On the convergence rate of a numerical method for the Hunter-Saxton equation

TL;DR

This paper establishes convergence rates for a numerical method solving the Hunter-Saxton equation, showing how initial data regularity affects the rate and providing improved rates for specific cases.

Contribution

The paper derives explicit convergence rates for a numerical scheme for the Hunter-Saxton equation, including special cases with improved rates and energy measure convergence.

Findings

Convergence rate of O(Δx^{β/8}) under certain initial regularity conditions.

Improved convergence rate of O(Δx^{1/4}) when α=0 without additional assumptions.

Energy measure converges with order O(Δx^{1/2}) in the bounded Lipschitz metric.

Abstract

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.