标题： Sapphire++：结合球谐展开与间断伽辽金方法的粒子输运代码 显示英文标题

标题： Sapphire++: A particle transport code combining a spherical harmonic expansion and the discontinuous Galerkin method

摘要： 我们介绍了Sapphire++，这是一个开源代码，旨在为天体物理应用数值求解Vlasov-Fokker-Planck方程。 Sapphire++采用基于分布函数球谐展开的数值算法，将Vlasov-Fokker-Planck方程表示为一组控制展开系数演化的偏微分方程系统。该代码结合间断Galerkin方法与隐式和显式时间步进方法来计算这些系数，从而在空间和时间精度的选择上提供了显著的灵活性。 我们通过示例验证了代码的有效性。特别是，我们模拟了平行激波中测试粒子的加速，并将其结果与解析预测进行了比较。 Sapphire++代码（https://github.com/sapphirepp/sapphirepp）作为免费且开源的工具向社区提供。 显示英文摘要

摘要： We present Sapphire++, an open-source code designed to numerically solve the Vlasov-Fokker-Planck equation for astrophysical applications. Sapphire++ employs a numerical algorithm based on a spherical harmonic expansion of the distribution function, expressing the Vlasov-Fokker-Planck equation as a system of partial differential equations governing the evolution of the expansion coefficients. The code utilises the discontinuous Galerkin method in conjunction with implicit and explicit time stepping methods to compute these coefficients, providing significant flexibility in its choice of spatial and temporal accuracy. We showcase the code's validity using examples. In particular, we simulate the acceleration of test particles at a parallel shock and compare the results to analytical predictions. The Sapphire++ code (https://github.com/sapphirepp/sapphirepp) is available as a free and open-source tool for the community.