Finite difference scheme on graded meshes to the time-fractional neutron diffusion equation with non-smooth solutions

Abstract

In this paper, we construct and analyze an efficient numerical scheme based on graded meshes in time for solving the fractional neutron diffusion equation with delayed neutrons and non-smooth solutions, which can be found everywhere in nuclear reactors. Using the L1 discretization of each time fractional derivatives on graded meshes and the classical finite difference for the spatial derivatives on uniform meshes, we prove the order of convergence in time is at best (2−2α) instead of 2α under non-smooth solutions, where 0<α<1/2 is the anomalous diffusion order. Numerical experiments are designed to verify our theoretical analysis. Although we can pick any mesh parameter r provided r≥(2−2α)/2α to get the optimal order, we choose the minimum in consideration of both accuracy and convergence.