Local error estimate of L1 scheme for time-fractional convection–diffusion–reaction equation on a star-shaped pipe network

Abstract

It is a significant issue to study the discrete method for the time-fractional parabolic problem on the metric graph in scientific computation. In this paper, the time-fractional convection–diffusion–reaction problem with initial singularity on the metric star-shaped graph is considered. In order to overcome the weak singularity of the solution numerically, a L1-finite difference method over the metric graph is constructed by using the graded temporal mesh and uniform spacial mesh. More specific, L1 scheme is employed for temporal discretization of the Caputo fractional-derivative. Also, the finite difference method is proposed for spacial discretization by applying the technique of ghost points to discretize the Robin–Kirchhoff condition at the intersection point and the boundary conditions. Through this method, the second-order truncation accuracy in space is obtained. Over the entire star graph, a sharp error estimate of this full-discrete scheme at each time level is demonstrated. Meanwhile, the convergence analysis for a preserving Robin–Kirchhoff condition scheme at each time step is given. Numerical results are presented to confirm the sharpness of the theoretical analysis.