Alikhanov Scheme: Fractional Time Discretization

Updated 3 February 2026

Alikhanov scheme is a fractional time discretization method that approximates the Caputo derivative using a piecewise-quadratic reconstruction, achieving second-order accuracy even with weak singularities.
It employs convolution quadrature on uniform or graded meshes, ensuring stability via positivity, monotonicity of weights, and discrete maximum principles in numerical solutions.
The method supports extensions for variable coefficients, nonlinearities, and fast solvers (e.g., SOE acceleration), making it effective for applications like subdiffusion, diffusion-wave, and fractional Black–Scholes models.

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 (Hou et al., 26 Jan 2026).

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 (Hou et al., 28 Jan 2026, Lyu et al., 2021).

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 (Hou et al., 26 Jan 2026).
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 ( $H^1$ , $L^2$ ) (Lyu et al., 2021, Zhao et al., 2017).
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 (Hou et al., 26 Jan 2026).

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 $r\ge2$ (Lyu et al., 2021, Hou et al., 28 Jan 2026).
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 (Hou et al., 26 Jan 2026, Hou et al., 28 Jan 2026, Lyu et al., 2021).

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 (Lyu et al., 2021).
Nonuniform meshes: The generalized Alikhanov formula accommodates arbitrary nonuniform timesteps $\tau_k$ , with precise construction of convolution weights to ensure stability and accuracy (Song et al., 2021).
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 (Zhao et al., 2017).

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 (Song et al., 2021).
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 (Zhao et al., 2021).
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 (Lyu et al., 2021, Lyu et al., 2021).

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 (Hou et al., 26 Jan 2026).
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 (Song et al., 2021).
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 (Zhao et al., 2021, Zhao et al., 2017).
Time-fractional sine-Gordon: Achieves provably optimal local and global error rates for nonlinear oscillatory PDEs (Hou et al., 28 Jan 2026).

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 (Hou et al., 26 Jan 2026, Song et al., 2021, Zhao et al., 2017, Lyu et al., 2021).