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

arXiv comment45 pages, 10 figures; Corrected some typos. An additional example and some more detailed calculations than the article

References in corpus (5)

• Global conservative solutions of the Camassa-Holm equation for initial data nonvanishing asymptotics
• Existence and Lipschitz stability for $\alpha$-dissipative solutions of the two-component Hunter-Saxton system
• Numerical conservative solutions of the Hunter--Saxton equation
• A Convergent Numerical Algorithm for $\alpha$-Dissipative Solutions of the Hunter-Saxton Equation
• A Lipschitz metric for $\alpha$-dissipative solutions to the Hunter-Saxton equation