Solving the Linear Advection Equation for the BGK Method

The complexity of the collision term in the Boltzmann equation for rarefied flows is such that it is often replaced by a model equation such as the Bhatnagar-Gross-Krook (BGK) model [5]. To solve the BGK equation numerically a number of different advection equations - one for each molecular velocity range which is included in the numerical method - must be solved in parallel. For a fixed time step, the CFL numbers for these different advection equations can differ by an order of magnitude or more. For a feasible time-step, and a reasonable computational time, it becomes necessary to solve at least some of these linear advection equations with a CFL number much greater than 1. It is the purpose of this work to provide the groundwork for solving the BGK method by presenting a fast, accurate solution to the linear advection equation that is stable for CFL numbers greater than 1