A compact LOD method and its extrapolation for two-dimensional modified anomalous fractional sub-diffusion equations

Abstract

A Crank–Nicolson-type compact locally one-dimensional (LOD) finite difference method is proposed for a class of two-dimensional modified anomalous fractional sub-diffusion equations with two time Riemann–Liouville fractional derivatives of orders (1−α) and (1−β)(0<α,β<1). The resulting scheme consists of simple tridiagonal systems and all computations are carried out completely in one spatial direction as for one-dimensional problems. This property evidently enhances the simplicity of programming and makes the computations more easy. The unconditional stability and convergence of the scheme are rigorously proved. The error estimates in the standard H1- and L2-norms and the weighted L∞-norm are obtained and show that the proposed compact LOD method has the accuracy of the order 2min{α,β} in time and 4 in space. A Richardson extrapolation algorithm is presented to increase the temporal accuracy to the order min{α+β,4min{α,β}} if α≠β and min{1+α,4α} if α=β. A comparison study of the compact LOD method with the other existing methods is given to show its superiority. Numerical results confirm our theoretical analysis, and demonstrate the accuracy and the effectiveness of the compact LOD method and the extrapolation algorithm.