High-order discretization errors for the Caputo derivative in Hölder spaces
Xiangyi Peng, Lisen Ding, Dongling Wang
TL;DR
The paper develops a unified theory for high-order discretizations of the Caputo derivative in Hölder spaces, extending existing L1 error results to L2, L1-2, and general L_k-type schemes (k ≤ 6). By exploiting Hölder regularity $u∈C^{m,β}([0,T])$ with $m+β>α$, it derives truncation-error bounds of the form $\mathcal{O}(τ^{m+β-α})$ for all schemes, with optimal rates such as $3-α$ when $u∈C^{2,1}$. The analysis hinges on high-order interpolation errors and backward-difference constructions, yielding a universal convergence law: the error order equals the degree of smoothness minus the derivative order. Numerical experiments on synthetic Hölder-regular functions confirm the theory and guide scheme choice based on solution regularity, facilitating more accurate and efficient fractional-time simulations.
Abstract
Building upon the recent work of Teso and Plociniczak (2025) regarding L1 discretization errors for the Caputo derivative in Hölder spaces, this study extends the analysis to higher-order discretization errors within the same functional framework. We first investigate truncation errors for the L2 and L1-2 methods, which approximate the Caputo derivative via piecewise quadratic interpolation. Then we generalize the results to arbitrary high-order discretization. Theoretical analyses reveal a unified error structure across all schemes: the convergence order equals the difference between the smoothness degree of the function space and the fractional derivative order, i.e., order of error = degree of smoothness - order of the derivative. Numerical experiments validate these theoretical findings.