On the convergence rate of a numerical method for the Hunter-Saxton equation

We derive a robust error estimate for a recently proposed numerical method for $\alpha$-dissipative solutions of the Hunter-Saxton equation, where $\alpha \in [0,1]$. In particular, if the following two conditions hold: i) there exist a constant $C>0$ 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 $h\in (0,2]$, and ii), the singular continuous part of the initial energy measure is zero, then the numerical wave profile converges with order $O(\mathrm{\Delta }{x}^{\frac{\beta }{8}})$ in $L^{\mathrm{\infty }}(\mathbb{R})$. Moreover, if $\alpha =0$, then the rate improves to $O(\mathrm{\Delta }{x}^{\frac{1}{4}})$ without the above assumptions, and we also obtain a convergence rate for the associated energy measure - it converges with order $O(\mathrm{\Delta }{x}^{\frac{1}{2}})$ in the bounded Lipschitz metric. These convergence rates are illustrated by several examples.