Maximum norm error analysis of difference schemes for fractional diffusion equations

In this study, we present several numerical approximation methods for solving the fractional diffusion equations with the Riesz space fractional derivative. Based on the fractional centered difference approximation to the Riesz fractional derivative, an explicit and an implicit finite difference schemes for solving the space fractional diffusion equations are obtained. The stability and global convergence of these two schemes in the maximum norm are analyzed rigorously. The convergence order is O(τ+h2), where τ is the temporal grid size and h is spatial grid size, respectively. Furthermore, a numerical scheme for the one-dimensional multi-term time-space fractional diffusion equations is obtained. The unconditionally stability and convergence of the scheme in maximum norm are established. The convergence orders are 2-α in the temporal direction and two in the spatial direction, where α is the maximum time fractional derivative orders in the equation. In addition, the extension to the two-dimensional case is also discussed. Finally, several numerical examples are provided to show the effectiveness and accuracy of our methods.