  A fractional advection-dispersion equation (fADE) has been advocated for
heavy-tailed flows where the usual Brownian diffusion models fail. A stochastic
differential equation (SDE) driven by a stable L\'{e}vy process gives a forward
equation that matches the space-fractional advection-dispersion equation and
thus gives the stochastic framework of particle tracking for heavy-tailed
flows. For constant advection and dispersion coefficient functions, the
solution to such SDE itself is a stable process and can be derived easily by
least square parameter fitting from the observed flow concentration data.
However, in a more generalized scenario, a closed form for the solution to a
stable SDE may not exist. We propose a numerical method for solving/generating
a stable SDE in a general set-up. The method incorporates a discretized finite
volume scheme with the characteristic line to solve the fADE or the forward
equation for the Markov process that solves the stable SDE. Then we use a
numerical scheme to generate the solution to the governing SDE using the fADE
solution. Also, often the functional form of the advection or dispersion
coefficients are not known for a given plume concentration data to start with.
We use a Levenberg--Marquardt (L-M) regularization method to estimate advection
and dispersion coefficient function from the observed data (we present the case
for a linear advection) and proceed with the SDE solution construction
described above.
