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

TL;DR

The paper establishes a robust convergence rate for numerical approximations of $\alpha$-dissipative solutions to the Hunter–Saxton equation, extending previous results to $α\in W^{1,\infty}(\mathbb{R},[0,1))$. By combining a generalized method of characteristics with a piecewise-linear projection and an NVWE-inspired pair of variable transformations, the authors control energy concentration and wave-breaking effects across Eulerian and Lagrangian coordinates. They prove that the numerical solution converges with an error of $\mathcal{O}(Δx^{1/8}+Δx^{β/4})$ in $L^{\infty}(\mathbb{R})$ under a Besov-type regularity $\bar{u}_x\in B_2^{β}$, and provide explicit convergence rates for representative data, including multipeakon and cusp profiles, supported by numerical experiments. This offers a robust, wave-breaking-tolerant framework for simulating HS dynamics with energy dissipation guided by spatially varying $α$.

Abstract

We prove that $α$-dissipative solutions to the Cauchy problem of the Hunter-Saxton equation, where $α\in W^{1, \infty}(\mathbb{R}, [0, 1))$, can be computed numerically with order $\mathcal{O}(Δx^{{1}/{8}}+Δx^{β/{4}})$ in $L^{\infty}(\mathbb{R})$, provided there exist constants $C > 0$ and $β\in (0, 1]$ such that the initial spatial derivative $\bar{u}_{x}$ satisfies $\|\bar{u}_x(\cdot + h) - \bar{u}_x(\cdot)\|_2 \leq Ch^β$ for all $h \in (0, 2]$. The derived convergence rate is exemplified by a number of numerical experiments.

Figures (8)

• Figure 1: The left picture visualizes $\bar{y}(\xi)$ and $\bar{y}_{{\Delta x}}(\xi)$. Since $\bar{y}$ and $\bar{y}_{{\Delta x}}$ are strictly increasing on $[\xi_{3j}, \xi_{3j+3})$, $\bar{y}(\xi)$ and $\bar{y}_{{\Delta x}}(2r-\xi)$ intersect in the unique point $\phi(r)$, marked by a red dot in the right picture.
• Figure 2: These plots illustrate what happens when $\bar{y}_{{\Delta x}}(\xi)$ is constant on the interval $[\xi_{3j}, \xi_{3j+1}]$. Then either $\bar{y}(\xi)$ and $\bar{y}_{{\Delta x}}(2r - \xi)$ intersect in a unique point, as in the left plot, or they coincide on a whole subinterval $[\xi_{3j}, \xi_{\mathrm{R}}] \subseteq [\xi_{3j}, \xi_{3j+1}]$ as in the right plot.
• Figure 3: The very left plot depicts $\bar{y}$ and $\bar{y}_{{\Delta x}}$, while $\bar{y}(\phi)$ and $\bar{y}_{{\Delta x}}(\psi)$, with $\phi$ and $\psi$ from Definition \ref{['def:maps']}, are shown in the middle, whereas $\bar{y}$ and $\bar{y}_{{\Delta x}}(f)$, with $f$ defined in AlphaRate, are displayed in the rightmost plot.
• Figure 4: A comparison of $u$ (top row) and $F$ (bottom row), both with dotted red lines, to that of $u_{{\Delta x}_j}$ (top row) and $F_{{\Delta x}_j}$ (bottom row) for ${\Delta x}_c = 4^{-2}$ (dashed blue) and ${\Delta x}_f = 4^{-4}$ (solid black) for Example \ref{['ex:Multipeakon']}. The solutions are compared from left to right at $t=0$, $1$, and $2$.
• Figure 5: A comparison of $u$ (top row) and $F$ (bottom row), both with dotted red lines, to that of $u_{{\Delta x}_j}$ (top row) and $F_{{\Delta x}_j}$ (bottom row) for ${\Delta x}_c = 4^{-1}$ (dashed blue) and ${\Delta x}_f = 4^{-4}$ (solid black ) for Example \ref{['ex:complexMultipeakon']}. The solutions are compared from left to right at $t=0$, $\frac{40}{19}$, and $3$.

Theorems & Definitions (45)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Proposition 2.8: PhdThesisNordli
• Lemma 2.9: LipschitzAlpha
• Definition 2.10