Finite difference approximation of time-fractional advection-diffusion equation on a metric star graph

This article proposes a numerical scheme based on finite difference approximation for solving the time-fractional advection-diffusion equation on a metric star graph comprising initial, boundary, and transmission conditions on the nodes of the graph. The choice of considering a star graph is motivated by the fact that any arbitrary graph can be decomposed into a star graph. The fractional derivative is considered in the Caputo sense and the well-known $ L1 $ L1 method is applied for its discrete approximation. The unconditional stability of the difference scheme for $ \Delta x_i\leq 2 $ Delta xi <= 2 is analyzed by employing the discrete energy method. The unique solution behaviour of the scheme is also studied. An overall convergence of order $ \mathcal {O}(\Delta t<^>{2-\alpha }+\sum _{i=1}<^>k\Delta x_i<^>2) $ O(Delta t2-alpha+& sum;i=1k Delta xi2) is obtained from the proposed scheme. Finally, a few experiments over test examples are demonstrated to verify the accuracy and convergence of the proposed scheme.