[Paper Review] Fast IMEX Time Integration of Nonlinear Stiff Fractional Differential Equations

Yongtao Zhou, Jorge L. Suzuki+2 more|arXiv (Cornell University)|Sep 9, 2019

Fractional Differential Equations Solutions63 references4 citations

TL;DR

This paper proposes first- and second-order implicit-explicit (IMEX) time integration methods for stiff nonlinear fractional differential equations (FDEs) with fractional order α ∈ (0,1]. The methods combine a fractional Adams-Moulton method with Lubich-like corrections to handle initial singularities and use a fast Toeplitz solver to achieve O(N log N) complexity. The schemes demonstrate global first- and second-order accuracy and improved linear stability regions, especially for smaller α values.

ABSTRACT

Efficient long-time integration of nonlinear fractional differential equations is significantly challenging due to the integro-differential nature of the fractional operators. In addition, the inherent non-smoothness introduced by the inverse power-law kernels deteriorates the accuracy and efficiency of many existing numerical methods. We develop two efficient first- and second-order implicit-explicit (IMEX) methods for accurate time-integration of stiff/nonlinear fractional differential equations with fractional order $α\in (0,1]$ and prove their convergence and linear stability properties. The developed methods are based on a linear multi-step fractional Adams-Moulton method (FAMM), followed by the extrapolation of the nonlinear force terms. In order to handle the singularities nearby the initial time, we employ Lubich-like corrections to the resulting fractional operators. The obtained linear stability regions of the developed IMEX methods are larger than existing IMEX methods in the literature. Furthermore, the size of the stability regions increase with the decrease of fractional order values, which is suitable for stiff problems. We also rewrite the resulting IMEX methods in the language of nonlinear Toeplitz systems, where we employ a fast inversion scheme to achieve a computational complexity of $\mathcal{O}(N \log N)$, where $N$ denotes the number of time-steps. Our computational results demonstrate that the developed schemes can achieve global first- and second-order accuracy for highly-oscillatory stiff/nonlinear problems with singularities.

Motivation & Objective

• To address the challenge of long-time, accurate integration of stiff, nonlinear fractional differential equations with singular initial behavior.
• To develop efficient time integration schemes that maintain high accuracy despite the non-smooth solutions caused by inverse power-law kernels in fractional operators.
• To improve the linear stability regions of existing IMEX methods, particularly for small fractional orders α ∈ (0,1], which are common in anomalous diffusion and viscoelastic models.
• To reduce computational cost from O(N²) to O(N log N) by reformulating the scheme as a nonlinear Toeplitz system and applying fast inversion techniques.
• To establish convergence and stability proofs for the proposed first- and second-order IMEX methods under general nonlinearities and singularities.

Proposed method

• The method uses a linear multistep fractional Adams-Moulton method (FAMM) as the base for the stiff (implicit) part of the FDE.
• Nonlinear force terms are treated explicitly using extrapolation to preserve accuracy and avoid solving nonlinear systems at each step.
• Lubich-like correction terms are applied to the fractional operators to capture the singular behavior near t=0, improving accuracy for non-smooth solutions.
• The resulting linear system is reformulated as a nonlinear Toeplitz system, enabling fast matrix-vector multiplication via FFT-based solvers.
• The scheme achieves O(N log N) computational complexity by exploiting the Toeplitz structure and using fast inversion algorithms.
• Stability and convergence are proven using discrete Gronwall inequality and bounds on the correction weights, with rigorous error estimates for both first- and second-order schemes.

Experimental results

Research questions

• RQ1 Can an IMEX method be constructed for stiff nonlinear FDEs that maintains first- and second-order accuracy despite initial singularities?
• RQ2 How can the linear stability region of IMEX schemes for FDEs be enhanced, particularly for small α values?
• RQ3 Can the computational cost of solving the resulting dense linear systems be reduced from O(N²) to O(N log N) without sacrificing accuracy?
• RQ4 What is the impact of Lubich-like corrections on the convergence rate and stability of IMEX methods for nonlinear FDEs?
• RQ5 How do the proposed schemes perform on highly oscillatory, stiff, and non-smooth FDE problems compared to existing methods?

Key findings

• The proposed first- and second-order IMEX methods achieve global first- and second-order accuracy for stiff, nonlinear FDEs with singular initial behavior.
• The linear stability regions of the schemes are larger than those of existing IMEX methods and increase as the fractional order α decreases, which is advantageous for stiff problems.
• The computational complexity is reduced to O(N log N) by reformulating the scheme as a nonlinear Toeplitz system and applying fast inversion techniques.
• The schemes maintain high accuracy even for highly oscillatory problems due to the incorporation of Lubich-like corrections for initial singularities.
• Theoretical convergence and stability are rigorously proven, with error bounds depending on the regularity of the solution and the order of the correction terms.
• Numerical experiments confirm the theoretical findings, showing robust performance across a range of α ∈ (0,1] and nonlinearities.