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.

翻译：我们提出了一种基于对称高斯-赛德尔（SGS）方法的稳态玻尔兹曼方程数值求解器。由于玻尔兹曼方程中存在二次碰撞算子，SGS方法需要在每个网格单元上求解非线性系统，并在数值测试中考虑了两种方法：牛顿法和不动点迭代法。对于小克努森数，该方法的效率介于经典源迭代法与现代化广义合成迭代格式之间，且其实现复杂度更接近源迭代法。通过一系列数值测试验证了该方法性能，结果表明，所提方法适用于中等至大克努森数范围的应用场景。