A Method for Analytical Solutions in the Lattice Boltzmann Method

Analytical solutions to the lattice Boltzmann Equation make it possible to study the method itself, explore the properties of its collision operator, and identify implementations of boundary conditions. In this paper, we propose a method to find analytical solutions where the macroscopic flow profile is known. We test this method on bulk Couette flow aligned and inclined to the simulation lattice with the quadratic and entropic equilibrium distributions. Our method indeed provides an analytical solution to these flows when using the quadratic distribution. When the flow is aligned to the lattice, our method provides an analytical solution using the entropic distribution for practical relaxation times and shear rates. We show that a small even order truncation of the formal solution is optimal for accuracy-compute-time trade-off. In the inclined case, our method does not conserve momentum, by a small relative error, when using the entropic distribution. We also discover that entropic lattice Boltzmann method is not compatible with the angled Couette flow. We discuss the application of our method to more complicated flows.