Stability and stabilisation of the lattice Boltzmann method: Magic steps and salvation operations

Details Stability and stabilisation of the lattice Boltzmann method: Magic steps and salvation operations Stability and stabilisation of the lattice Boltzmann method: Magic steps and salvation operations

2006-11-16

arxiv.org/abs/cond-mat/0611444v3

Phys. Rev. E 75, 036711 (2007)

Physics

Condensed Matter

Statistical Mechanics

Significantly extended version with new LBM accuracy analysis and extended bibliography

Scientific paper

10.1103/PhysRevE.75.036711

We revisit the classical stability versus accuracy dilemma for the lattice Boltzmann methods (LBM). Our goal is a stable method of second-order accuracy for fluid dynamics based on the lattice Bhatnager--Gross--Krook method (LBGK). The LBGK scheme can be recognised as a discrete dynamical system generated by free-flight and entropic involution. In this framework the stability and accuracy analysis are more natural. We find the necessary and sufficient conditions for second-order accurate fluid dynamics modelling. In particular, it is proven that in order to guarantee second-order accuracy the distribution should belong to a distinguished surface -- the invariant film (up to second-order in the time step). This surface is the trajectory of the (quasi)equilibrium distribution surface under free-flight. The main instability mechanisms are identified. The simplest recipes for stabilisation add no artificial dissipation (up to second-order) and provide second-order accuracy of the method. Two other prescriptions add some artificial dissipation locally and prevent the system from loss of positivity and local blow-up. Demonstration of the proposed stable LBGK schemes are provided by the numerical simulation of a 1D shock tube and the unsteady 2D-flow around a square-cylinder up to Reynolds number $\mathcal{O}(10000)$.

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