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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.
|Issue Date: ||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|
|Appears in Collections:||Reports, Dept. of Mathematics|
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