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Numerical stability of explicit off-lattice Boltzmann schemes: a comparative study. (English) Zbl 1351.76180
Summary: 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 A. Bardow, I. V. Karlin and A. A. Gusev [“General characteristic-based algorithm for off-lattice Boltzmann simulations”, Europhys. Lett. 75, No. 3, 434 (2006; doi:10.1209/epl/i2006-10138-1)] is the most numerically stable and computationally efficient scheme compared to similar schemes, for the flow problems tested here.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
Software:
Armadillo
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