We introduce an extension of the particle-in-cell (PIC) method that captures the Landau collisional effects in the Vlasov-Maxwell-Landau equations. The method arises from a regularisation of the variational formulation of the Landau equation, leading to a discretisation of the collision operator that conserves mass, charge, momentum, and energy, while increasing the (regularised) entropy. The collisional effects appear as a fully deterministic effective force, thus the method does not require any transport-collision splitting. The scheme can be used in arbitrary dimension, and for a general interaction, including the Coulomb case. We validate the scheme on scenarios such as the Landau damping, the two-stream instability, and the Weibel instability, demonstrating its effectiveness in the numerical simulation of plasma.

翻译：针对Vlasov-Maxwell-Landau方程组，我们提出了一种能够捕获Landau碰撞效应的粒子-网格（PIC）方法扩展。该方法源于对Landau方程变分形式的正则化处理，从而得到一种离散化的碰撞算子，该算子守恒质量、电荷、动量与能量，同时增加（正则化）熵。碰撞效应表现为完全确定性的有效力，因此该方法无需进行输运-碰撞分裂。该方案适用于任意维度及一般相互作用（包括库仑相互作用）。我们通过Landau阻尼、双流不稳定性和Weibel不稳定性等场景验证了该方案的有效性，证明了其在等离子体数值模拟中的高效性。