A symmetric Gauss-Seidel method for the steady-state Boltzmann equation

We introduce numerical solvers for the steady-state Boltzmann equation based on the symmetric Gauss-Seidel (SGS) method. Due to the quadratic collision operator in the Boltzmann equation, the SGS method requires solving a nonlinear system on each grid cell, and we consider two methods, namely Newton's method and the fixed-point iteration, in our numerical tests. For small Knudsen numbers, our method has an efficiency between the classical source iteration and the modern generalized synthetic iterative scheme, and the complexity of its implementation is closer to the source iteration. A variety of numerical tests are carried out to demonstrate its performance, and it is concluded that the proposed method is suitable for applications with moderate to large Knudsen numbers.