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

翻译：我们证明，对于初始空间导数$\bar{u}_{x}$满足：存在常数$C > 0$和$\beta \in (0, 1]$，使得对所有$h \in (0, 2]$有$\|\bar{u}_x(\cdot + h) - \bar{u}_x(\cdot)\|_2 \leq Ch^{\beta}$，则Cauchy问题Hunter-Saxton方程的$\alpha$-耗散解（其中$\alpha \in W^{1, \infty}(\mathbb{R}, [0, 1))$）可以在$L^{\infty}(\mathbb{R})$范数下以$\mathcal{O}(\Delta x^{{1}/{8}}+\Delta x^{{\beta}/{4}})$的阶数进行数值计算。所推导的收敛速率通过一系列数值实验得到了例证。