An Efficient Finite Volume Method for Nonlinear Distributed-Order Space-Fractional Diffusion Equations in Three Space Dimensions

Abstract

A Crank–Nicolson finite volume approximation for the nonlinear distributed-order space-fractional diffusion equations in three space dimensions is developed, and a resulting nonlinear algebra system with Kronecker-product type coefficient matrix is formulated. The finite volume scheme is proved to be unconditionally stable and convergent with second-order accuracy in terms of the step sizes in time and distributed orders, and $\min \{3-{\hat{\alpha }}, 3-{\hat{\beta }}, 3-{\hat{\gamma \}}}$-order accuracy in space with respect to a discrete norm. At each time step, the Newton’s iterative method is employed as a nonlinear solver to handle the resulting nonlinear algebra system. Moreover, during each Newton’s iteration, an efficient biconjugate gradient stabilized method (BiCGSTAB) is developed, in which both the matrix storage and matrix-vector multiplications are efficiently conducted by using the Toeplitz sub-structure of the coefficient matrices. It is proved that the BiCGSTAB method only requires the storage of $\mathcal {O}(N)$ and the computational cost of $\mathcal {O}(N \log N)$ per iteration, while no accuracy is lost compared with the Gaussian elimination method. Thus, an efficient finite volume method is developed, and it is well suitable for large-scale modeling and simulations, especially for multi-dimensional problems. Numerical experiments are presented to verify the theoretical results and show strong potential of the efficient method.