---
title: 'Alikhanov Scheme: Fractional Time Discretization'
url: https://www.emergentmind.com/topics/alikhanov-scheme
type: topic
---

# Alikhanov Scheme: Fractional Time Discretization

The Alikhanov scheme denotes a class of high-order, convolution-based fractional-step time discretizations for Caputo-type time-fractional partial differential equations (PDEs) on uniform or nonuniform temporal meshes. Its core is an L2-type, piecewise-quadratic reconstruction for the time-derivative, attaining second-order accuracy in time even in the presence of weak initial singularities, as seen in subdiffusion or diffusion-wave-type problems. The scheme is central to fractional numerical analysis, underpinning optimal, unconditionally stable solvers for a broad spectrum of fractional parabolic and hyperbolic equations and their nonlinear variants.

## 1. Discrete Caputo Derivatives via the Alikhanov Approach

The Alikhanov scheme implements a discrete approximation to the Caputo fractional derivative $D_t^\alpha u(t)$ of order $\alpha \in (0,2)$ on (possibly nonuniform) meshes. For $0<t_0<t_1<\dots<t_N=T$, the key ingredients are:

- **Time mesh and shifted nodes:** Adopts a general quasi-graded or fully nonuniform mesh with grading parameter $r\ge 1$, introducing fractional offset points $t_n^* = t_n - (1-\sigma)\tau_n$, where $\sigma=1-\alpha/2$ for $\alpha\in(0,1)$ and $\sigma=1-\beta/2$ for higher order ($\beta=\alpha/2$).
- **Piecewise interpolation:** Utilizes quadratic Lagrange interpolants on $[t_{k-1},t_{k+1}]$ and linear interpolants on $[t_{n-1},t_n]$.
- **Convolution quadrature:** The discrete Caputo derivative at $t_n^*$ is expressed as a convolution sum:
  $$
  \delta_t^{\alpha,*} u^{n} = \sum_{j=1}^n g_{n-1,j-1}(u^{j}-u^{j-1}),
  $$
  where the weights $g_{n-1,j-1}$ derive from explicit integrals over the mesh subintervals involving the singular kernel $(t_n^*-s)^{-\alpha}$.

This structure ensures positivity and monotonicity of weights under mild mesh-ratio constraints, which is crucial for stability and maximum principles [2601.18505].

## 2. Construction of Fully Discrete Schemes

The Alikhanov temporal discretization fits naturally within fully discrete frameworks for time-fractional PDEs via:

- **Spatial discretization:** Commonly finite differences (five-point, central difference) or finite elements, yielding spatial error $O(h^2)$ (second-order).
- **Linearization of nonlinearities:** The Newton/Fréchet linearization about $U^{n-1}$, leading to
  $$
  f(u^n) \approx f(U^{n-1}) + f'(U^{n-1})(u^n-U^{n-1}),
  $$
  is combined with the implicit/explicit time blending at $t_n^*$ (shifted Crank–Nicolson).
- **Linear algebraic system:** Each time step requires inversion of sparse matrices of the form $A_n = g_{n-1,n-1}I + \sigma L_h - \sigma f'(U^{n-1})$.

For $\alpha\in(1,2)$ (diffusion-wave regime), symmetric fractional-order reduction (SFOR) reformulates the problem as a coupled system involving Caputo derivatives of order $\beta=\alpha/2$ applied to auxiliary variables, with a corresponding Alikhanov discretization for each equation [2601.20566, 2101.09678].

## 3. Stability Theory and Discrete Comparison Principles

Foundational stability of Alikhanov schemes is established through:

- **Maximum/comparison principles:** Under mesh conditions and $\sigma\ge 1/2$, the scheme satisfies discrete comparison and maximum principles, enabling monotonicity-based $L^\infty$ control and vital barriers [2601.18505].
- **Energy methods:** Discrete fractional integration-by-parts and Grönwall-type inequalities, leveraging positivity and monotonicity of weights, underpin unconditional stability results in various spatial norms ($H^1$, $L^2$) [2102.09396, 1704.06733].
- **Barrier functions:** For pointwise in time control, constructing auxiliary (often piecewise power-type) “barrier” sequences yields sharp estimates for the inhomogeneous error in the discrete fractional equation [2601.18505].

These principles are robust under both uniform and graded meshes, the latter being essential for resolving initial singular layers.

## 4. Error Analysis and Convergence Rates

Alikhanov schemes achieve optimal temporal accuracy, contingent on the grading parameter $r$ and fractional order $\alpha$:

| Temporal Mesh Grading $r$            | Local Order (pointwise)         | Global Order (maximal in time)        |
|--------------------------------------|---------------------------------|---------------------------------------|
| $r = 1$ (uniform)                   | $\min\{1,2\}=1$                 | $\min\{\alpha,2\} = \alpha$           |
| $r = 2$                             | $2$                             | $\min\{\alpha\,r,2\}$                 |
| $r = 2/\alpha$                      | $2$                             | $2$                                   |

- **Subdiffusion ($\alpha\in(0,1)$):** Local $L^2$ temporal error is $O(\tau^{\min\{r,2\}})$; global error is $O(\tau^{\min\{\alpha r,2\}})$.
- **Diffusion-wave ($\alpha\in(1,2)$):** Via SFOR, local and global $H^1$-or $L^2$-errors are $O(\tau^{\min\{2,r\}} + h^2)$, with full second-order attainable for $r\ge2$ [2102.09396, 2601.20566].
- **Nonuniform (graded) meshes:** Achieve second-order in time away from $t=0$ once $r\ge2$; uniform meshes are limited by solution singularity at $t=0$ (only $O(\tau^\alpha)$ global).

The sharpness of these rates has been confirmed by numerical experiments for 2D subdiffusion, sine-Gordon, and semilinear diffusion-wave equations [2601.18505, 2601.20566, 2101.09678].

## 5. Extensions: General Variable Coefficients, Nonuniform Grids, Nonlinearity

Alikhanov schemes have been extended to:

- **Variable coefficients:** Schemes retain second-order accuracy under mild assumptions on time-space dependence of elliptic and lower-order coefficients, provided coefficient variations are smooth and mesh grading is compatible [2102.09396].
- **Nonuniform meshes:** The generalized Alikhanov formula accommodates arbitrary nonuniform timesteps $\tau_k$, with precise construction of convolution weights to ensure stability and accuracy [2109.02028].
- **Nonlinearities:** Newton or other fixed-point type linearization strategies are supported; discrete energy methods yield stability even for fully nonlinear source terms under Lipschitz conditions [1704.06733].

## 6. Fast Solvers, Memory Reduction, and Preconditioning

The nonlocal "history" in time-fractional systems incurs $\mathcal{O}(N^2)$ complexity. Advances leveraging the Alikhanov framework include:

- **Sum-of-exponentials (SOE) acceleration:** Replaces power-law kernels with exponential sums, reducing memory and computational cost to $\mathcal{O}(N\log N)$ while retaining $O(\tau^2)$ accuracy if the SOE tolerance matches the Alikhanov discretization error [2109.02028].
- **All-at-once systems and preconditioning:** Block-structured Kronecker product systems arising from high-order Alikhanov schemes may be solved efficiently using FFT-accelerated Krylov methods paired with circulant or bilateral preconditioners, yielding mesh-independent condition numbers and rapid convergence [2109.06510].
- **Adaptive and graded meshes:** Adaptive step-size strategies and mesh grading further optimize accuracy near $t=0$, essential in resolving initial singularities typically induced by fractional dynamics [2101.09678, 2102.09396].

## 7. Representative Applications and Numerical Results

The Alikhanov scheme serves as the backbone in various application contexts:

- **2D nonlinear subdiffusion:** Rigorous pointwise and global convergence for nonlinear PDEs with weak singularities [2601.18505].
- **Time-fractional Black–Scholes equation:** Efficient, high-order solvers (second in time, fourth in space) for anomalous finance models, with SOE acceleration and graded meshes [2109.02028].
- **Time-space fractional Bloch–Torrey and advection–diffusion:** Second-order implicit schemes, effective all-at-once linear solvers, and robust preconditioning methods for fractional transport in neural imaging and materials [2109.06510, 1704.06733].
- **Time-fractional sine-Gordon:** Achieves provably optimal local and global error rates for nonlinear oscillatory PDEs [2601.20566].

Numerical experiments consistently confirm theoretical convergence orders up to $O(\tau^2 + h^2)$ (and $O(h^4)$ in some settings) and showcase the efficiency and robustness of Alikhanov-based methods under practical workload and memory constraints [2601.18505, 2109.02028, 1704.06733, 2102.09396].

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**References**:  
[2601.18505], [2601.20566], [2101.09678], [2102.09396], [2109.02028], [1704.06733], [2109.06510]

Source: https://www.emergentmind.com/topics/alikhanov-scheme