Source-linked AI summary
A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations
Hong-lin Liao, Tao Tang, Tao Zhou
TL;DR
The paper addresses the need for accurate and stable time discretization of the time-fractional Allen-Cahn equation with initial singularities and multiple time scales. It develops a second-order nonuniform Alikhanov scheme, proves discrete maximum-principle preservation and sharp maximum-norm convergence, and adds graded-mesh and adaptive-stepping strategies. Numerical experiments support the scheme’s effectiveness, while the authors note that energy stability is not established.
Problem
The time-fractional Allen-Cahn equation has an intrinsic initial singularity, motivating schemes that accommodate limited temporal regularity.
Method
The paper develops a second-order Alikhanov-type scheme on nonuniform time grids, with graded meshes and an adaptive time-stepping strategy.
Results
The scheme preserves the discrete maximum principle and yields maximum-norm convergence under the stated assumptions.
Takeaways & Limitations
Nonuniform analysis enables graded meshes for the initial singularity and adaptive stepping for long-time simulations.
Takeaways & Limitations
Energy stability is not established because a discrete energy dissipation law for the second-order scheme is unavailable.
Abstract
from arXiv · showhide
In this work, we present a second-order nonuniform time-stepping scheme for the time-fractional Allen-Cahn equation. We show that the proposed scheme preserves the discrete maximum principle, and by using the convolution structure of consistency error, we present sharp maximum-norm error estimates which reflect the temporal regularity. As our analysis is built on nonuniform time steps, we may resolve the intrinsic initial singularity by using the graded meshes. Moreover, we propose an adaptive time-stepping strategy for large time simulations. Numerical experiments are presented to show the effectiveness of the proposed scheme. This seems to be the first second-order maximum principle preserving scheme for the time-fractional Allen-Cahn equation.
1 Introduction
The paper studies numerical schemes for the time-fractional Allen-Cahn equation, whose solutions can have an initial singularity and multiple time scales. It proposes a second-order nonuniform scheme with maximum-principle preservation, sharp maximum-norm error estimates, and adaptive stepping.
- The time-fractional Allen-Cahn equation generalizes the classical Allen-Cahn equation and has an associated energy law and maximum principle.
- Its solution may exhibit an intrinsic initial singularity, motivating graded meshes near the initial time.
- The solution may evolve across fast initial dynamics and a much slower later coarsening stage, motivating adaptive time grids.
- The paper presents a second-order Alikhanov-type scheme on nonuniform time grids that preserves the discrete maximum principle.
- It also derives sharp maximum-norm error estimates, proposes adaptive stepping for long-time simulations, and reports numerical experiments.
2 Preliminaries
The preliminaries establish nonuniform time-grid notation and mesh conditions for resolving initial singularities and multiple time scales. They also introduce assumptions and convolution-kernel tools used in the analysis.
- The paper uses nonuniform time grids 0 = t_0 < t_1 < ··· < t_N = T with step sizes τ_k and adjacent step ratios ρ_k.
- A graded mesh concentrates time levels near t = 0 to resolve the initial behavior u_t ∼ O(t^(α−1)).
- Adaptive time stepping is motivated by the need to capture fast dynamics away from t = 0 and slow coarsening near the steady state.
- The grading parameter γ controls concentration near t = 0: γ = 1 gives a quasi-uniform mesh, while larger γ makes initial steps smaller.
- The analysis assumes sufficient spatial smoothness and introduces generic constants independent of temporal and spatial mesh sizes.
- Discrete fractional Grönwall tools use convolution kernels satisfying positivity, monotonicity, and related bounds under stated step-size restrictions.
3 A second-order maximum principle preserving scheme
The paper constructs a second-order fully discrete Alikhanov scheme on nonuniform grids and establishes the kernel, solvability, and matrix properties needed for maximum-principle preservation. Under stated mesh conditions, the scheme preserves the discrete maximum principle and is unconditionally stable.
- 3.1 The Alikhanov formula under nonuniform grids: The time discretization uses a second-order Alikhanov-type formula with interpolation operators and discrete convolution kernels on nonuniform grids.
- 3.1 The Alikhanov formula under nonuniform grids: The nonuniform formula extends the uniform-mesh Alikhanov formula and is designed to resolve the initial singularity with graded meshes.
- 3.1 The Alikhanov formula under nonuniform grids: The discrete convolution kernels satisfy boundedness and monotonicity properties that are essential for verifying the discrete maximum principle.
- 3.2 The second order fully discrete scheme: The fully discrete method combines the time discretization with a centered spatial Laplacian under periodic boundary conditions and a pointwise nonlinear force.
- 3.2 The second order fully discrete scheme: The nonlinear difference scheme is uniquely solvable because its associated objective function is strictly convex.
- 3.3 Discrete maximum principle: Assuming the ratio restriction and maximum-step condition, the second-order scheme preserves the discrete maximum principle and is unconditionally stable.
4 Error convolution structure and convergence analysis
The analysis represents consistency errors through discrete convolution structures and combines these bounds with fractional Grönwall arguments to establish maximum-norm convergence. Under graded-like meshes, the scheme achieves optimal second-order accuracy when the grading resolves the temporal singularity.
- Consistency-error convolution structure: The consistency error of the nonuniform Alikhanov formula is analyzed using discrete convolution kernels and complementary convolution weights.The temporal error from the time-weighted approximation is controlled by the Alikhanov approximation error.
- Consistency-error convolution structure: The global consistency error is bounded under step-ratio and graded-like mesh conditions for solutions with an initial temporal singularity.The regularity assumptions use a parameter σ and bounds on higher time derivatives near t = 0.
- Consistency-error convolution structure: The first-level consistency error can be worse when the regularity parameter satisfies 0 < σ ≤ α, while the global estimate retains a convolution-based bound.The analysis identifies superconvergence behavior for the nonuniform Alikhanov formula.
- Convergence analysis: The discrete maximum principle enables convergence analysis without assuming Lipschitz continuity of the nonlinear term f(u).The error equation incorporates temporal consistency, time-weighting, nonlinear, and spatial truncation contributions.
- Convergence analysis: The numerical solution converges in the maximum norm under the ratio restriction and a maximum-step-size condition.The proof combines the discrete fractional Grönwall inequality with the consistency estimates.
- Convergence analysis: The scheme achieves optimal accuracy O(τ^2) when the graded parameter satisfies γ ≥ max {1, 2/σ}.This condition links the mesh grading directly to the solution’s temporal regularity.
5 Numerical implementations
The numerical section describes fast Alikhanov evaluation, adaptive time stepping, temporal-accuracy tests, and experiments demonstrating long-time behavior and maximum-principle preservation.
- Fast implementation: The Alikhanov formula is accelerated with a sum-of-exponentials approximation of the history kernel, reducing long-time memory cost and storage requirements.The Caputo derivative is split into history and local parts; the history contribution is evaluated recursively.
- Adaptive time stepping: The adaptive strategy updates time steps using a safety coefficient, reference tolerance, and relative error estimated at each time level.It first computes solutions with a first-order scheme and the proposed scheme, then uses their difference within the adaptive procedure.
- Temporal accuracy: The temporal experiments report O-order convergence and achieve optimal second-order accuracy when the grading parameter satisfies γ ≥ γopt = 2/σ.Tests use graded meshes near the initial time to resolve the initial singularity, with random small cells afterward.
- Long-time simulations: For α = 0.7 and T = 10, adaptive stepping agrees well with a fine uniform mesh while using 108 steps instead of 970.The example combines a graded initial mesh with adaptive stepping over the remaining interval.
- Equilibration: For α = 0.4, 0.7, and 0.9 over T = 100, larger α accelerates approach of the maximum norm to 1 and energy dissipation, while the norm remains bounded by 1.The four drops merge into one progressively shrinking drop, with larger α producing greater shrinkage.
- Discrete maximum principle: The maximum norm remains bounded by 1 under tested time-step limits, and graded meshes can preserve this bound with larger later time steps by resolving the initial singularity.For ε = 0.02 and 0.08, uniform-step tests require τ < 0.67 and τ < 0.57, respectively; a graded mesh permits τ = 0.73 for ε = 0.02.
6 Conclusions
The work proposes a second-order maximum-principle-preserving scheme on nonuniform time steps, with sharp error estimates, graded meshes for initial singularities, and adaptive long-time simulation. However, energy stability remains unresolved on general nonuniform meshes.
- The proposed scheme is second-order, preserves the discrete maximum principle, and uses nonuniform time steps.
- Sharp maximum-norm error estimates reflect the temporal regularity of the solution.
- Graded meshes can resolve the intrinsic initial singularity, while adaptive time stepping targets long-time simulations.
- Energy stability is not addressed because a discrete energy dissipation law has not been established for the second-order scheme.
- On general nonuniform meshes, the restrictions needed to establish positive semidefiniteness of the discrete-kernel quadratic form remain open.