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In the present work, a box-type difference scheme with convergence order \(O(\tau^{2}+h^{2})\) is proposed for the fractional sub-diffusion equation with spatially variable coefficient under Neumann boundary conditions. Here h, τ are space and temporal step length, respectively. The method is based on applying the \(L2-1_{\sigma}\) formula to approximate the time Caputo fractional derivative and introducing the auxiliary variable. By virtue of the special properties of the \(L2-1_{\sigma}\) formula and the mathematical induction method, the unconditional stability and convergence for our scheme are proved by the discrete energy method. Numerical examples are given to verify the theoretical analysis and efficiency of the box-type scheme.

A second order box-type scheme for fractional sub-diffusion equation with spatially variable coefficient under Neumann boundary conditions