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.
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Acknowledgements
This work was supported in part by the National Natural Science Foundation of China (Nos. 91630207 and 11571115), by the Natural Science Foundation of Shandong Province (No. ZR2017 MA006), by the Fundamental Research Funds for the Central Universities (No. 18CX02044A), by the National Science Foundation (No. DMS-1620194) and by the OSD/ARO MURI Grant (No. W911NF-15-1-0562). The authors would like to express their most sincere thanks to the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper.
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Zheng, X., Liu, H., Wang, H. et al. An Efficient Finite Volume Method for Nonlinear Distributed-Order Space-Fractional Diffusion Equations in Three Space Dimensions. J Sci Comput 80, 1395–1418 (2019). https://doi.org/10.1007/s10915-019-00979-2
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DOI: https://doi.org/10.1007/s10915-019-00979-2