Segment explicit-implicit difference method on nonuniform mesh for parabolic equation with discontinuous coefficients

In this paper, the convergence of the segment explicit-implicit difference scheme on a nonuniform mesh for parabolic equation with discontinuous coefficients is discussed. The truncation error on each net point is O(1), but the solution of the scheme tends to the solution of the parabolic equation in the sense of the maximum norm and the rate of convergence is O(τ+h). Moreover, the numerical flux of the difference scheme tends to the flux of the differential equation in the mean, and the rate of convergence is also O(τ+h).