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

2024/11/12 by Thomas Christiansen, Christiansen, Thomas, Katrin Grunert +1
Mathematics · Physics and Astronomy · #65M15 #65M25 (Primary) 35Q35 (Secondary) #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Waves and Solitons #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2411.07712

openalex publication_date 2024/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02

Abstract

We prove that α-dissipative solutions to the Cauchy problem of the Hunter-Saxton equation, where α∈ W^{1, ∞}(ℝ, [0, 1)), can be computed numerically with order O(Δx^1/8+Δx^β/4) in L^∞(ℝ), provided there exist constants C > 0 and β∈ (0, 1] such that the initial spatial derivative u_x satisfies ‖u_x(⋅ + h) - u_x(⋅)‖₂ ≤ Ch^β for all h ∈ (0, 2]. The derived convergence rate is exemplified by a number of numerical experiments.

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