Optimal error analysis of space–time second-order difference scheme for semi-linear non-local Sobolev-type equations with weakly singular kernel

Abstract

In this paper, we construct and analyze a Crank–Nicolson difference scheme for solving the semilinear Sobolev-type equation with the Riemann–Liouville fractional integral of order α∈(0,1). The proposed scheme consists of two main stages. First, to compensate for the singular behavior of the exact solution at the initial time t=0, a Crank–Nicolson scheme and the product trapezoidal integration rule are proposed on graded meshes for time discretization. Second, a classical finite difference operator on uniform meshes is used for space discretization. Under suitable assumptions of the regularity condition, the stability and convergence analysis of this scheme in a new norm are given by the energy argument. It is shown that how the mesh grading exponent γ affects the temporal convergence rate of the presented scheme in the final convergence result. This scheme can attain a second-order accuracy in both time and space by choosing an optimal grading parameter γ. Numerical results confirm that the error analysis is sharp, and comparisons with results on uniform temporal meshes are included to indicate the effectiveness of our method.