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 $O(\Delta x^{\frac{\beta}{8}})$ in $L^{\infty}(\mathbb{R})$. Moreover, if $\alpha=0$, then the rate improves to $O(\Delta x^{\frac{1}{4}})$ without the above assumptions, and we also obtain a convergence rate for the associated energy measure - it converges with order $O(\Delta x^{\frac{1}{2}})$ in the bounded Lipschitz metric. These convergence rates are illustrated by several examples.

翻译：本文针对Hunter-Saxton方程$\alpha$-耗散解的一种近期提出的数值方法，推导了其鲁棒误差估计，其中$\alpha \in [0, 1]$。特别地，若满足以下两个条件：i) 存在常数$C > 0$和$\beta \in (0, 1]$，使得初始空间导数$\bar{u}_{x}$对所有$h \in (0, 2]$满足$\|\bar{u}_x(\cdot + h) - \bar{u}_x(\cdot)\|_2 \leq Ch^{\beta}$；ii) 初始能量测度的奇异连续部分为零，则数值波剖面在$L^{\infty}(\mathbb{R})$范数下以$O(\Delta x^{\frac{\beta}{8}})$阶收敛。此外，若$\alpha=0$，则无需上述假设即可获得$O(\Delta x^{\frac{1}{4}})$的改进收敛速率，同时我们得到了关联能量测度的收敛速率——其在有界Lipschitz度量下以$O(\Delta x^{\frac{1}{2}})$阶收敛。这些收敛速率通过若干算例得到了验证。