A Preconditioned Fast Finite Volume Method for Distributed-Order Diffusion Equation and Applications

A Crank-Nicolson finite volume scheme for the modeling of the Riesz space distributed-order diffusion equation is proposed. The corresponding linear system has a symmetric positive definite Toeplitz matrix. It can be efficiently stored in O(NK) memory. Moreover, for the finite volume scheme, a fast version of conjugate gradient (FCG) method is developed. Compared with the Gaussian elimination method, the computational complexity is reduced from O(MN3 + NK) to O(l(A) MN logN + NK), where l(A) is the average number of iterations at a time level. Further reduction of the computational cost is achieved due to use of a circulant preconditioner. The preconditioned fast finite volume method is combined with the Levenberg-Marquardt method to identify the free parameters of a distribution function. Numerical experiments show the efficiency of the method.