We propose an Hermite spectral method for the Fokker-Planck-Landau (FPL) equation. Both the distribution functions and the collision terms are approximated by series expansions of the Hermite functions. To handle the complexity of the quadratic FPL collision operator, a reduced collision model is built by adopting the quadratic collision operator for the lower-order terms and the diffusive Fokker-Planck operator for the higher-order terms in the Hermite expansion of the reduced collision operator. The numerical scheme is split into three steps according to the Strang splitting, where different expansion centers are employed for different numerical steps to take advantage of the Hermite functions. The standard normalized Hermite basis [36] is adopted during the convection and collision steps to utilize the precalculated coefficients of the quadratic collision terms, while the one constituted by the local macroscopic velocity and temperature is utilized for the acceleration step, by which the effect of the external force can be simplified to an ODE. Projections between different expansion centers are achieved by an algorithm proposed in [29]. Several numerical examples are studied to test and validate our new method.

我们为Fokker-Planck-Landau（FPL）方程提出了一种Hermite光谱方法。分布函数和碰撞项都通过Hermite函数的级数展开来近似。为了处理二次FPL碰撞算子的复杂性，在减少碰撞的Hermite展开中，通过对低阶项采用二次碰撞算子，对高阶项采用扩散Fokker-Planck算子，建立了简化的碰撞模型。操作员。根据斯特朗分裂，将数值方案分为三个步骤，其中对不同的数值步骤采用不同的扩展中心，以利用Hermite函数。在对流和碰撞步骤中采用标准归一化Hermite基础[36]来利用二次碰撞项的预先计算的系数，而由局部宏观速度和温度构成的系数用于加速步骤，从而获得外力可以简化为ODE。通过[29]中提出的算法可以实现不同扩展中心之间的投影。研究了几个数值示例，以测试和验证我们的新方法。