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Rate of convergence for numerical $α$-dissipative solutions of the Hunter-Saxton equation (2411.07712v3)
Published 12 Nov 2024 in math.NA, cs.NA, and math.AP
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.