A second-order accurate numerical method with graded meshes for an evolution equation with a weakly singular kernel

Abstract

A second-order accurate numerical method with graded meshes is proposed and analyzed for an evolution equation with a weakly singular kernel. The graded meshes are employed to compensate for the singular behavior of the exact solution at t = 0 . For the time discretization, the product integration rule is used to approximate the Riemann–Liouville fractional integral, a generalized Crank–Nicolson time-stepping is considered and shown that the error is of order k 2 , where k denotes the maximum time step. A fully discrete difference scheme is constructed with space discretization by compact difference method. Numerical experiment is carried out to support the theoretical results. The comparison between the method on uniform grids and graded grids shows the efficiency of our method.