2024/09/27 by Thomas Christiansen, Christiansen, Thomas
Mathematics · Physics and Astronomy · #65M12 #65M15 #65M25 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Waves and Solitons #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2409.18903
openalex publication_date 2024/09/27 · openalex created_date 2024/10/12 · openalex updated_date 2026/07/28
We derive a robust error estimate for a recently proposed numerical method for α-dissipative solutions of the Hunter-Saxton equation, where α∈ [0, 1]. In particular, if the following two conditions hold: i) there exist a constant C > 0 and β∈ (0, 1] such that the initial spatial derivative ux satisfies ‖ux(⋅ + h) - ux(⋅)‖2 ≤ Chβ for all h ∈ (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^\fracβ8) in L∞(ℝ). Moreover, if α=0, then the rate improves to O(Δx(1)/(4)) without the above assumptions, and we also obtain a convergence rate for the associated energy measure - it converges with order O(Δx(1)/(2)) in the bounded Lipschitz metric. These convergence rates are illustrated by several examples.