4,677 research outputs found
Stable L\'{e}vy diffusion and related model fitting
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.Comment: Published at https://doi.org/10.15559/18-VMSTA106 in the Modern
Stochastics: Theory and Applications (https://vmsta.org/) by VTeX
(http://www.vtex.lt/
Modelling solute transport in soil columns using advective-dispersive equations with fractional spatial derivatives
Solute transport in soils is commonly simulated with the advective–dispersive equation, or ADE. It has been reported that this model cannot take into account several important features of solute movement through soil. Recently, a new model has been suggested that results in a solute transport equation with fractional spatial derivatives, or FADE. We have assembled a database on published solute transport experiments in soil columns to test the new model. The FADE appears to be a useful generalization of the ADE. The order of the fractional differentiation reflects differences in physical conditions of the solute transport in soi
Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics
A fractional time derivative is introduced into the Burger's equation to
model losses of nonlinear waves. This term amounts to a time convolution
product, which greatly penalizes the numerical modeling. A diffusive
representation of the fractional derivative is adopted here, replacing this
nonlocal operator by a continuum of memory variables that satisfy local-in-time
ordinary differential equations. Then a quadrature formula yields a system of
local partial differential equations, well-suited to numerical integration. The
determination of the quadrature coefficients is crucial to ensure both the
well-posedness of the system and the computational efficiency of the diffusive
approximation. For this purpose, optimization with constraint is shown to be a
very efficient strategy. Strang splitting is used to solve successively the
hyperbolic part by a shock-capturing scheme, and the diffusive part exactly.
Numerical experiments are proposed to assess the efficiency of the numerical
modeling, and to illustrate the effect of the fractional attenuation on the
wave propagation.Comment: submitted to Siam SIA
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