1. Fast upwind and Eulerian-Lagrangian control volume schemes for time-dependent directional space-fractional advection-dispersion equations.
- Author
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Du, Ning, Guo, Xu, and Wang, Hong
- Subjects
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ADVECTION-diffusion equations , *KRYLOV subspace , *EQUATIONS , *DIFFERENTIAL operators , *POROUS materials , *PROBABILITY measures , *DIRECTIONAL derivatives - Abstract
• Consider a multidimensional time-dependent space-fractional equation to model solute transport in heterogeneous porous media. • The equation involves all directions in its fractional derivative rather than only coordinate directions. • We derive the fast control-volume schemes, these schemes naturally have second order accuracy in space. • A fast Krylov subspace iterative solver for control volume schemes is proposed. 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. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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