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

Abstract 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.