A Low-Dimensional Compact Finite Difference Method on Graded Meshes for Time-Fractional Diffusion Equations

Abstract

Abstract In this paper, we propose an improvement of the classical compact finite difference (CFD) method by using a proper orthogonal decomposition (POD) technique for time-fractional diffusion equations in one- and two-dimensional space. A reduced CFD method is constructed with lower dimensions such that it maintains the accuracy and decreases the computational time in comparison with classical CFD method. Since the solution of time-fractional diffusion equation typically has a weak singularity near the initial time t = 0 {t=0} , the classical L1 scheme on uniform meshes may obtain a scheme with low accuracy. So, we use the L1 scheme on graded meshes for time discretization. Moreover, we provide the error estimation between the reduced CFD method based on POD and classical CFD solutions. Some numerical examples show the effectiveness and accuracy of the proposed method.