Variable-step L2 Formula for Caputo Derivatives

• Variable-step L2 formula is a nonuniform, locally quadratic discretization method for Caputo-type fractional derivatives that achieves second-order or (3-α) temporal accuracy.
• It employs n-dependent convolution weights and a local–nonlocal split to enforce stability properties such as positivity, discrete gradient structures, and energy dissipation.
• Adaptive mesh strategies with precise step-ratio conditions enable effective application in time-fractional models like Allen–Cahn and Cahn–Hilliard equations while optimizing computational efficiency.

Variable-step L2 formula denotes a family of nonuniform, locally quadratic time-discretization formulas for Caputo-type fractional derivatives on meshes $0=t_0<t_1<\cdots<t_N=T$, with coefficients that depend on the current time level and the local step ratios. In recent arXiv literature, the term covers both the exact nonuniform L2 formula evaluated at $t_n$ and the nonuniform L2-$1_\sigma$ or Alikhanov-type formula evaluated at shifted points $t_{n-\sigma}$ or $t_{n-\sigma_n}$. These formulas are designed to preserve second-order or $3-\alpha$ temporal accuracy on graded or adaptive meshes while supporting positivity, discrete gradient structures, maximum bound principles, energy dissipation laws, and $L^2$-stability estimates (Li et al., 24 Aug 2025, Liao et al., 2023, Huang et al., 26 Jun 2026). The same phrase also appears in an unrelated $L^2$ Hilbert-space fitting problem for multi-step functions (Bossu et al., 17 Oct 2025).

1. Core definition and notation

On a nonuniform time mesh, the standard notation is

$$\tau_k:=t_k-t_{k-1}>0,\qquad \rho_k:=\frac{\tau_k}{\tau_{k-1}},$$

together with backward differences $\Delta_\tau w(t_k):=w(t_k)-w(t_{k-1})$ or $t_n$0. The underlying continuous operator is the Caputo derivative

$t_n$1

or, in the variable-order setting,

$t_n$2

The distinctive feature of a variable-step L2 formula is that the history term is represented by $t_n$3-dependent discrete convolution weights derived from local quadratic interpolation rather than by uniform-grid coefficients (Li et al., 24 Aug 2025, Liao et al., 2023, Huang et al., 26 Jun 2026).

Formulation	Evaluation point	Representative operator
Variable-step L2 type	$t_n$4	$t_n$5
Nonuniform L2-$t_n$6	$t_n$7, $t_n$8	$t_n$9
Variable-order L2-$1_\sigma$0	$1_\sigma$1, $1_\sigma$2	$1_\sigma$3

2. Exact nonuniform constructions

For the exact variable-step L2-type approximation on a nonuniform mesh, the Caputo derivative at $1_\sigma$4 is written in a “$1_\sigma$5–$1_\sigma$6” split: $1_\sigma$7 with

$1_\sigma$8

$1_\sigma$9

Collecting terms yields the unified convolution representation

$t_{n-\sigma}$0

Under $t_{n-\sigma}$1, the truncation error is $t_{n-\sigma}$2 at the first step and $t_{n-\sigma}$3 at interior steps (Li et al., 24 Aug 2025).

For the nonuniform L2-$t_{n-\sigma}$4 formula for the time-fractional Allen–Cahn model, the derivative is evaluated at

$t_{n-\sigma}$5

and approximated by

$t_{n-\sigma}$6

The weights are built from exact integrals

$t_{n-\sigma}$7

$t_{n-\sigma}$8

and assembled into

$t_{n-\sigma}$9

$t_{n-\sigma_n}$0

$t_{n-\sigma_n}$1

The last interval uses a linear interpolant on $t_{n-\sigma_n}$2, while earlier intervals use quadratic interpolation (Liao et al., 2023).

For variable-order subdiffusion, the variable-step L2-$t_{n-\sigma_n}$3 operator is evaluated at the superconvergent point

$t_{n-\sigma_n}$4

with $t_{n-\sigma_n}$5, and the discrete operator is

$t_{n-\sigma_n}$6

The coefficients $t_{n-\sigma_n}$7 and $t_{n-\sigma_n}$8 are kernel integrals obtained from local quadratic interpolation, and $t_{n-\sigma_n}$9 has an explicit closed form (Huang et al., 26 Jun 2026).

3. Splitting, kernel monotonicity, and discrete gradient structure

A major analytical development is the replacement of the raw convolution form by a local–nonlocal split. For the nonuniform L2-$3-\alpha$0 scheme,

$3-\alpha$1

with

$3-\alpha$2

This isolates a local term analogous to the trapezoid rule of the first derivative and a nonlocal summation analogous to the L1 formula of the Caputo derivative. Under the weak step-ratio condition $3-\alpha$3, the auxiliary kernels

$3-\alpha$4

are positive, decreasing, temporally monotone, and satisfy a discrete convexity inequality. These properties lead to the discrete gradient structure

$3-\alpha$5

with nonnegative functionals $3-\alpha$6 and $3-\alpha$7 (Liao et al., 2023).

The relaxed L2-type analysis for the time-fractional Cahn–Hilliard equation uses a different split: $3-\alpha$8 where

$3-\alpha$9

The first two terms form a local two-step stabilizer, and the remaining $L^2$0-kernels are organized to recover a discrete gradient identity. The key one-step inequality is

$L^2$1

where $L^2$2 on the admissible ratio range. This inequality is the cornerstone of the subsequent energy dissipation theory (Li et al., 24 Aug 2025).

4. Step-ratio conditions and shifted evaluation points

The admissible step-ratio condition depends on the specific L2 family. For the relaxed L2-type formula for time-fractional Cahn–Hilliard equations, the requirement is

$L^2$3

where $L^2$4 is the unique root of

$L^2$5

and satisfies

$L^2$6

The same paper compares this with the earlier Liao–Liu–Zhao framework, which required

$L^2$7

The new result removes the lower bound $L^2$8 and enlarges the admissible upper bound beyond $L^2$9 (Li et al., 24 Aug 2025).

For the nonuniform L2-$L^2$0 Allen–Cahn analysis, the condition is instead a lower-ratio constraint,

$L^2$1

where $L^2$2 is the unique positive root of

$L^2$3

Numerically, $L^2$4 and $L^2$5, and $L^2$6 increases monotonically with $L^2$7 (Liao et al., 2023).

For the variable-order L2-$L^2$8 discretization, the emphasis shifts from step-ratio admissibility to the choice of $L^2$9 and the superconvergent point $$\tau_k:=t_k-t_{k-1}>0,\qquad \rho_k:=\frac{\tau_k}{\tau_{k-1}},$$0. The analysis assumes

$$\tau_k:=t_k-t_{k-1}>0,\qquad \rho_k:=\frac{\tau_k}{\tau_{k-1}},$$1

with $$\tau_k:=t_k-t_{k-1}>0,\qquad \rho_k:=\frac{\tau_k}{\tau_{k-1}},$$2. The paper also states that numerical results show the second inequality can be relaxed or omitted without degrading the observed accuracy or stability, so many superconvergent points are admissible at each time level (Huang et al., 26 Jun 2026). These results suggest that there is no single universal step-ratio or shift rule for all variable-step L2 formulas.

5. Stability, energy laws, and convergence

For the time-fractional Allen–Cahn model, the nonuniform L2-$$\tau_k:=t_k-t_{k-1}>0,\qquad \rho_k:=\frac{\tau_k}{\tau_{k-1}},$$3 scheme

$$\tau_k:=t_k-t_{k-1}>0,\qquad \rho_k:=\frac{\tau_k}{\tau_{k-1}},$$4

preserves the discrete maximum bound principle under the ratio constraint $$\tau_k:=t_k-t_{k-1}>0,\qquad \rho_k:=\frac{\tau_k}{\tau_{k-1}},$$5 and a mild step-size restriction: $$\tau_k:=t_k-t_{k-1}>0,\qquad \rho_k:=\frac{\tau_k}{\tau_{k-1}},$$6 It also satisfies the asymptotically compatible modified energy law

$$\tau_k:=t_k-t_{k-1}>0,\qquad \rho_k:=\frac{\tau_k}{\tau_{k-1}},$$7

where

$$\tau_k:=t_k-t_{k-1}>0,\qquad \rho_k:=\frac{\tau_k}{\tau_{k-1}},$$8

As $$\tau_k:=t_k-t_{k-1}>0,\qquad \rho_k:=\frac{\tau_k}{\tau_{k-1}},$$9, $\Delta_\tau w(t_k):=w(t_k)-w(t_{k-1})$0, which recovers the Crank–Nicolson energy dissipation law (Liao et al., 2023).

For the time-fractional Cahn–Hilliard equation, the relaxed L2-type temporal approximation combined with a compact fourth-order spatial discretization yields a fully discrete scheme with four stated structural properties. First, unique solvability holds if

$\Delta_\tau w(t_k):=w(t_k)-w(t_{k-1})$1

Second, the scheme has exact discrete volume conservation: $\Delta_\tau w(t_k):=w(t_k)-w(t_{k-1})$2 Third, the modified compatible energy

$\Delta_\tau w(t_k):=w(t_k)-w(t_{k-1})$3

is nonincreasing if

$\Delta_\tau w(t_k):=w(t_k)-w(t_{k-1})$4

Fourth, the error satisfies the temporal order $\Delta_\tau w(t_k):=w(t_k)-w(t_{k-1})$5 and spatial order $\Delta_\tau w(t_k):=w(t_k)-w(t_{k-1})$6: $\Delta_\tau w(t_k):=w(t_k)-w(t_{k-1})$7 As $\Delta_\tau w(t_k):=w(t_k)-w(t_{k-1})$8, the $\Delta_\tau w(t_k):=w(t_k)-w(t_{k-1})$9-kernels vanish and $t_n$00 (Li et al., 24 Aug 2025).

For variable-order time-fractional subdiffusion, the nonuniform L2-$t_n$01 operator satisfies the positivity estimate

$t_n$02

This underlies stability in the discrete $t_n$03 norm and an a priori estimate with temporal rate $t_n$04 and spatial order $t_n$05. In particular, if $t_n$06, the temporal accuracy is second order; if $t_n$07, the temporal rate is $t_n$08 (Huang et al., 26 Jun 2026).

6. Implementation, limitations, and distinct usages

Straightforward implementation is history-based. For the variable-order L2-$t_n$09 operator, one stores the increments $t_n$10 and assembles

$t_n$11

which costs $t_n$12 work per step because the weights are $t_n$13-dependent. The paper emphasizes that no global recomputation is required, and it provides a graded mesh

$t_n$14

with practical choices of $t_n$15 that avoid solving a nonlinear equation (Huang et al., 26 Jun 2026).

For the relaxed L2-type Cahn–Hilliard scheme, the paper proposes the explicit nonuniform mesh

$t_n$16

which gives

$t_n$17

The straightforward history sum costs $t_n$18 per step and $t_n$19 in total, while fast history techniques such as sum-of-exponentials, adaptive windowing, and short-memory can be incorporated without changing the local stabilizer analysis (Li et al., 24 Aug 2025).

For the nonuniform L2-$t_n$20 Allen–Cahn scheme, adaptive time stepping is made compatible with the lower-ratio condition by enforcing

$t_n$21

and long-time simulation can be accelerated by a sum-of-exponentials approximation to $t_n$22, reducing the time-convolution cost from $t_n$23 to $t_n$24 for fixed tolerance (Liao et al., 2023).

A common misconception is that the positivity arguments available for the uniform-grid L2 operator automatically extend to nonuniform grids. The uniform-grid phase-field analysis of the classical L2 operator explicitly does not provide variable-step L2 weights or the necessary positivity or Cholesky arguments for nonuniform meshes; all derivations there assume $t_n$25 and $t_n$26 (Quan et al., 2021).

The phrase “variable-step L2 formula” is also used in distinct contexts. In $t_n$27 Hilbert space approximation, it denotes the formula for the globally optimal $t_n$28-step “escalier” fit to a curve, where the best step heights are interval means and the interior boundaries satisfy the midpoint condition

$t_n$29

with convergence

$t_n$30

under $t_n$31 with bounded $t_n$32 and $t_n$33 (Bossu et al., 17 Oct 2025). In another neighboring but distinct usage, the variable-step BDF3 literature refers to a “variable-step $t_n$34 formula” as a mesh-robust $t_n$35 stability and convergence inequality for third-order BDF time stepping, rather than as an L2-type Caputo discretization (Liao et al., 2022).