A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations

作者：Wang Hong^*; Du Ning

来源：Journal of Computational Physics, 2013, 240: 49-57.

DOI：10.1016/j.jcp.2012.07.045

• Circulant matrices
• Conjugate gradient squared method for dense matrices
• Dense matrix preconditioners
• Steady-state space-fractional diffusion equations
• Toeplitz matrices

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the classical second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the corresponding numerical methods have full coefficient matrices which require storage of O(N-2) and computational cost of O(N-3) for a problem of size N. %26lt;br%26gt;We develop a superfast-preconditioned conjugate gradient squared method for the efficient solution of steady-state space-fractional diffusion equations. The method reduces the computational work from O(N-2) to O(N log N) per iteration and reduces the memory requirement from O(N-2) to O(N). Furthermore, the method significantly reduces the number of iterations to be mesh size independent. %26lt;br%26gt;Preliminary numerical experiments for a one-dimensional steady-state diffusion equation with 213 nodes show that the fast method reduces the overall CPU time from 3 h and 27 min for the Gaussian elimination to 0.39 s for the fast method while retaining the accuracy of Gaussian elimination. In contrast, the regular conjugate gradient squared method diverges after 2 days of simulations and more than 20,000 iterations.

• 出版日期2013-5-1
• 单位山东大学