Numerical Stability of Explicit Off-lattice Boltzmann Schemes: A comparative study

TL;DR

This study compares the numerical stability of explicit off-lattice Boltzmann schemes, finding characteristics-based methods more stable than Runge-Kutta-based ones, with Bardow et al.'s scheme being the most stable and efficient.

Contribution

It provides a systematic comparison of stability limits for various explicit off-lattice Boltzmann schemes on benchmark flow problems.

Findings

Characteristics-based schemes are more stable than Runge-Kutta-based schemes.

Bardow et al.'s scheme offers the highest stability and efficiency.

Explicit schemes' stability depends on the flow problem and scheme choice.

Abstract

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.