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 operator that conserves mass, charge, momentum, and energy, while dissipating 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：我们提出了一种扩展的粒子网格（PIC）方法，用于捕捉Vlasov-Maxwell-Landau方程中的Landau碰撞效应。该方法源于Landau方程变分形式的正则化处理，从而得到一种能够守恒质量、电荷、动量和能量，同时耗散（正则化）熵的算子离散格式。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。该格式适用于任意维度及一般相互作用（包括库仑情况）。我们在朗道阻尼、双流不稳定性和韦贝尔不稳定性等场景下验证了该格式，证明了其在等离子体数值模拟中的有效性。