Decomposing the collision operator in the lattice Boltzmann method

In transport theory, physical phenomena are well described using the Boltzmann equation, which is efficiently simulated and discretized with the lattice Boltzmann method. The collision step defines the microscopic molecules behavior, and thus the simulated physical phenomena. For complex phenomena, the collision step becomes complex as well. In this paper, we propose a framework to systematically decompose the collision step into individual collision rules. Each collision rule is easier to understand, and thus a faster understanding of the whole is achieved. By inverting the process, i.e., composing multiple collision rules together, one can create collision steps that can better describe the underlying complex phenomena. This framework's applications are manyfold, from both a theoretical and an application standpoint. Shown here is the decomposition of the Robin boundary condition into the Dirichlet and Neumann boundary conditions, extending it to a partial Robin boundary condition and semipermeable reactive membranes.