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Correction of high-order BDF convolution quadrature for fractional evolution equations

Bangti Jin, Buyang Li, Zhi Zhou

arXiv:1703.08808v1math.NA

TL;DR

High-order BDF convolution quadrature loses accuracy for fractional evolution equations when weak initial-time singularities arise from incompatible source and initial data. The paper develops algebraic corrections at the first k − 1 steps, proving kth-order accuracy at fixed positive times without compatibility conditions and supporting it numerically, while noting deterioration near t = 0 without compatibility.

  • Problem

    Weak solution singularities caused by source-term incompatibility generally reduce direct high-order BDF convolution quadrature to first-order accuracy.

  • Method

    The paper systematically constructs algebraic corrections for the starting k − 1 steps of BDF convolution quadrature, with coefficients derived using Laplace transforms and generating functions.

  • Results

    The corrected BDFk achieves kth-order accuracy at fixed positive times without source compatibility conditions or extra solution regularity, supported by numerical experiments.

  • Takeaways & Limitations

    The correction framework provides robust high-order BDF schemes for subdiffusion and diffusion-wave equations while modifying only their initial steps.

  • Takeaways & Limitations

    Without compatibility conditions, the error estimate deteriorates as t approaches 0 because the solution and its derivatives exhibit weak singularities there.

Abstract

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We develop proper correction formulas at the starting $k-1$ steps to restore the desired $k^{\rm th}$-order convergence rate of the $k$-step BDF convolution quadrature for discretizing evolution equations involving a fractional-order derivative in time. The desired $k^{\rm th}$-order convergence rate can be achieved even if the source term is not compatible with the initial data, which is allowed to be nonsmooth. We provide complete error estimates for the subdiffusion case $α\in (0,1)$, and sketch the proof for the diffusion-wave case $α\in(1,2)$. Extensive numerical examples are provided to illustrate the effectiveness of the proposed scheme.

1 Introduction

The paper addresses order reduction in high-order BDF convolution quadrature caused by weak initial-time singularities, especially when source and initial data are incompatible. It develops starting-step corrections that recover kth-order accuracy for fractional evolution equations.

  • Methodological context: The framework builds on Lubich convolution quadrature, whose stability properties inherit from the underlying linear multistep methods.The paper applies high-order BDF-generated convolution quadrature to fractional evolution equations.
  • Motivation: Weak solution singularities at t = 0 generally reduce direct high-order BDF convolution quadrature to first-order accuracy, even with smooth data.The singularity arises when the source term is incompatible with the initial data.
  • Motivation: Starting-weight corrections can capture leading singularities and recover a uniform O(τ k) rate, but their PDE extension requires source-term compatibility conditions.This limitation motivates a different correction strategy for fractional evolution PDEs.
  • Contribution: The proposed strategy systematically corrects only the starting k − 1 steps of high-order BDF methods using algebraic criteria derived from solution representations.Explicit correction coefficients are provided for BDFs up to order 6.
  • Contribution: Corrected BDFk schemes cover subdiffusion, α ∈ (0, 1), and diffusion-wave problems, α ∈ (1, 2), with stability behavior characterized across fractional orders.For conditionally stable diffusion-wave cases, the paper gives an explicit CFL condition on the time step.
  • Results: The corrected schemes achieve kth-order accuracy at fixed positive times without compatibility conditions on the source term or extra solution regularity, with numerical support.The stated error bound depends only on data regularity.

2 BDFs for Subdiffusion and its Correction

For subdiffusion, the paper modifies high-order BDF convolution quadrature at its first k − 1 steps to cancel singularity-induced error terms. Algebraic correction criteria yield kth-order accuracy at fixed positive times, while uniform-in-time accuracy requires compatibility conditions.

  • Standard BDF convolution quadrature: BDFk convolution quadrature is posed on a uniform time grid and approximates the fractional derivative for orders k = 1, ..., 6.The coefficients can be computed efficiently by fast Fourier transform or recursion.
  • Accuracy loss: Direct BDFk discretization generally achieves only first-order accuracy because weak solution singularities at t = 0 persist through later numerical steps.Smooth initial data and source terms do not generally remove this order reduction.
  • Correction scheme: The corrected scheme adds starting-step terms with coefficients chosen to achieve O(τ k) accuracy for general initial data and possibly incompatible right-hand sides.Its only difference from the standard scheme is at the first k − 1 steps, making implementation straightforward.
  • Derivation of correction criteria: The correction coefficients are derived by comparing continuous and discrete kernel representations obtained through Laplace transforms and generating functions.The resulting algebraic criteria cancel the leading singularities for BDFk with k = 3, ..., 6.
  • Coefficient computation: For k = 4 and 6, the coefficients b_(k−2),j vanish identically for j = 1, ..., k − 1.The coefficients are computed recursively from the correction relations and tabulated for the relevant BDF orders.
  • Error estimates: Uniform O(τ k) convergence requires compatibility conditions; without them, the estimate deteriorates as t approaches 0 because the solution and derivatives remain weakly singular.The theorem’s stated estimate assumes Av ∈ L2(Ω), while analogous bounds can be obtained for v ∈ L2(Ω) with proof modifications.

3 Corrected BDF for diffusion-wave problem

The corrected BDF scheme extends systematic starting-step corrections to diffusion-wave equations, achieving kth-order accuracy under explicit algebraic criteria and stability conditions. The approach also relaxes source-term regularity requirements and identifies when high-order BDFs require a CFL restriction.

  • Motivation and corrected scheme: The diffusion-wave correction targets the failure of straightforward BDFk discretization to attain O(τ^k) convergence and avoid unnecessarily high regularity assumptions on f.The corrected scheme rewrites f as a time derivative and uses corrected starting steps.
  • Motivation and corrected scheme: The corrected scheme uses ¯∂τg^n instead of f^n, with correction terms designed to ensure the desired O(τ^k) rate.
  • Stability: For α<α*(k)=π/(π−ϑ_k), the scheme is unconditionally stable; for α≥α*(k), stability requires τ^αr(A)≤c(α,k)−γ.The critical values α*(k) are 1.91, 1.68, 1.40, and 1.11 for k=3,...,6.
  • Stability: The CFL constant c(α,k) is computed numerically, and its dependence on BDF order is non-monotone: BDF6 remains nonzero as α approaches 2, while BDF4 can be less restrictive than BDF3.
  • Coefficient selection: The correction coefficients are selected through algebraic criteria derived from solution representations, with coefficients b^(k) computed analogously to the subdiffusion case.The resulting b^(k) coefficients are reported in Table 3.
  • Accuracy and regularity: Under the stated criteria and stability condition, Theorem 3.2 provides an error estimate for any t_n>0, while requiring only the (k−1)th time derivative of f.This relaxes the regularity requirement compared with the subdiffusion theorem.

4 Numerical experiments and discussions

Numerical experiments show that corrected high-order BDF schemes recover the desired temporal convergence for subdiffusion and diffusion-wave problems, while diffusion-wave stability can require a step-size restriction.

  • Subdiffusion: Corrected BDFk schemes converge steadily at O(τ^k) for subdiffusion case (a), confirming the predicted rates.The experiments use piecewise linear Galerkin finite elements in space and focus on temporal convergence.
  • Subdiffusion: BDF5 is preferred over BDF6 in the preasymptotic regime because BDF6 reaches asymptotic convergence only at a relatively small time step.For case (a), BDF6’s asymptotic regime begins at N = 50.
  • Subdiffusion: Initial correction is necessary: uncorrected BDFk schemes achieve only first-order convergence, with nearly identical accuracy across orders.The low-order behavior is attributed to poor approximation during the initial steps.
  • Subdiffusion: Corrected BDFk also achieves O(τ^k) for the inhomogeneous subdiffusion case, whereas uncorrected BDF, L1, and L1-2 schemes reach only O(τ).This result holds despite smooth problem data in the reported inhomogeneous example.
  • Diffusion-wave: For diffusion-wave problems with α ≥ α*(k), the corrected scheme is stable only under the CFL condition τ ≤ τ0; at α = 1.5 and BDF5, τ0 ≈ 5.60 × 10^-4.The scheme is unstable for τ = 5.88 × 10^-4 and stable for τ = 5.55 × 10^-4, confirming the threshold’s sharpness.
  • Diffusion-wave: For diffusion-wave problems with α < α*(k), the corrected BDFk scheme is unconditionally stable and converges at O(τ^k).The reported numerical convergence agrees with Theorem 3.2.

A An alternative view on the correction scheme (2.4)

The appendix relates the proposed correction to an alternative convolution-quadrature construction and characterizes the correction as a minimal modification of the starting steps.

  • The alternative construction provides the desired accuracy for the corresponding single source component.
  • The convolution-quadrature weights are coefficients in the power-series expansion of δτ(ζ)^−ℓ−1.
  • The approach replaces δ(ζ)^−ℓ−1 with a close approximation γℓ(ζ) whose error is O(|ζ−1|^{k−ℓ−1}).
  • The proposed scheme differs from the alternative only in the starting k−1 steps.
  • The correction is unique among schemes modifying only those starting steps while retaining O(τ^k) accuracy.

B Proof of Theorem 2.1

The proof establishes analytic and sectorial estimates for the BDF symbol and uses generating functions and contour representations to derive the corrected-scheme analysis.

  • The contour estimates use sector choices for θ and δ together with bounds on δτ(e^−zτ).
  • The proof converts the discrete equations into generating-function identities by multiplying by ζ^n, summing, and collecting convolution terms.
  • Cauchy’s integral formula and the change of variables ζ=e^−zτ produce contour representations over Γτ.
  • The resolvent-related estimates rely on the zero-free polynomial η(ζ)=δτ(ζ)/(1−ζ) near the unit circle.
  • The BDF symbol δτ(e^−zτ) remains in suitable sectors, enabling analyticity of the operator kernel K(δτ(e^−zτ)).

C Proof of Theorem 2.2

The proof of Theorem 2.2 decomposes the error into contour-integral terms and bounds the discrete kernel approximation using symbol and resolvent estimates.

  • The kernel approximation satisfies ∥µ(e^−zτ)K(δτ(e^−zτ))−K(z)∥≤cτ^k|z|^{k−1−α}.
  • The symbol estimate |µ(e^−zτ)−1|≤cτ^k|z|^k supplies one component of the kernel bound.
  • The proof splits U^n−u(t_n) into terms I_1 through I_4 and estimates them separately.
  • The estimate for I_3 follows by direct computation after selecting δ=t_n^−1.
  • The I_4 term is treated as the error for a numerical solution with the compatible right-hand side R_k.

D Proof of Theorem 3.1

The diffusion-wave proof applies the source-term splitting and contour argument under alternative stability conditions, ensuring analyticity of the relevant operator kernel.

  • The functions W^n satisfy a discrete equation obtained from the source splitting, with the correction term entering for k≤n≤N.
  • Multiplication by ζ^n and summation convert the discrete system into a generating-function representation.
  • Under Condition 3.1(i), the transformed symbol lies in a sector that preserves analyticity of K(δτ(e^−τz)).
  • Under Condition 3.1(ii), analyticity follows from a positive distance between δ(e^−zτ)^α and the scaled spectrum of A.
  • Cauchy’s integral formula and ζ=e^−zτ complete the contour-based representation in both cases.

E Proof of Theorem 3.2

Theorem 3.2 is established by adapting the proof of Theorem 2.2 and verifying resolvent estimates under the stated conditions. The argument uses spectral properties of the discrete Laplacian, contour geometry, and CFL-based bounds.

  • Under Condition 3.1(i), Theorem 3.2 follows by the same approach as Theorem 2.2 using estimates (3.5) and (3.6).
  • Under Condition 3.1(ii), the proof is analogous provided the required resolvent estimate holds.
  • For the discrete Laplacian A = ∆h, the closure of the spectrum is S(A) = [−r(A), 0], enabling the contour argument through a positive angle condition.
  • The CFL condition and estimate (B.1) control the difference between the relevant δ-values when θ is close to π/2.
  • Choosing a sufficiently small contour parameter δ yields the needed bound, completing the proof of (E.1).
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