A numerical view on α-dissipative solutions of the Hunter-Saxton equation

TL;DR

The paper advances numerical treatment of α-dissipative Hunter–Saxton solutions by integrating a projection-based Lagrangian discretization with a generalized method of characteristics and an energy-removal iteration; a nonuniform, finite sequence of breaking times {τ_k^*} is extracted to manage clustered wave-breaking events and minimize unnecessary recomputation. It provides a rigorous convergence theory: convergence of the projected initial data, convergence of the Lagrangian solution X_{Δx}(t) to X(t) for all t in [0,T], and, via the M-map, convergence of the Eulerian solution (u_{Δx}, μ_{Δx}, ν_{Δx}) to (u, μ, ν). The numerical results on multipeakon and cusp-like data illustrate substantial computational gains from the minimal-time stepping and confirm accurate capture of wave breaking and energy dissipation across α ∈ [0,1). The framework thus offers a robust, convergent, and efficient approach for simulating α-dissipative HS dynamics in both theoretical and applied contexts.

Abstract

We propose a new numerical method for $α$-dissipative solutions of the Hunter-Saxton equation, where $α$ belongs to $W^{1, \infty}(\mathbb{R}, [0, 1))$. The method combines a projection operator with a generalized method of characteristics and an iteration scheme, which is based on enforcing minimal time steps whenever breaking times cluster. Numerical examples illustrate that these minimal time steps increase the efficiency of the algorithm substantially. Moreover, convergence of the wave profile is shown in $C([0, T], L^{\infty}(\mathbb{R}))$ for any finite $T \geq 0$.

Figures (5)

• Figure 1: The relation between the Eulerian and Lagrangian gridpoints. Note, that points, where $G_{{\Delta x}}$ has a jump, are mapped to intervals where $y_{{\Delta x}}$ is constant.
• Figure 2: Comparison of $u$ (top row, red dotted line) and $F$ (bottom row, red dotted line) with $u_{{\Delta x}}$ (upper row) and $F_{{\Delta x}}$ (lower row) for ${\Delta x} = 10^{-1}$ (dashed blue line) and ${\Delta x}=10^{-4}$ (solid black line) for Example \ref{['ex:MultipeakonExample']}. The solutions are compared from left to right, at the times $t=0$, $\frac{20}{9}$, and $3$. Moreover, also the conservative ($\alpha=0$, gray dashdotted) and dissipative ($\alpha=1$, gray dotted) solutions are displayed at $t=\frac{20}{9}$ and $3$.
• Figure 3: In the two leftmost figures, the errors $\sup_{t \in [0, 3]} \|u(t) - u_{{\Delta x}}(t)\|_{\infty}$ and $|F_{\infty}(3) - F_{{\Delta x}, \infty}(3)|$ are plotted as functions of ${\Delta x}$ for Example \ref{['ex:MultipeakonExample']}, while the time evolution of the total numerical energy $F_{{\Delta x}, \infty}(t)$ for ${\Delta x}=5\cdot10^{-1}$ (dashed blue) and ${\Delta x}=10^{-4}$ (solid black) is compared with $F_{\infty}(t)$ (dotted red) in the right figure.
• Figure 4: A comparison of $u_{\mathrm{ref}}$ (top row, red dotted line) and $F_{\mathrm{ref}}$ (bottom row, red dotted line) with that of $u_{{\Delta x}}$ (top row) and $F_{{\Delta x}}$ (bottom row) for ${\Delta x}=10^{-1}$ (blue dashed) and ${\Delta x} = 10^{-4}$ (black solid) for Example \ref{['ex:CuspExample']}. The solutions are compared from left to right, at the times $t =0$, $\frac{3}{2}$, and $3$, with $\beta =\frac{19}{20}$. Moreover, also the conservative ($\alpha=0$, gray dashdotted) and dissipative ($\alpha=1$, gray dotted) solutions are displayed at $t=\frac{3}{2}$ and $3$.
• Figure 5: The errors $\sup_{t \in [0, 3]} \|u_{\mathrm{ref}}(t) - u_{{\Delta x}}(t)\|_{\infty}$ and $|F_{\mathrm{ref}, \infty}(3) - F_{{\Delta x}, \infty}(3)|$ are plotted as functions of ${\Delta x}$ for Example \ref{['ex:CuspExample']} in the two left figures, while the time evolution of the total numerical energy $F_{{\Delta x}, \infty}(t)$ for ${\Delta x}=10^{-1}$ (dashed blue) and ${\Delta x}=10^{-4}$ (solid black) is compared to $F_{\mathrm{ref}, \infty}(t)$ (dotted red) in the right plot.

Theorems & Definitions (39)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Theorem 2.8: AlphaHS
• Definition 3.1
• Remark 3.2