Uniformly Stable Explicitly Solvable Finite Difference Method for Fractional Diffusion Equations

Abstract

A finite difference scheme for the one-dimensional space fractional diffusion equation is presented and analysed. The scheme is constructed by modifying the shifted Grünwald approximation to the spatial fractional derivative and using an asymmetric discretisation technique. By calculating the unknowns in differential nodal point sequences at the odd and even time levels, the discrete solution of the scheme can be obtained explicitly. We prove that the scheme is uniformly stable. The error between the discrete solution and the analytical solution in the discrete l2 norm is optimal in some cases. Numerical results for several examples are consistent with the theoretical analysis.

A Second‐Order Uniformly Stable Explicit Asymmetric Discretization Method for One‐Dimensional Fractional Diffusion Equations

“…We prove that the scheme is uniformly stable when ∈ [( √ 17 − 1)/2, 2], and the error between the numerical and analytical solutions in discrete 2 norm is of order (Δ 2 ℎ −2( −1) +Δ +ℎ 2 ), where is the order of spatial fractional derivative, and ℎ and Δ are the space and time mesh sizes. The error estimate here is better than that in [19], where the error in discrete 2 norm is of order (Δ 2 ℎ −2( −1) +Δ +ℎ). Some examples are presented to show that the numerical results match our theoretical analysis.…”

“…In conclusion, the discrete solutions of (19) and 20can be obtained explicitly one by one instead of solving the linear algebraic system.…”

“…, − 1. Now, the following semi-implicit finite difference scheme is constructed using the asymmetric technique [18,19].…”

“…The results in Tables 9-12 are listed by using the finite difference scheme developed by Rui and Huang [19] for Examples 1 and 2. We set Δ = ℎ −0.5 and have the same spatial grid sizes ℎ from 1/10 to 1/320, since their error estimates between the numerical and analytical solutions are optimal under the condition Δ = ℎ −0.5 for ∈ (1,2).…”

“…This asymmetric technique has been used to construct parallel algorithms in several research results [20][21][22][23]. Within the scope of our knowledge, there are only two papers including this paper and the one Rui and Huang presented [19] to analyze the error for the asymmetric technique and get the error estimates between the discrete and analytical solutions. Former papers presented the stability property, 2 Complexity mentioned that the truncation error is of order (Δ ℎ −1 + Δ + ℎ) for parabolic problems [20,21], or presented the asymmetric technique [22,23] in numerical calculations.…”