[Paper Review] A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem

Hong-lin Liao, William McLean+1 more|arXiv (Cornell University)|Jan 1, 2018

Fractional Differential Equations Solutions16 references31 citations

TL;DR

This paper proposes an error convolution structure (ECS) analysis for a second-order accurate, nonuniform time-stepping scheme based on the Alikhanov approximation to the Caputo fractional derivative. By leveraging piecewise quadratic and linear interpolation on graded, nonuniform time meshes, the authors establish a sharp L²-norm error estimate of order O(τ² + h²) for linear reaction-subdiffusion problems, proving stability and convergence even with initial singularities, and demonstrating that graded meshes effectively resolve the t^{α−1} singularity in the solution's time derivative.

ABSTRACT

Stability and convergence of a time-weighted discrete scheme with nonuniform time steps are established for linear reaction-subdiffusion equations. The Caupto derivative is approximated at an offset point by using linear and quadratic polynomial interpolation. Our analysis relies on two tools: a discrete fractional Grönwall inequality and the global consistency analysis. The new consistency analysis makes use of an interpolation error formula for quadratic polynomials, which leads to a convolution-type bound for the local truncation error. To exploit these two tools, some theoretical properties of the discrete kernels in the numerical Caputo formula are crucial and we investigate them intensively in the nonuniform setting. Taking the initial singularity of the solution into account, we obtain a sharp error estimate on nonuniform time meshes. The fully discrete scheme generates a second-order accurate solution on the graded mesh provided a proper grading parameter is employed. An example is presented to show the sharpness of our analysis.

Motivation & Objective

• To develop a second-order time discretization scheme for linear reaction-subdiffusion problems with improved accuracy on nonuniform time meshes.
• To address the initial singularity in the solution's time derivative, which limits convergence on uniform meshes.
• To establish a rigorous stability and convergence theory for the fractional Crank–Nicolson method on general nonuniform grids with variable step sizes.
• To demonstrate that graded meshes (e.g., tk = T(k/N)^γ) restore second-order convergence despite the solution's low regularity at t=0.
• To extend the ECS analysis framework to higher-order approximations, specifically the Alikhanov formula, enabling error localization and global consistency analysis.

Proposed method

• Proposes an error convolution structure (ECS) analysis to localize and simplify the error analysis of discrete convolution approximations to the Caputo derivative on nonuniform grids.
• Uses piecewise quadratic interpolation in subintervals [tk−1, tk] for k = 1 to n−1 and linear interpolation in [tn−1, tn−θ] to construct the Alikhanov approximation with offset θ = α/2.
• Derives a new interpolation error formula with integral remainder for quadratic polynomials to bound the local truncation error in the ECS framework.
• Establishes a global consistency error bound that reveals the error distribution behavior over long-time integration, showing that the error becomes effectively local when mesh grading follows the error equidistribution principle.
• Applies the ECS framework to prove stability and convergence of the fractional Crank–Nicolson scheme under a mild restriction on the local step-size ratio ρ ≤ 7/4.
• Uses the Galerkin finite element method in space and a Crank–Nicolson-like time-stepping scheme to achieve second-order accuracy in time and second-order in space.

Experimental results

Research questions

• RQ1 Can a second-order time discretization be constructed for linear reaction-subdiffusion problems that remains stable and convergent on general nonuniform time meshes with variable step sizes?
• RQ2 How can the initial singularity in the solution (with ∂tu ∼ t^{α−1}) be resolved to restore second-order convergence in time?
• RQ3 What is the role of mesh grading (e.g., tk = T(k/N)^γ) in achieving optimal convergence for nonuniform time-stepping schemes?
• RQ4 Can the error convolution structure (ECS) framework be effectively applied to higher-order approximations like the Alikhanov formula to simplify and localize error analysis?
• RQ5 What is the optimal stability condition on the local step-size ratio ρ = max(τk/τk+1) for the nonuniform Alikhanov scheme?

Key findings

• The ECS analysis successfully localizes the error analysis of the nonuniform Alikhanov approximation, simplifying the treatment of general nonuniform time grids.
• A sharp L²-norm error estimate of order O(τ² + h²) is derived for the fractional Crank–Nicolson scheme, confirming second-order convergence in time.
• The scheme is proven stable and convergent under the condition that the local step-size ratio satisfies ρ ≤ 7/4, allowing adaptive time stepping with τ_next / τ_current ≥ 4/7.
• Graded meshes with γ ≥ 1 effectively resolve the initial singularity, restoring second-order convergence even when the solution is not smooth at t=0.
• The global consistency error bound reveals that the nonlocal nature of the fractional derivative leads to a localized error distribution when the mesh is graded according to the error equidistribution principle.
• The analysis is validated through a numerical example demonstrating the sharpness of the theoretical error bounds.