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 $\|\bar{u}_{x}(\cdot + h) - \bar{u}_{x}(\cdot )\|_{2} \leq Ch^{\beta }$ 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 $\mathscr{O}({\varDelta x}^{{\beta }/{8}})$ in $L^{\infty }(\mathbb{R})$. Moreover, if $\alpha =0$ then the rate improves to $\mathscr{O}({\varDelta x}^{{1}/{4}})$ without the above assumptions, and we also obtain a convergence rate for the associated energy measure—it converges with order $\mathscr{O}({\varDelta x}^{{1}/{2}})$ in the bounded Lipschitz metric. These convergence rates are illustrated by several examples.