An unconditionally stable maximum-principle-preserving numerical method for a normalized time-fractional diffusion equation

In this paper, we present the mathematical and numerical analysis of a numerical method for a normalized time-fractional diffusion equation. The method is unconditionally stable and preserves the maximum principle. The normalized derivative was defined using a weighting function. The integral of this function is always equal to one for all fractional orders and times. This property makes it possible to compare the diffusion behavior under different fractional orders in a fair way. We considered a recently proposed finite difference scheme with an implicit time discretization for the equation. The resulting algebraic system has a tridiagonal matrix. The Thomas algorithm was applied to solve the system efficiently. We proved that the numerical scheme is unconditionally stable, satisfies the maximum principle, and converges correctly. The convergence order is $O(\mathrm{\Delta }{t}^{2-\alpha })$ in time and $O(h^2)$ in space. Several numerical examples were also presented. The results agree with the theoretical analysis and further validate the influence of different fractional orders on the diffusion behavior.