Physical and numerical limitations of stationary Vlasov-Poisson solvers based on backward Liouville methods are investigated with five solvers that combine different meshes, numerical integrators, and electric field interpolation schemes. Since some of the limitations arise when moving from an integrable to a non-integrable configuration, an elliptical Langmuir probe immersed in a Maxwellian plasma was considered and the eccentricity ($e_p$) of its cross-section used as integrability-breaking parameter. In the cylindrical case, $e_p=0$, the energy and angular momentum are both conserved. The trajectories of the charged particles are regular and the boundaries that separate trapped from non-trapped particles in phase space are smooth curves. However, their computation has to be done carefully because, albeit small, the intrinsic numerical errors of some solvers break these conservation laws. It is shown that an optimum exists for the number of loops around the probe that the solvers need to classify a particle trajectory as trapped. For $e_p\ne 0$, the angular momentum is not conserved and particle dynamics in phase space is a mix of regular and chaotic orbits. The distribution function is filamented and the boundaries that separate trapped from non-trapped particles in phase space have a fractal geometry. The results were used to make a list of recommendations for the practical implementation of stationary Vlasov-Poisson solvers in a wide range of physical scenarios.

使用五个结合了不同网格，数值积分器和电场插值方案的求解器，研究了基于后向Liouville方法的平稳Vlasov-Poisson求解器的物理和数值限制。由于从可积分配置变为不可积分配置时会出现一些限制，因此考虑了浸入麦克斯韦等离子体中的椭圆Langmuir探头，并且偏心率（${Ë}_p$的横截面）用作可打破积分的参数。在圆柱形的情况下，${Ë}_p=0$，能量和角动量都守恒。带电粒子的轨迹是规则的，并且在相空间中将捕获的粒子与未捕获的粒子分开的边界是平滑曲线。但是，它们的计算必须谨慎进行，因为尽管很小，但某些求解器的内在数值误差违反了这些守恒定律。结果表明，对于求解器需要将粒子轨迹分类为捕获轨迹的探测器周围的回路数量而言，存在一个最佳值。为了${Ë}_p\ne 0$，角动量不守恒，相空间中的粒子动力学是规则轨道和混沌轨道的混合。分布函数是细丝状的，并且在相空间中将被捕获粒子与未捕获粒子分开的边界具有分形几何形状。结果被用来为在广泛的物理场景中实际实施固定式Vlasov-Poisson求解器提供一系列建议。