Local error analysis of L1 scheme for time-fractional diffusion equation on a star-shaped pipe network

The numerical analysis for differential equations on networks has become a significant issue in theory and diverse fields of applications. Nevertheless, solving time-fractional diffusion problem on metric graphs has been less studied so far, as one of the major challenging tasks of this problem is the weak singularity of solution at initial moment. In order to overcome this difficulty, a new L1-finite difference method considering the weak singular solution at initial time is proposed in this paper. Specifically, we utilize this method on temporal graded meshes and spacial uniform meshes, which has a new treatment at the junction node of metric graph by employing Taylor expansion method, Neumann-Kirchhoff and continuity conditions. Over the whole star graph, the optimal error estimate of this fully discrete scheme at each time step is given. Also, the convergence analysis for a discrete scheme that preserves the Neumann-Kirchhoff condition at each time level is demonstrated. Finally, numerical results show the effectiveness of proposed full-discrete scheme, which can be applied to star graphs and even more general graphs with multiple cross points.