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On using random walks to solve the space-fractional advection-dispersion equations. (English) Zbl 1092.82038
The solution of space-fractional advection-dispersion equations (fADE) by random walks depends on the analogy between the fADE and the forward equation for the associated Markov process. The forward equation, which provides a Lagrangian description of particles moving under specific Markov processes, is derived here by the adjoint method. The fADE, however, provides an Eulerian description of solute fluxes. There are two forms of the fADE, based on fractional-flux (FF-ADE) and fractional divergence (FD-ADE). The FF-ADE is derived by taking the integer-order mass conservation of non-local diffusive flux into account, while the FD-ADE is derived by considering the fractional-order mass conservation of local diffusive flux. The analogy between the fADE and the forward equation depends on which form of the fADE is used and on the spatial variability of the dispersion coefficient \(D\) in the fADE. If \(D\) does not vary in space, then the fADEs can be solved by tracking particles following a Markov process with a simple drift and an \(\alpha\)-stable Lévy noise with index \(\alpha\) that corresponds to the fractional order of the fADE. If \(D\) varies smoothly in space and the solute concentration at the upstream boundary remains zero, the FD-ADE can be solved by simulating a Markov process with a simple drift, an \(\alpha\)-stable Lévy noise and an additional term with the dispersion gradient and an additional Lévy noise of order \(\alpha-1\). However, a non-Markov process might be needed to solve the FF-ADE with a space-dependent \(D\), except for specific \(D\) such as a linear function of space.
In the present article, the authors concentrate on the solution of fADEs with space-dependent velocity and dispersion coefficients since they are more realistic and there are no analytical solutions. Numerical examples are also presented as demonstrations.

MSC:
82C70 Transport processes in time-dependent statistical mechanics
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