[Paper Review] A time-spectral algorithm for fractional wave problems

Binjie Li, Hao Luo+1 more|arXiv (Cornell University)|Aug 9, 2017

Fractional Differential Equations Solutions8 references3 citations

TL;DR

This paper presents a high-order time-spectral algorithm for time-fractional wave problems with $1<\gamma<2$, combining spectral spatial discretization in time and finite element methods in space. The method achieves exponential convergence in temporal error when the solution is smooth, significantly outperforming traditional $O(\tau^{3-\gamma})$ schemes, with rigorous stability and convergence analysis confirmed by numerical experiments.

ABSTRACT

This paper develops a high-accuracy algorithm for time fractional wave problems, which employs a spectral method in the temporal discretization and a finite element method in the spatial discretization. Moreover, stability and convergence of this algorithm are derived, and numerical experiments are performed, demonstrating the exponential decay in the temporal discretization error provided the solution is sufficiently smooth.

Motivation & Objective

• To develop a high-order numerical method for time-fractional wave problems with $1<\gamma<2$ that overcomes the limited temporal accuracy of standard schemes like L1 or Grünwald-Letnikov.
• To address the high computational cost and memory demand of fractional derivatives due to their nonlocal nature by employing spectral methods in time for exponential convergence.
• To establish rigorous stability and convergence analysis for the proposed algorithm in both temporal and spatial discretizations.
• To demonstrate through numerical experiments that the temporal error decays exponentially with increasing polynomial degree $M$, provided the solution is sufficiently smooth.

Proposed method

• The algorithm uses a spectral method for temporal discretization, approximating the Riemann-Liouville fractional derivative $D_{0+}^{\gamma}$ via orthogonal polynomials in time.
• Spatial discretization is performed using the standard Galerkin finite element method with $H_0^1$-conforming elements.
• The method constructs a weak formulation in space-time, leveraging Sobolev spaces $H^{\alpha}(0,T;X)$ and weighted $B^j$ spaces to handle the nonlocal nature of fractional derivatives.
• Stability is proven via energy estimates in a suitable weighted function space, ensuring boundedness of the discrete solution.
• Convergence analysis is conducted in the norms $\|u-U\|_{H^{1+\gamma_0}(0,T;L^2(\Omega))}$ and $\|u(T)-U(T)\|_{H_0^1(\Omega)}$, showing optimal convergence rates.
• Numerical experiments validate the theoretical findings, showing exponential decay in error as the polynomial degree $M$ increases.

Experimental results

Research questions

• RQ1 Can a spectral method in time achieve exponential convergence for time-fractional wave problems with $1<\gamma<2$, surpassing the $O(\tau^{3-\gamma})$ order of traditional schemes?
• RQ2 What is the stability behavior of a space-time spectral-finite element scheme for fractional wave equations with nonlocal time derivatives?
• RQ3 How does the regularity of the solution affect the convergence rate of the time-spectral method in both $L^2$ and $H^1$ norms?
• RQ4 Can the proposed algorithm maintain high accuracy while reducing memory and computational cost compared to classical time-stepping methods?
• RQ5 What is the actual convergence order in practice, especially when the solution has limited smoothness, such as $|1-2t|^\beta$?

Key findings

• The temporal discretization error decays exponentially with respect to the polynomial degree $M$, as demonstrated by numerical experiments with smooth solutions.
• For Example 1, the error in $\|u-U\|_{H^{1+\gamma_0}(0,T;L^2(\Omega))}$ converges with order approximately $O(M^{-3})$ when $m=4$, indicating high-order accuracy.
• In Example 2 with $\beta=2.5$, the error $\|u(T)-U(T)\|_{H_0^1(\Omega)}$ converges with order $O(M^{-4.99})$, approaching exponential decay.
• The convergence order for $\|u-U\|_{H^{1+\gamma_0}(0,T;L^2(\Omega))}$ in Example 2 with $\beta=2.5$ is approximately $O(M^{-1.85})$, consistent with theoretical prediction $O(M^{0.75-\beta+\epsilon})$.
• The method achieves significantly higher accuracy than classical schemes: while $L1$ schemes are limited to $O(\tau^{3-\gamma})$, this algorithm achieves exponential convergence in $M$, the number of temporal basis functions.
• Even when the solution has limited regularity (e.g., $|1-2t|^\beta$), the method maintains high convergence rates, with observed orders exceeding theoretical predictions in some norms.