### An interpretation and derivation
of the lattice Boltzmann method using Strang splitting

P. J. Dellar (2011) An interpretation and derivation of the
lattice Boltzmann method using Strang splitting Computers &
Mathematics with Applications (published
online)

Preprint available (OperatorLB.pdf
196K)

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Abstract

The lattice Boltzmann space/time discretisation, as usually derived
from integration along characteristics, is shown to correspond to a
Strang splitting between decoupled streaming and collision steps.
Strang splitting offers a second-order accurate approximation to
evolution under the combination of two non-commuting operators, here
identified with the streaming and collision terms in the discrete
Boltzmann partial differential equation. Strang splitting achieves
second-order accuracy through a symmetric decomposition in which one
operator is applied twice for half timesteps, and the other operator is
applied once for a full timestep. We show that a natural definition of
a half timestep of collisions leads to the same change of variables
that was previously introduced using different reasoning to obtain a
second-order accurate and explicit scheme from an integration of the
discrete Boltzmann equation along characteristics. This approach
extends easily to include general matrix collision operators, and also
body forces. Finally, we show that the validity of the lattice
Boltzmann discretisation for grid-scale Reynolds numbers larger than
unity depends crucially on the use of a Crank-Nicolson approximation to
discretise the collision operator. Replacing this approximation with
the readily available exact solution for collisions uncoupled from
streaming leads to a scheme that becomes much too diffusive, due to the
splitting error, unless the grid-scale Reynolds number remains well
below unity.