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：我们证明了对于初值问题中的Hunter-Saxton方程的α-耗散解（其中α ∈ W^{1, ∞}(ℝ, [0, 1))），在L^{∞}(ℝ)范数下可达到𝒪(Δx^{1/8} + Δx^{β/4})阶的数值计算精度，前提是存在常数C > 0和β ∈ (0, 1]，使得初始空间导数ū_x满足对所有h ∈ (0, 2]均有‖ū_x(· + h) - ū_x(·)‖₂ ≤ Ch^{β}。通过一系列数值实验验证了所推导的收敛速率。