  The off-lattice Boltzmann (OLB) method consists of numerical schemes which
are used to solve the discrete Boltzmann equation. Unlike the commonly used
lattice Boltzmann method, the spatial and time steps are uncoupled in the OLB
method. In the currently proposed schemes, which can be broadly classified into
Runge-Kutta-based and characteristics-based, the size of the time-step is
limited due to numerical stability constraints. In this work, we systematically
compare the numerical stability of the proposed schemes in terms of the maximum
stable time-step. In line with the overall LB method, we investigate the
available schemes where the advection approximation is explicit, and the
collision approximation is either explicit or implicit. The comparison is done
by implementing these schemes on benchmark incompressible flow problems such as
Taylor vortex flow, Poiseuille flow and, lid-driven cavity flow. It is found
that the characteristics-based OLB schemes are numerically more stable than the
Runge-Kutta-based schemes. Additionally, we have observed that, with respect to
time-step size, the scheme proposed by Bardow et al. [1] is the most
numerically stable and computationally efficient scheme compared to similar
schemes, for the flow problems tested here.
