Rate of convergence for numerical $$\alpha$$-dissipative solutions of the Hunter–Saxton equation

Abstract 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.