2006/11/30 by R. A. Brownlee, Alexander N. Gorban, A. N. Gorban +2 · 3 citations
Engineering · Mathematics · Physics and Astronomy · #Boltzmann distribution #Classical mechanics #Dissipation #Fluid Dynamics and Turbulent Flows #Fluid Dynamics and Vibration Analysis #Instability #Lattice (music) #Lattice Boltzmann Simulation Studies #Lattice Boltzmann methods #Mathematical analysis #Mathematics #Mechanics #Physics #Quantum mechanics #Reynolds number #Statistical physics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.75.036711
published as Phys. Rev. E 75, 036711 (2007) · Significantly extended version with new LBM accuracy analysis and extended bibliography
arxiv created 2007/01/19 · openalex publication_date 2007/03/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 recognized 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 modeling. 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 stabilization 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 blowup. Demonstration of the proposed stable LBGK schemes are provided by the numerical simulation of a one-dimensional (1D) shock tube and the unsteady 2D flow around a square cylinder up to Reynolds number Re approximately 20,000.