided there exist constants $C>0$ and $\beta \in (0,1]$ such that the initial spatial derivative ${\bar{u}}_x$ satisfies $\mathrm{\parallel }{\bar{u}}_x(\cdot +h)-{\bar{u}}_x(\cdot ){\mathrm{\parallel }}_2\le C{h}^{\beta }$ for all $h\in (0,2]$. The derived convergence rate is exemplified by a number of numerical experiments.