A novel compact ADI scheme for the time-fractional subdiffusion equation in two space dimensions

Abstract

In this paper, a novel compact alternating direction implicit (ADI) scheme is proposed for solving the time-fractional subdiffusion equation in two space dimensions. The established scheme is based on the modified L1 method in time and the compact finite difference method in space. The unique solvability, unconditionally stability and convergence of the scheme are proved. The derived compact ADI scheme is coincident with the one for 2D integer order parabolic equation when the β → 1, where 1 − β (∈ (0, 1)) is the order of the Riemann-Liouville derivative operator. In addition, the novel ADI scheme is used to solve the 2D modified fractional diffusion equation, and the corresponding stability and convergence results are also given. Numerical results are provided to verify the theoretical analysis.

Fast High-Order Difference Scheme for the Modified Anomalous Subdiffusion Equation Based on Fast Discrete Sine Transform

“…In this paper, we focus on the numerical method for the timefractional modified subdiffusion equation [1]:…”

“…The analytical solution the author obtained contains the infinite series of Fox special functions, which is of complex form that makes it difficult to apply to practical numerical simulations. So, one needs to resort to the numerical methods for efficiently solving equation (1). Many efficient numerical methods for solving fractional models have emerged in recent years, see the book [5] and the two review papers [2,6].…”

“…For the two-dimensional case, Chen and Li employed the modified L1 method and compact difference method and proposed a compact alternating direction implicit scheme. By utilizing the energy method, they proved that their scheme is stable with an accuracy of Oðτ 2 min ðα,βÞ + h 4 Þ in the new defined norm which is equivalent with H 1 -norm, under the assumption that the solution is sufficiently smooth [1]. Such assumption may be too restrictive to limit the scope of application of their scheme.…”

Parallel finite-difference algorithms for three-dimensional space-fractional diffusion equation with $$\psi $$-Caputo derivatives

“…In multidimensional case, lowering of computational complexity is often achieved using splitting schemes (Chen and Li 2016;Yin et al 2018;Samarskii 2001;Bogaenko et al 2017) that reduce multidimensional problems to the series of one-dimensional ones. In addition to the direct effect of complexity reduction, significant reduction in the connectivity between computation blocks makes it possible to apply parallelization techniques efficiently in this case.…”

Mathematical Modeling of Solutes Migration under the Conditions of Groundwater Filtration by the Model with the k-Caputo Fractional Derivative

“…Such schemes for the classical models of diffusion were thoroughly studied in [14]. The unique solvability, stability and convergence of a splitting scheme for the fractional diffusion equation with the Caputo derivative with respect to the time variable was proved in [15]. Compared with the conventional methods (e.g., [16]) that discretize fractional differential equation to a single system of linear or non-linear algebraic equations, splitting schemes result in a series of independent equation systems of lower size that allows developing simple and efficient computational algorithms [17].…”