A stable spherical harmonics ($P_N$) method for solving the problem of a multi-region sphere with internal sources subject to an externally incident angular flux is developed. The method consists in solving the problem locally (region by region) and using an iterative sweep technique to connect and update the solutions for the different regions until global convergence is attained. A post-processing step which is very effective in improving the $P_N$ solutions is also developed. Since calculating the current from its definition in terms of the angular flux is prone to loss of precision in spherical geometry, an alternative scheme for computing accurate post-processed currents is devised.

Because the developed method does not rely on space and angle discretizations, it can achieve very accurate results even for problems with solutions that display strong flux gradients. This kind of problem is challenging for methods built on discretization schemes. An example of such a problem is solved and the results obtained demonstrate the good performance of the method. Accurate numerical results are also given for two variants of the example problem that involve changes in the driving term and in the scattering law.

稳定的球谐（$P_{ñ}$）解决了内部区域受到外部入射角通量影响的多区域球体问题的方法。该方法包括局部（逐个区域）解决问题，并使用迭代扫描技术连接和更新不同区域的解决方案，直到实现全局收敛。后处理步骤对于改善$P_{ñ}$还开发了解决方案。由于根据角通量的定义来计算电流容易降低球面几何形状的精度，因此提出了一种用于计算精确后处理电流的替代方案。

因为开发的方法不依赖于空间和角度离散，所以即使对于显示强磁通梯度的解决方案，它也可以实现非常准确的结果。对于基于离散化方案的方法而言，这种问题具有挑战性。解决了此类问题的一个示例，所获得的结果证明了该方法的良好性能。还为示例问题的两个变体提供了精确的数值结果，这些变体涉及驱动项和散射定律的变化。