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 \(\mathcal {O}({\varDelta x}^{1/8}+{\varDelta x}^{\beta /4})\) 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 \(\Vert {\bar{u}}_x(\cdot + h) - {\bar{u}}_x(\cdot )\Vert _2 \le Ch^{\beta }\) for all \(h \in (0, 2]\) . The derived convergence rate is exemplified by a number of numerical experiments.
