A Weighted ADI Scheme for Subdiffusion Equations

A weighted ADI scheme is proposed for solving two-dimensional anomalous diffusion equations with the fractional Caputo derivative. The Alikhanov formula (J Comput Phys 280:424–438, 2015) with a weaker assumption is applied to approximate the fractional derivative and a high-order perturbed term of temporal order $$1+2\alpha $$1+2α is added to the pure implicit approach. By using the discrete energy method, it is proven that the ADI scheme is stable and convergent with the temporal order of $$\min \{1+2\alpha ,2\}$$min{1+2α,2} such that it achieves second-order time accuracy when $$\frac{1}{2}\le \alpha <1$$12≤α<1. Numerical experiments are included to support the theoretical analysis. Application of suggested method to the solution which lacks the smoothness near the initial time is examined by employing a class of nonuniform meshes refined near the singular point.

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