$H^1$-analysis of H3N3-2\textbf{$_σ$}-based difference method for fractional hyperbolic equations
Rui-lian Du, Changpin Li, Zhi-zhong Sun
TL;DR
The paper develops a direct high-order discretization for time-fractional hyperbolic equations with Caputo derivatives of order $1<\alpha<2$ by formulating the $H3N3-2_\sigma$ interpolation, which achieves second-order accuracy in time on a uniform grid. A second-order finite difference scheme is built using this derivative approximation and central spatial differences, with unconditional $H^1$-stability and convergence proven via energy estimates. To address initial weak regularity, the authors introduce a graded-time-mesh version of the scheme and provide regularity results that justify the graded approach, along with fast algorithms based on sum-of-exponentials to accelerate computations. Numerical experiments compare the proposed method to L2C-based schemes, confirming the predicted convergence rates and demonstrating effectiveness and efficiency, including on graded meshes. The work advances stability analysis for fractional hyperbolic equations and offers practical high-order tools for simulating memory-affected wave phenomena.
Abstract
A novel H3N3-2$_σ$ interpolation approximation for the Caputo fractional derivative of order $α\in(1,2)$ is derived in this paper, which improves the popular L2C formula with (3-$α$)-order accuracy. By an interpolation technique, the second-order accuracy of the truncation error is skillfully estimated. Based on this formula, a finite difference scheme with second-order accuracy both in time and in space is constructed for the initial-boundary value problem of the time fractional hyperbolic equation. It is well known that the coefficients' properties of discrete fractional derivatives are fundamental to the numerical stability of time fractional differential models. We prove the related properties of the coefficients of the H3N3-2$_σ$ approximate formula. With these properties, the numerical stability and convergence of the difference scheme are derived immediately by the energy method in the sense of $H^1$-norm. Considering the weak regularity of the solution to the problem at the starting time, a finite difference scheme on the graded meshes based on H3N3-2$_σ$ formula is also presented. The numerical simulations are performed to show the effectiveness of the derived finite difference schemes, in which the fast algorithms are employed to speed up the numerical computation.