body {
background-color: #444;
padding: 0 10px;
margin: 0;
min-width: fit-content;
}
.page-container {
margin: 10px auto;
width: fit-content;
}
.page {
overflow: hidden;
position: relative;
background-color: white;
}
.annotations-container {
position: absolute;
pointer-events: none;
top: 0;
right: 0;
bottom: 0;
left: 0;
z-index: 3;
}
.annotations-container > div {
position: absolute; pointer-events: auto; -webkit-user-select: none;
}
.annotations-container > div:hover {
background-color: rgba(255, 255, 0, 0.25);
cursor: pointer;
}
.t {
transform-origin: bottom left;
z-index: 2;
position: absolute;
white-space: pre;
overflow: visible;
line-height: 1.5;
}
.text-container {
white-space: pre;
}
@supports (-webkit-touch-callout: none) {
.text-container {
white-space: normal;
}
}
@font-face {
font-family: Helvetica_v;
src: url(data:application/font-woff;charset=utf-8;base64,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) format("woff");
}
@font-face {
font-family: Times-Bold_s;
src: url(data:application/font-woff;charset=utf-8;base64,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) format("woff");
}
@font-face {
font-family: Times-Italic_u;
src: 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GLOBAL RANDOM WALK SIMULATIONS
OF DIFFUSION
Calin Vamos, Nicolae Suciu
"Tiberiu Popoviciu" Institute of Numaerical Analysis,
Cluj Napoca Branch of Romanian Academy,
cvamos@ ictp-acad. math. ubbcluj. ro, nsuciu@ @ ictp-acad. math. ubbcluj. ro
Harry Vereecken, Olaf Nitzsche, Horst Hardelauf
lnstitutfiir Agrosphlire (ICG-IV), Forschungszentrum Jiilich GmbH,
h. vereecken @fz-juelich.de, o.nitzsche @fz-juelich.de, h.hardelauj@fz-juelich.de
Abstract Random walk methods are suitable to build up convergent solutions for reaction-
diffusion problems and were successfully applied to simulations of transport
processes in a random environment. The disadvantage is that, for realistic cases,
these methods become time and memory expensive. To increase the compu-
tation speed and to reduce the required memory, we derived a "global random
walk" method in which the particles at a given site of the grid are simultaneously
scattered following the binomial Bernoulli repartition. It was found that the
computation time is reduced three orders of magnitude with respect to individual
random walk methods. Moreover, by suitable "microscopic balance" boundary
conditions, we obtained good simulations of transport in unbounded domains,
using normal size grids. The global random walk improves the statistical quality
of simulations for diffusion processes in random fields. The method was tested
by comparisons with analytical and finite difference solutions as well as with
concentrations measured in "column experiments", used in laboratory study of
soils' hydrogeological and chemical properties.
Keywords: diffusion, unbounded domains, random fields
1. Introduction
In this article we present a new numerical method capable of performing sim-
ulations for complex transport processes, faster than other similar approaches,
and using realistic initial and boundary conditions.
Scientific Computing, Validated Numerics, Interval Methods, Edited by Kramer and
Wolff von Gudenberg, Kluwer Academic/Plenum Publishers, New York, 2001 343
344 SCIENTIFIC COMPUTING, VALIDATED NUMERICS, INTERVAL METHODS
Numerical simulations for diffusion can be obtained by both discrete solutions
of diffusion equations and by random walk methods. In the first class enter the
well known finite difference and finite element/volume methods. In the second,
we quote "fractional random walk method" used by Chorin [1] to solve the
viscous terms in Navier-Stokes equations, "particle tracking" methods, as that
used by Tompson and Gelhar [10] as model of solute transport in porous media,
as well as the more recent "cellular automata" [5] and "lattice gas models" [14].
The common denominator of random walk methods is the use of statistical
ensembles of distinct random walkers trajectories to compute the concentration
field as average number of particles in a reference volume. For more details and
reviews on these methods we refer to [10, 11] and to the handbook in stochastic
methods of Gardiner [3]. Besides the simplicity of their algorithms, random
walk methods are not affected by numerical oscillations at large Peclet numbers
(Pe), as it is the case for finite differences [11], and many diffusion problems
benefit from strong proofs of convergence [3, 13].
Related to these methods, we remark: 1) the computation of concentrations
requires only statistical ensembles of positions, at given time, for systems of
fictitious particles, and 2) the states of this ensemble are invariant under the
particles' permutation. Thus, the knowledge of the individual trajectories is not
necessary to compute concentrations. This enables us to use a global random
walk law: all the particles from a given site are simultaneously scattered fol-
lowing the binomial Bernoulli repartition (the sum of individual random walk
probabilities). We propose for this approach the name of "global random walk",
hereinafter referred· to as GRW.
As a mathematical object, GRW is similar to a "lattice gas model", where
"the state of the system is described by specifying the number of particles
present at the site x, n ( x) ", the states space is the space of configurations
n = { w = n (·)}, and the dynamics is a stochastic process in n [ 12]. "Lattice
gas" deals with Markov processes in n. It is the same for diffusion processes
modeled with GRW, with the difference that the simpler rules (random walk or,
more generally, random biased jumps between the grid points) allow the use of
Bernoulli repartition to achieve global change of state.
The advantages of the new method are: the computing time is reduced by
several orders of magnitude, microscopic processes (reactions, decay, retention)
and variable diffusion coefficients can be easily implemented, it describes the
diffusion fronts, it allows the simulation of transport in unbounded domains
using normal size grids, and it is numerically stable and not sensitive to great
Pe numbers.
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