Properties of the cascaded lattice Boltzmann automaton

The theory of the lattice Boltzmann automaton is based on a moment transform which is not Galilean invariant. It is explained how the central moments transform, used in the cascaded lattice Boltzmann method, overcomes this problem by choosing the center of mass coordinate system as the frame of reference. Galilean invariance is restored and the form of the kinetic theory is unaffected. Conservation laws are not compromised by the high order polyinomials in the equilibrium distribution arising from the central moment transform. Two sources of instabilities in lattice Boltzmann simulations are discussed: negative numerical viscosity due to insufficient Galilean invariance and aliasing. The cascaded lattice Boltzmann automaton overcomes both problems. It is discussed why aliasing is unavoidable in lattice Boltzmann methods that rely on a single relaxation time. An appendix lists the complete scattering operator of the D2Q9 cascaded lattice Boltzmann automaton.

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