\documentclass{article}
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\title{A quasirandom walk method \\
for advection-diffusion problems}
\author{Christian L\'ecot
\thanks{Corresponding author.
E-mail: Christian.Lecot@univ-savoie.fr},
Ibrahim Coulibaly, and Afiss Koudiraty \\
Laboratoire de Math\'ematiques,
Universit\'e de Savoie \\
73376 Le Bourget-du-Lac cedex, France}
\date{}
\maketitle

The random walk technique is commonly used
to model advection and diffusion in the
environment. A step towards improving the
accuracy of the method is to replace
pseudorandom numbers by quasirandom numbers.
Quasi-Monte Carlo methods are known to be
particularly powerful in the area of
numerical integration but the scope of
these methods was widened in recent years.

This paper examines a quasirandom walk
method for the $s$-dimensional
advection-diffusion equation
with constant coefficients. Spatial and time
derivatives are replaced with finite
differences. Discretization is upwind in
space and forward Euler in time. The solution
is approximated as a linear combination of
Dirac measures (particles). Particles are
sampled from some known initial distribution.
Then they move according to the dynamics
described in the difference equation. The
particle movement is regarded as a
quasi-Monte Carlo approximation in the
$(s+1)$-dimensional unit cube.

Quasi-Monte Carlo methods rely
on low-discrepancy point sets. Currently,
the most effective constructions are obtained
from the theory of $(t,m,s)$-nets and
$(t,s)$-sequences. However quasirandom points
cannot be blindly used in place of pseudorandom
points for particle simulations. A way to
increase the effectiveness of
quasirandom sequences is to use a scrambling
technique. At every time step, the number
order of the particles is scrambled according
to their positions before assigning a new
quasirandom point to each particle.
This technique is also used in applying
pseudorandom sequences to simulations for
kinetic equations.

Convergence of the scheme is proved for a
quasirandom walk method using a
$(t,s+1)$-sequence with this scrambling
technique. Computational results are presented
for a simple demonstration problem
in one and two dimensions. The results indicate
that an improvement in both magnitude of error
and convergence rate is achieved over a
standard random walk simulation using
pseudorandom numbers.

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