Doubling Smith Method for a Class of Large-Scale Generalized Fractional Diffusion Equations

The implicit difference approach is used to discretize a class of generalized fractional diffusion equations into a series of linear equations. By rearranging the equations as the matrix form, the separable forcing term and the coefficient matrices are shown to be low-ranked and of nonsingular M-matrix structure, respectively. A low-ranked doubling Smith method with determined optimally iterative parameters is presented for solving the corresponding matrix equation. In comparison to the existing Krylov solver with Fast Fourier Transform (FFT) for the sequence Toeplitz linear system, numerical examples demonstrate that the proposed method is more effective on CPU time for solving large-scale problems.

Automated summary

The implicit difference approach discretizes a class of generalized fractional diffusion equations into a series of linear equations. By rearranging the equations as the matrix form, the separable forcing term and the coefficient matrices are shown to be low-ranked and of nonsingular M-matrix structure, respectively. A low-ranked doubling Smith method with optimally determined iterative parameters is presented for solving the corresponding matrix equation. Numerical examples demonstrate that the proposed method is more effective on CPU time for solving large-scale problems compared to the existing Krylov solver with Fast Fourier Transform (FFT) for the sequence Toeplitz linear system.