Fast upwind and Eulerian-Lagrangian control volume schemes for time-dependent directional space-fractional advection-dispersion equations

JOURNAL OF COMPUTATIONAL PHYSICS

卷:405

Article

Du, Ning¹  Guo, Xu^2,3  Wang, Hong⁴ 

[1] Shandong Univ, Sch Math, Jinan 250100, Shandong, Peoples R China

[2] Shandong Univ, Geotech & Struct Engn Res Ctr, Jinan 250100, Shandong, Peoples R China

[3] Chinese Univ Hong Kong, Dept Math, Shatin, Hong Kong, Peoples R China

[4] Univ South Carolina, Dept Math, Columbia, SC 29208 USA

关键词: Directional space-fractional advection-dispersion equation;    Superdiffusive solute transport;    Krylov subspace iterative solver;    Control volume method;   

DOI  :  10.1016/j.jcp.2019.109127

来源: Elsevier

【 摘 要 】

We develop control volume methods for two-dimensional time-dependent advection-dominated directional space-fractional advection-dispersion equations with the directional space-fractional derivative weighted in all the directions by a probability measure in the unit circle, which are used to model the anisotropic superdiffusive transport of solutes in groundwater moving through subsurface heterogeneous porous media. We develop a fast upwind control volume method for the governing equation to eliminate the spurious numerical oscillations that often occur in space-centered numerical discretizations of advection term, which are relatively straightforward to implement. We also develop a Eulerian-Lagrangian control-volume method for the governing equation, which symmetrizes the governing equation by combining the time-derivative term and the advection term into a material derivative term along characteristic curves. Both methods are locally mass-conservative, which are essential in these applications. Due to the nonlocal nature of the directional space-fractional differential operators, corresponding numerical discretizations usually generate full stiffness matrices. Conventional direct solvers tend to require O (N-2) memory requirement and have O (N-3) computational complexity per time step, where N is the number of spatial unknowns, which is computationally significantly more expensive than the numerical approximations of integer-order advection-diffusion equations. Based on the analysis of the structure of stiffness matrix, we propose a fast Krylov subspace iterative solver to accelerate the numerical approximations of both the upwind and Eulerian-Lagrangian control volume methods, which reduce computational complexity from O (N-3) by a direct solver to O (N log N) per Krylov subspace iteration per time step and a memory requirement from O (N-2) to O (N). Numerical results are presented to show the utility of the methods. (C) 2019 Elsevier Inc. All rights reserved.

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