Lattice Boltzmann schemes replace Boltzmann's binary collision operator with a model using linear relaxation of the distribution functions towards their equilibrium values. Examples of these models include linearisations of lattice gas collision operators, the single-relaxation-time or BGK collision operator, and various multiple-relaxation-time collision operators designed to optimise different measures of accuracy or stability. The two-relaxation-time collision operator groups the discrete particles velocities into anti-parallel pairs, called "dumb-bells" in the theory of lattice gas collision operators, and assigns different relaxation times to the odd and even combinations of anti-parallel velocities. This is equivalent to assigning different relaxation times to the odd and even moments of the distribution functions. For example, momentum flux is an even moment while heat or energy flux is an odd moment. A particular "magic" combination of the two relaxation rates has good properties for computing Poiseuille flow parallel to one of the coordinate axes. The point of zero velocity is then located precisely half-way between lattice points.

We will describe a different interpretation of the two-relaxation-time collision operator that assigns different relaxation times to the forward-propagating and backward-propagating parts of each anti-parallel pair of discrete velocities. The "magic" combination sets the forward-propagating distribution function to equilibrium in this interpretation. The distribution function at any lattice point and time level thus depends only on the distribution function propagating in the reverse direction at the previous time level, and on the equilibrium distributions that are known functions of the fluid density and velocity. By considering two successive lattice Boltzmann timesteps, and hence two reversals of direction, we can extend this to show that the distribution function at any lattice point and time level depends on the same distribution function at the same lattice point two time levels earlier, and on the equilibrium distributions. This allows us to construct closed finite difference schemes for evolving the fluid density and velocity alone across three time levels.

The discrete evolution across three time levels can be thought of as a discrete approximation to partial differential equations with second derivatives with respect to time. However, we prefer to intepret them as discrete approximation to a first order system, the expected conservation laws for mass and momentum, and separate evolution equations for the mass and momentum fluxes. The latter depend only on the fluid density and velocity, so we retain just enough kinetic behaviour to simulate a Maxwell fluid with a finite stress relaxation time. Our approach also extends to include bounce-back boundary conditions with a one timestep delay, the natural implementation based on a halo of "ghost" cells outside the computational domain. We will show numerical experiments including a vortex dipole colliding with a rigid boundary, and the flow driven by an oscillating boundary.