--- title: Variable-step L2 Formula for Caputo Derivatives url: https://www.emergentmind.com/topics/variable-step-l2-formula type: topic --- # Variable-step L2 Formula for Caputo Derivatives Variable-step L2 formula denotes a family of nonuniform, locally quadratic time-discretization formulas for Caputo-type fractional derivatives on meshes \(0=t_00,\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 \(\nabla_\tau u^k:=u^k-u^{k-1}\). The underlying continuous operator is the Caputo derivative \[ {}^C D_t^\alpha w(t_n)=\frac{1}{\Gamma(1-\alpha)}\int_0^{t_n}(t_n-s)^{-\alpha}w'(s)\,ds, \] or, in the variable-order setting, \[ {}_0^C D_t^{\alpha(t)} u(t) =\frac{1}{\Gamma(1-\alpha(t))}\int_0^t \frac{u'(\xi)}{(t-\xi)^{\alpha(t)}}\,d\xi. \] The distinctive feature of a variable-step L2 formula is that the history term is represented by \(n\)-dependent discrete convolution weights derived from local quadratic interpolation rather than by uniform-grid coefficients [2508.17178] [2311.13216] [2606.28443]. | Formulation | Evaluation point | Representative operator | |---|---|---| | Variable-step L2 type | \(t_n\) | \(\sum_{k=1}^n B_{n-k}^{(n)}\,\Delta_\tau w(t_k)\) | | Nonuniform L2-\(1_\sigma\) | \(t_{n-\theta}\), \(\theta=\alpha/2\) | \(\sum_{k=1}^n a_{n,k}\,\Delta u^k\) | | Variable-order L2-\(1_\sigma\) | \(t_{n-\sigma_n}\), \(\sigma_n=\alpha_n/2\) | \(\sum_{k=1}^n c_{n-k,n}^{(\alpha_n^*)}\,\nabla_\tau u^k\) | ## 2. Exact nonuniform constructions For the exact variable-step L2-type approximation on a nonuniform mesh, the Caputo derivative at \(t_n\) is written in a “\(c\)–\(d\)” split: \[ {}^C D_t^\alpha w(t_n) = \sum_{k=1}^{n} c_{n-k}^{(n)} \,\Delta_\tau w(t_k) + \frac{\rho_n}{1+\rho_n}\,d_0^{(n)}\big(\Delta_\tau w(t_n)-\rho_n\Delta_\tau w(t_{n-1})\big) + \sum_{k=1}^{n-1} d_{n-k}^{(n)}\,\frac{\Delta_\tau w(t_{k+1})-\rho_{k+1}\Delta_\tau w(t_k)}{\rho_{k+1}(1+\rho_{k+1})} + R_t^n, \] with \[ c_{n-k}^{(n)} := \frac{(t_n - t_{k-1})^{1-\alpha} - (t_n - t_k)^{1-\alpha}}{\tau_k\,\Gamma(2-\alpha)}, \] \[ d_{n-k}^{(n)} := \frac{2\big((t_n - t_{k-1})^{2-\alpha} - (t_n - t_k)^{2-\alpha}\big)}{\tau_k^2\,\Gamma(3-\alpha)} -\frac{(t_n - t_{k-1})^{1-\alpha} + (t_n - t_k)^{1-\alpha}}{\tau_k\,\Gamma(2-\alpha)}. \] Collecting terms yields the unified convolution representation \[ {}^C D_t^\alpha w(t_n)=\sum_{k=1}^n B_{n-k}^{(n)}\,\Delta_\tau w(t_k)+R_t^n. \] Under \(w\in C^3[0,T]\), the truncation error is \(2-\alpha\) at the first step and \(3-\alpha\) at interior steps [2508.17178]. For the nonuniform L2-\(1_\sigma\) formula for the time-fractional Allen–Cahn model, the derivative is evaluated at \[ t_{n-\theta}=\theta t_{n-1}+(1-\theta)t_n,\qquad \theta=\frac{\alpha}{2}, \] and approximated by \[ (\partial_\tau^\alpha u)^{n-\theta}\approx \sum_{k=1}^n a_{n,k}\,\Delta u^k. \] The weights are built from exact integrals \[ a^{(n)}_{\,n-k} =\frac{\omega_{2-\alpha}(t_{n-\theta}-t_{k-1})-\omega_{2-\alpha}(t_{n-\theta}-\min\{t_k,t_{n-\theta}\})}{\tau_k}, \] \[ \zeta^{(n)}_{\,n-k} =\frac{2}{\tau_k^2}\int_{t_{k-1}}^{t_k}\big(s-t_{k-\frac12}\big)\,\omega_{1-\alpha}(t_{n-\theta}-s)\,ds, \] and assembled into \[ a_{n,n}=a^{(n)}_0+\frac{1}{r_n(1+r_n)}\,\zeta^{(n)}_1, \] \[ a_{n,k}=a^{(n)}_{n-k}+\frac{1}{r_k(1+r_k)}\,\zeta^{(n)}_{\,n-k+1}-\frac{1}{1+r_{k+1}}\,\zeta^{(n)}_{\,n-k}, \quad 2\le k\le n-1, \] \[ a_{n,1}=a^{(n)}_{\,n-1}-\frac{1}{1+r_2}\,\zeta^{(n)}_{\,n-1}. \] The last interval uses a linear interpolant on \([t_{n-1},t_{n-\theta}]\), while earlier intervals use quadratic interpolation [2311.13216]. For variable-order subdiffusion, the variable-step L2-\(1_\sigma\) operator is evaluated at the superconvergent point \[ t_{n-\sigma_n}:=\sigma_n t_{n-1}+(1-\sigma_n)t_n,\qquad \sigma_n=\frac{\alpha_n}{2}, \] with \(\alpha_n^*:=\alpha(t_{n-\sigma_n})\), and the discrete operator is \[ \big(D_\tau^{\alpha_n^*} u\big)^{n-\sigma_n} = a_0^{(\alpha_n^*)}\,\nabla_\tau u^n + \sum_{k=1}^{n-1}\Big(a_{n-k}^{(\alpha_n^*)}\,\nabla_\tau u^k - b_{n-k}^{(\alpha_n^*)}\,\nabla_\tau u^k + \rho_k\,b_{n-k}^{(\alpha_n^*)}\,\nabla_\tau u^{k+1}\Big) = \sum_{k=1}^{n} c_{n-k,n}^{(\alpha_n^*)}\,\nabla_\tau u^k. \] The coefficients \(a_{n-k}^{(\alpha_n^*)}\) and \(b_{n-k}^{(\alpha_n^*)}\) are kernel integrals obtained from local quadratic interpolation, and \(a_{n-k}^{(\alpha_n^*)}\) has an explicit closed form [2606.28443]. ## 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-\(1_\sigma\) scheme, \[ (\partial_{\tau}^{\alpha}u)^{n-\theta}=J^n_{\mathrm{CN}}+J^n_{\mathrm{L1}}, \] with \[ J^n_{\mathrm{CN}}:=\frac{\alpha}{2-\alpha}\,a^{(n)}_0\,\Delta u^n,\qquad J^n_{\mathrm{L1}}:=\sum_{k=1}^n\hat a^{(n)}_{\,n-k}\,\Delta u^k. \] 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 \(r_k\ge r_\star(\alpha)\), the auxiliary kernels \[ A_0^{(n)}:=2\hat a_0^{(n)},\qquad A_{n-k}^{(n)}:=\hat a_{n-k}^{(n)} \] are positive, decreasing, temporally monotone, and satisfy a discrete convexity inequality. These properties lead to the discrete gradient structure \[ 2\big(\Delta u^n\big)\,(\partial_{\tau}^{\alpha}u)^{n-\theta} =\mathcal{G}[\Delta u^n]-\mathcal{G}[\Delta u^{n-1}] +\mathcal{R}[\Delta u^n] +\frac{2\alpha}{2-\alpha}\,a^{(n)}_0\,\|\Delta u^n\|^2, \] with nonnegative functionals \(\mathcal G\) and \(\mathcal R\) [2311.13216]. The relaxed L2-type analysis for the time-fractional Cahn–Hilliard equation uses a different split: \[ {}^C D_t^\alpha w(t_n) = \Big(\theta\,c_0^{(n)} + \frac{\rho_n}{1+\rho_n}\,d_0^{(n)}\Big)\,\Delta_\tau w^n - \frac{\rho_n^2}{1+\rho_n}\,d_0^{(n)}\,\Delta_\tau w^{n-1} + \sum_{k=1}^n \widetilde{c}_{n-k}^{(n)}\,\Delta_\tau w^k, \] where \[ \theta = \frac{1}{2-\alpha} +\frac{2^{1-\alpha}\alpha^2+\alpha-2\alpha^2}{2(2-\alpha)(1+\overline{\rho})}, \qquad \overline{\rho}\approx 4.7476114. \] The first two terms form a local two-step stabilizer, and the remaining \(\widetilde c\)-kernels are organized to recover a discrete gradient identity. The key one-step inequality is \[ {}^C D_t^\alpha w(t_n)\,\Delta_\tau w^n \ge \mathcal{G}[\Delta_\tau w^n]-\mathcal{G}[\Delta_\tau w^{n-1}] +\frac{q(\rho_n,\rho_{n+1},\alpha)}{2\tau_n^\alpha\Gamma(3-\alpha)}\big(\Delta_\tau w^n\big)^2, \] where \(q(\rho_n,\rho_{n+1},\alpha)>0\) on the admissible ratio range. This inequality is the cornerstone of the subsequent energy dissipation theory [2508.17178]. ## 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 \[ 1\le \rho_k=\frac{\tau_k}{\tau_{k-1}}\le \rho^*(\alpha),\qquad k\ge 2, \] where \(\rho^*(\alpha)\) is the unique root of \[ q_2(\rho,\alpha):=\frac{1+\rho}{\alpha}+\rho-\rho^{\,2-\frac{\alpha}{2}} +2^{-\alpha}\alpha+\frac12-\alpha=0, \] and satisfies \[ \rho^*(\alpha)>\overline{\rho}\approx 4.7476114,\qquad \rho^*(1)\approx 4.864,\qquad \rho^*(\alpha)\to+\infty\ \text{as}\ \alpha\to0^+. \] The same paper compares this with the earlier Liao–Liu–Zhao framework, which required \[ 0.3960\le \frac{\tau_k}{\tau_{k-1}}\le r^*(\alpha),\qquad r^*(\alpha)\ge 4.660. \] The new result removes the lower bound \(0.3960\) and enlarges the admissible upper bound beyond \(4.7476\) [2508.17178]. For the nonuniform L2-\(1_\sigma\) Allen–Cahn analysis, the condition is instead a lower-ratio constraint, \[ r_k=\frac{\tau_k}{\tau_{k-1}}\ge r_\star(\alpha),\qquad k\ge 2, \] where \(r_\star(\alpha)\in(0.3865,0.4037)\) is the unique positive root of \[ h(r,\alpha):= 2\sqrt{\frac{2(1-\alpha/2)\,r}{1+\alpha+(1-\alpha/2)\,r}} +\frac{r}{1+r}+3-\frac{1}{r^2(1+r)}=0. \] Numerically, \(r_\star(0)\approx0.3865\) and \(r_\star(1)\approx0.4037\), and \(r_\star\) increases monotonically with \(\alpha\in(0,1)\) [2311.13216]. For the variable-order L2-\(1_\sigma\) discretization, the emphasis shifts from step-ratio admissibility to the choice of \(\alpha_n\) and the superconvergent point \(t_{n-\sigma_n}\). The analysis assumes \[ \alpha_n\in\Big[\min_{t\in[t_{n-1},t_n]}\alpha(t),\ \max_{t\in[t_{n-1},t_n]}\alpha(t)\Big], \qquad \alpha(t_{n-\alpha_n/2})\ge \alpha_n, \] with \(\sigma_n=\alpha_n/2\). 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 [2606.28443]. 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-\(1_\sigma\) scheme \[ \frac{\alpha}{2-\alpha}\,a_0^{(n)}(u^n-u^{n-1}) +\sum_{k=1}^{n}\hat a_{n-k}^{(n)}(u^k-u^{k-1}) =\varepsilon^2\Delta_h u^{n-\theta}-f(u)^{n-\theta} \] preserves the discrete maximum bound principle under the ratio constraint \(r_k\ge r_\star(\alpha)\) and a mild step-size restriction: \[ \|u^n\|_\infty\le 1\quad\text{for all }n\ge 1\quad\text{if}\quad \|u^0\|_\infty\le 1. \] It also satisfies the asymptotically compatible modified energy law \[ \partial_{\tau}\mathcal{E}_{\alpha}[u^n] +\frac{\alpha}{2(2-\alpha)}\,a^{(n)}_0\,\tau_n\,\|u^{n-\frac12}\|^2\le 0, \] where \[ \mathcal{E}_{\alpha}[u^n]=E[u^n]+\frac12(\mathcal G[\Delta u^n],1). \] As \(\alpha\to1^-\), \(\mathcal E_\alpha[u^n]\to E[u^n]\), which recovers the Crank–Nicolson energy dissipation law [2311.13216]. 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 \[ \tau_{\max} \le \Bigg(\frac{(2-\alpha+2\rho_n)h^2}{12\,\kappa\,(1+\rho_n)\,\Gamma(3-\alpha)}\Bigg)^{1/\alpha}. \] Second, the scheme has exact discrete volume conservation: \[ \int_\Omega u(x,t_n)\,dx=\int_\Omega u(x,t_{n-1})\,dx. \] Third, the modified compatible energy \[ \mathcal E^n:=E^n+\frac{1}{\kappa}\big(\mathcal G[\Delta_\tau u^n],1\big)_{-\mathcal H} \] is nonincreasing if \[ \tau_n\le \Bigg(\frac{4\varepsilon^2\,q(\rho_n,\rho_{n+1},\alpha)}{\kappa\,\Gamma(3-\alpha)}\Bigg)^{1/\alpha}. \] Fourth, the error satisfies the temporal order \(3-\alpha\) and spatial order \(4\): \[ \|e^n\|\le C_1\,C\,\sqrt{b-a}\times \begin{cases} \tau^{2-\alpha}+h^4, & n=1,\\[2pt] \tau^{3-\alpha}+h^4, & 2\le n\le N. \end{cases} \] As \(\alpha\to1^-\), the \(J\)-kernels vanish and \(\mathcal E^n\to E^n\) [2508.17178]. For variable-order time-fractional subdiffusion, the nonuniform L2-\(1_\sigma\) operator satisfies the positivity estimate \[ \Big(\big(D_\tau^{\alpha_n^*}v\big)^{n-\sigma_n},\,v^{n-\sigma_n}\Big) \ge \frac12 \sum_{k=1}^n c_{n-k,n}^{(\alpha_n^*)}\Big(\|v^k\|^2-\|v^{k-1}\|^2\Big), \qquad 2\le n\le N. \] This underlies stability in the discrete \(L^2\) norm and an a priori estimate with temporal rate \(\min\{2,r\delta\}\) and spatial order \(\min\{s,k+1\}\). In particular, if \(r\ge 2/\delta\), the temporal accuracy is second order; if \(1\le r<2/\delta\), the temporal rate is \(r\delta\) [2606.28443]. ## 6. Implementation, limitations, and distinct usages Straightforward implementation is history-based. For the variable-order L2-\(1_\sigma\) operator, one stores the increments \(\nabla_\tau u^k\) and assembles \[ \big(D_\tau^{\alpha_n^*} u\big)^{n-\sigma_n} =\sum_{k=1}^{n} c_{n-k,n}^{(\alpha_n^*)}\,\big(u^k-u^{k-1}\big), \] which costs \(O(n)\) work per step because the weights are \(n\)-dependent. The paper emphasizes that no global recomputation is required, and it provides a graded mesh \[ t_n=T\Big(\frac{n}{N}\Big)^r,\qquad r\ge 1, \] with practical choices of \(\alpha_n\) that avoid solving a nonlinear equation [2606.28443]. For the relaxed L2-type Cahn–Hilliard scheme, the paper proposes the explicit nonuniform mesh \[ \tau_k =\frac{(2k+1)^3}{N(N+2)(2N^2+4N+3)}\,T,\qquad k=1,\dots,N, \] which gives \[ \rho_k=\Big(\frac{2k+1}{2k-1}\Big)^3\le \Big(\frac53\Big)^3\approx 4.6296296 <\overline\rho\approx 4.7476114<\rho^*(\alpha). \] The straightforward history sum costs \(O(n)\) per step and \(O(N^2)\) 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 [2508.17178]. For the nonuniform L2-\(1_\sigma\) Allen–Cahn scheme, adaptive time stepping is made compatible with the lower-ratio condition by enforcing \[ \tau_{n+1}=\max\{\tau_{\mathrm{ada}},\,r_\star\tau_n\}, \] and long-time simulation can be accelerated by a sum-of-exponentials approximation to \(\omega_{1-\alpha}\), reducing the time-convolution cost from \(O(N^2)\) to \(O(N\log N)\) for fixed tolerance [2311.13216]. 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 \(\Delta t=T/N\) and \(t_k=k\Delta t\) [2108.08437]. The phrase “variable-step L2 formula” is also used in distinct contexts. In \(L^2([a,b])\) Hilbert space approximation, it denotes the formula for the globally optimal \(n\)-step “escalier” fit to a curve, where the best step heights are interval means and the interior boundaries satisfy the midpoint condition \[ f(t_k)=\frac{c_k^*+c_{k+1}^*}{2},\qquad k=1,\dots,n-1, \] with convergence \[ \|f-s_n\|_2^2=O(1/n) \] under \(f\in C^1([a,b])\) with bounded \(f\) and \(f'\) [2510.16148]. In another neighboring but distinct usage, the variable-step BDF3 literature refers to a “variable-step \(L^2\) formula” as a mesh-robust \(L^2\) stability and convergence inequality for third-order BDF time stepping, rather than as an L2-type Caputo discretization [2204.12742]. Source: https://www.emergentmind.com/topics/variable-step-l2-formula