A Fast Temporal Second Order Difference Scheme for the Fractional Sub-Diffusion Equations on One Dimensional Space Unbounded Domain (Journal of Mathematical Study, 2023)
The numerical solution of the fractional sub-diffusion equations on one dimensional space unbounded domain is considered. Based on the high-order localartificial boundary conditions proposed in [Zhang W., et al., J. Math. Study., 2017,50(1): 28-53], the original space unbounded problem can be reformulated to an initial-boundary value problem on a bounded computational domain. By Alikhanov’s $L2-1_σ$ formula and sum-of-exponentials approximation, a fast temporal second order difference scheme for the reduced problem is presented. The unique solvability, stabilityand convergence order $O(τ^2+h^2 )$ of the proposed method are proved by means of energy method, where $τ$ and $h$ denote the time and space step sizes, respectively. Somenumerical examples are included to validate the theoretical results. To the best of ourknowledge, this is the first work that combines the high order numerical method withthe artificial boundary method for the time fractional diffusion problems on spatialunbounded domains.
Keywords: Fractional sub-diffusion equations; space unbounded domain; high-order local artificial boundary conditions; difference scheme; fast algorithm; energy method.