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 α-dissipative solutions of the Hunter–Saxton equation, where α ∈ [0, 1]. In particular, if the following two conditions hold: (i) there exist a constant $C> 0$ and β ∈ (0, 1] such that the initial spatial derivative ūₓ satisfies \|ūₓ(· + h) - ūₓ(· )\|₂ ≤ Chβ for all h ∈ (0, 2], and (ii) the singular continuous part of the initial energy measure is zero then the numerical wave profile converges with order O( x^β /8) in L∞(R). Moreover, if α =0 then the rate improves to O( x^1/4) without the above assumptions, and we also obtain a convergence rate for the associated energy measure—it converges with order O( x^1/2) in the bounded Lipschitz metric. These convergence rates are illustrated by several examples.