On the convergence rate of a numerical method for the Hunter-Saxton equation
<|reference_start|>On the convergence rate of a numerical method for the Hunter-Saxton equation: 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.<|reference_end|>
@article{christiansen2024on,
    title={On the convergence rate of a numerical method for the Hunter-Saxton
  equation},
    author={Thomas Christiansen},
    journal={arXiv preprint arXiv:2409.18903},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.18903},
    primaryClass={math.NA cs.NA math.AP}
}