A new difference scheme for the time fractional diffusion equation

By Anatoly A. Alikhanov. Journal of Computational Physics, 2014

834 citations recorded by the source corpus.

Abstract In this paper we construct a new difference analog of the Caputo fractional derivative (called the L 2 - 1 σ formula). The basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the second order in time for the time fractional diffusion equation with variable coefficients are considered. Stability of the suggested schemes and also their convergence in the grid L 2 -norm with the rate equal to the order of the approximation error are proved. The obtained results are supported by the numerical calculations carried out for some test problems.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/5743029f4a5de9c5983b4b5cad1a5e0d7936fd85

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In Applied Mathematics

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

A Discrete Grönwall Inequality with Applications to Numerical Schemes for Subdiffusion Problems (2018, methodology): “Included are the well-known L1 formula [14, 17, 20, 24, 25], the fast L1 formula [16], the Alikhanov approximation [2, 12, 15], and their applications for multi-term and distributed-order Caputo derivatives (see Remark 5).”
A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation (2019, methodology): “([1]) If the positive sequence {d k | 0 ≤ k ≤ n, n ≥ 1} is strictly decrease for k, and satisfies (2σ − 1)d n − σd n−1 > 0 for a constant σ ∈ (0, 1) .”
Numerical Approaches to Fractional Integrals and Derivatives: A Review (2020, background): “[82] shows that the L2-1σ formula refines the accuracy indeed.”
Dissipativity and Stability Analysis for Fractional Functional Differential Equations (2015, background): “([1]) Let y ∈ Rn be continuous and differentiable functions.”
Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview (2018, background): “There have been several important efforts in extending the L1 scheme to high-order schemes by using high-order local polynomials [73, 26, 81] and superconvergent points [3].”