Efficient and Accurate Numerical Methods Using the Accelerated Spectral Deferred Correction for Solving Fractional Differential Equations

We develop an efficient and accurate spectral deferred correction (SDC) method for fractional differential equations (FDEs) by extending the algorithm in [14] for classical ordinary differential equations (ODEs). Specifically, we discretize the resulted Picard integral equation by the SDC method and accelerate the convergence of the SDC iteration by using the generalized minimal residual algorithm (GMRES). We first derive the correction matrix of the SDC method for FDEs and analyze the convergence region of the SDC method. We then present several numerical examples for stiff and non-stiff FDEs including fractional linear and nonlinear ODEs as well as fractional phase field models, demonstrating that the accelerated SDC method is much more efficient than the original SDC method, especially for stiff problems. Furthermore, we resolve the issue of low accuracy arising from the singularity of the solutions by using a geometric mesh, leading to highly accurate solutions compared to uniform mesh solutions at almost the same computational cost. Moreover, for long-time integration of FDEs, using the geometric mesh leads to great computational savings as the total number of degrees of freedom required is relatively small.

Automated summary

In this paper, an efficient and accurate spectral deferred correction (SDC) method is developed for fractional differential equations (FDEs). The issue of low accuracy arising from the singularity of the solutions is resolved by using a geometric mesh, leading to highly accurate solutions at almost the same computational cost. Moreover, the use of the geometric mesh allows for great computational savings in long-time integration of FDEs.