On the convergence rate of a numerical method for the Hunter-Saxton equation

Abstract

We derive a robust error estimate for a recently proposed numerical method for $α$-dissipative solutions of the Hunter-Saxton equation, where $α\in [0, 1]$. In particular, if the following two conditions hold: i) there exist a constant $C > 0$ 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 $h \in (0, 2]$, and ii), the singular continuous part of the initial energy measure is zero, then the numerical wave profile converges with order $O(Δx^{\fracβ{8}})$ in $L^{\infty}(\mathbb{R})$. Moreover, if $α=0$, then the rate improves to $O(Δx^{\frac{1}{4}})$ without the above assumptions, and we also obtain a convergence rate for the associated energy measure - it converges with order $O(Δx^{\frac{1}{2}})$ in the bounded Lipschitz metric. These convergence rates are illustrated by several examples.

Figures (6)

• Figure 1: The solution $u$ (left) and the associated energy $F$ (right) from Example \ref{['ex:simpleBreaking']} at the times $t=0$, $1$, and $1.95$.
• Figure 2: A schematic representation of all the quantities involved when deriving a convergence rate for $\{u_{{\Delta x}}(t)\}_{{\Delta x} > 0}$, and their respective rates of convergence. Here $f$ (to be defined) denotes a bijection which maps $\mathcal{S}$ to $\mathcal{S}_{{\Delta x}}$.
• Figure 3: The mapping $f$ rearranges the points within $[\xi_{3j}, \xi_{3j+3})$ in such a way that $\bar{y}_{{\Delta x}}(f)$ is piecewise constant on $\mathcal{S}_{3j}$. As a consequence, $\bar{V}_{\xi}(\xi)\! -\! \bar{V}_{{\Delta x}, \xi}(f(\xi))\!=\!0$ for a.e. $\xi \in\! \mathcal{S}_{3j}$.
• 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 ${\Delta x}_c = 1/4$ (blue dashed) and ${\Delta x}_f=10^{-3}$ (black solid) for the multipeakon from Example \ref{['ex:simpleBreaking']}. The solutions are compared from left to right at $t=0$, $2$, and $4$ with $\alpha=1/2$.
• 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 ${\Delta x}_c = 1/4$ (blue dashed) and ${\Delta x}_f=10^{-3}$ (black solid) for Example \ref{['ex:cosinus']}. The solutions are compared from left to right at $t=0$, $\frac{2}{\pi}$, and $\frac{4}{\pi}$, with $\alpha=3/4$.

Theorems & Definitions (40)

• Example 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5: AlphaAlgorithm
• Definition 2.6: AlphaAlgorithm
• Proposition 3.1: AlphaAlgorithm
• Lemma 3.2
• proof