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

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 O(N), 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 O(Nlog2N). 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. [ABSTRACT FROM AUTHOR]