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Title: Stability and stabilisation of the lattice Boltzmann method Magic steps and salvation operations
Authors: Brownlee, R.A.
Gorban, Alexander N.
Levesley, Jeremy
First Published: 2006
Publisher: Dept. of Mathematics, University of Leicester.
Abstract: 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 O(10000).
Series/Report no.: MA 06-022
Type: Report
Appears in Collections:Reports, Dept. of Mathematics

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