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An efficient quadratic finite volume method for variable coefficient Riesz space‐fractional diffusion equations
Author(s) -
Li Fangli,
Fu Hongfei,
Liu Jun
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6306
Subject(s) - mathematics , toeplitz matrix , coefficient matrix , biconjugate gradient stabilized method , discretization , matrix (chemical analysis) , quadratic equation , mathematical analysis , linear system , pure mathematics , geometry , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , composite material
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 ( N log 2 N ) . 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.