When applying Hamiltonian operator splitting methods for the time integration of multi-species Vlasov-Maxwell-Landau systems, the reliable and efficient numerical approximation of the Landau equation represents a fundamental component of the entire algorithm. Substantial computational issues arise from the treatment of the physically most relevant three-dimensional case with Coulomb interaction. This work is concerned with the introduction and numerical comparison of novel approaches for the evaluation of the Landau collision operator. In the spirit of collocation, common tools are the identification of fundamental integrals, series expansions of the integral kernel and the density function on the main part of the velocity domain, and interpolation as well as quadrature approximation nearby the singularity of the kernel. Focusing on the favourable choice of the Fourier spectral method, their practical implementation uses the reduction to basic integrals, fast Fourier techniques, and summations along certain directions. Moreover, an important observation is that a significant percentage of the overall computational effort can be transferred to precomputations which are independent of the density function. For the purpose of exposition and numerical validation, the cases of constant, regular, and singular integral kernels are distinguished, and the procedure is adapted accordingly to the increasing complexity of the problem. With regard to the time integration of the Landau equation, the most expedient approach is applied in such a manner that the conservation of mass is ensured.

翻译：在采用哈密顿算子分裂方法对多物种Vlasov-Maxwell-Landau系统进行时间积分时，朗道方程的可靠高效数值逼近是整个算法的基础组成部分。处理具有库仑相互作用的物理上最重要的三维情况会引发大量计算难题。本文致力于介绍并数值比较用于评估朗道碰撞算子的新方法。基于配点法的思想，常用工具包括辨识基本积分、对积分核和速度域主要部分上的密度函数进行级数展开，以及在核奇点附近采用插值与求积近似。聚焦于傅里叶谱方法的优选方案，其实际实现依赖于基本积分的约化、快速傅里叶技术以及沿特定方向的求和。此外，一个重要观察是总体计算量中有显著比例可转移至与密度函数无关的预计算。为便于阐述与数值验证，本研究区分了常数核、正则核和奇异积分核三种情况，并相应调整流程以适应问题复杂度的递增。针对朗道方程的时间积分，采用最便捷的方法以确保质量守恒。