Abstract
In this paper, we propose an efficient compact finite difference method for a class of time-fractional subdiffusion equations with spatially variable coefficients. Based on the L2-1σ approximation formula of the time-fractional derivative and a fourth-order compact finite difference approximation to the spatial derivative, an efficient compact finite difference method is developed. The local truncation error and the solvability of the developed method are discussed in detail. The unconditional stability of the resulting scheme and also its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Numerical examples are provided to demonstrate the accuracy and the theoretical results.
Highlights
The motion time and the status of particles are described by the time drift term zu/zt
Various numerical methods are used to obtain the numerical solutions of fractional differential, such as the finite difference method [14,15,16,17,18], the finite element method
We propose some new methods to construct a high-order compact finite difference method for equation (1)
Summary
E proof follows from Lemma 2 of [29]. Α) du dt t0 + Oτ2, and for n ≥ 2, du dt tn−(α/2) 2τ (3 − α)u tn − (4 − 2α)u tn− 1 (14). E proof follows from (31)–(33) in [32]. We discretize (1) into a compact finite difference scheme. Suppose u(x, t) be the solution of (1), we define the grid functions as follows: Uni u xi, tn, Wni zu zt xi, tn, Zni z. (2 − α)HxδtU1i /2 + μ1c(0,α1)HxU1i − U0i δx ψδxU1i ,(α/2) +(1 − α)HxW0i + Hxf1i −(α/2) + Rαtx1i , 1 ≤ i ≤ M − 1,. We obtain the following compact finite difference scheme:.
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