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

TL;DR

This paper establishes a convergence rate for numerical solutions of the Hunter-Saxton equation with $ ext{alpha}$-dissipative solutions, demonstrating the order of accuracy depending on initial data regularity and providing numerical validation.

Contribution

It proves a specific convergence order for numerical $ ext{alpha}$-dissipative solutions of the Hunter-Saxton equation under certain regularity conditions, advancing numerical analysis for this PDE.

Findings

Convergence rate of order $ ext{O}( riangle x^{1/8} + riangle x^{eta/4})$ established.

Numerical experiments confirm the theoretical convergence rate.

Results depend on initial data regularity parameter $eta$.

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 $\|\bar{u}_x(\cdot + h) - \bar{u}_x(\cdot)\|_2 \leq Ch^{\beta}$ for all $h \in (0, 2]$. The derived convergence rate is exemplified by a number of numerical experiments.