An efficient quadratic finite volume method for variable coefficient Riesz space‐fractional diffusion equations

Abstract

A quadratic finite volume (FV) method for steady‐state Riesz space‐fractional diffusion equations (sFDEs) with variable diffusivity coefficient is developed using piecewise quadratic basis functions, and a resulting linear algebra system with two‐by‐two block‐type Toeplitz‐like coefficient matrix is formulated. It is proved that the method requires a minimum memory of order scriptOfalse(Nfalse), where N is the number of spatial partitions. Moreover, as two of the produced Toeplitz‐like submatrices are not square, a new fast nonsquare Toeplitz‐like matrix‐vector product is specially designed, which requires an almost linear computational complexity of order scriptOfalse(Nlog2Nfalse). Then, a fast version of biconjugate gradient stabilized (BiCGSTAB) solution algorithm, named FBiCGSTAB, is proposed for the FV scheme. The FV method combined with Crank‐Nicolson (CN) time discretization is applied to solve time‐dependent sFDEs, and an efficient CN‐FV scheme is developed and analyzed. Finally, numerical results are presented to show the utility of the fast FV and fast CN‐FV methods.

Numerical Solutions of Space-Fractional Advection–Diffusion–Reaction Equations

“…Fractional and tempered fractional [1] differential equations (FDEs) have proved to be strong tools in the modelling of many physical phenomena, including acoustics and thermal systems and rheology and modelling of materials, leading to significant developments of analytical and numerical methods for solving fractional ordinary and partial differential equations in recent times. They comprise, e.g., Laplace-Fourier transform techniques and Green function approach [2], Lie symmetries theory and group analysis [3][4][5][6], Adomian decomposition [7,8], and homotopy perturbation methods [9], as well as finite element [10,11] and finite volume schemes [12,13], finite difference methods [14], and spectral ones [15][16][17][18].…”