Structure-preserving particle methods have recently been proposed for a class of nonlinear continuity equations, including aggregation-diffusion equation in [J. Carrillo, K. Craig, F. Patacchini, Calc. Var., 58 (2019), pp. 53] and the Landau equation in [J. Carrillo, J. Hu., L. Wang, J. Wu, J. Comput. Phys. X, 7 (2020), pp. 100066]. One common feature to these equations is that they both admit some variational formulation, which upon proper regularization, leads to particle approximations dissipating the energy and conserving some quantities simultaneously at the semi-discrete level. In this paper, we formulate continuity equations with a density dependent bilinear form associated with the variational derivative of the energy functional and prove that appropriate particle methods satisfy a compatibility condition with its regularized energy. This enables us to utilize discrete gradient time integrators and show that the energy can be dissipated and the mass conserved simultaneously at the fully discrete level. In the case of the Landau equation, we prove that our approach also conserves the momentum and kinetic energy at the fully discrete level. Several numerical examples are presented to demonstrate the dissipative and conservative properties of our proposed method.

翻译：结构保持粒子方法最近被提出用于一类非线性连续性方程，包括[J. Carrillo, K. Craig, F. Patacchini, Calc. Var., 58 (2019), pp. 53]中的聚集-扩散方程以及[J. Carrillo, J. Hu., L. Wang, J. Wu, J. Comput. Phys. X, 7 (2020), pp. 100066]中的朗道方程。这些方程的一个共同特征是它们都允许某种变分形式，经过适当的正则化后，可在半离散层面导出同时耗散能量并守恒某些量的粒子近似。本文针对一类与能量泛函变分导数相关的密度依赖双线性形式所描述的连续性方程，证明了适当的粒子方法满足与其正则化能量的相容性条件。这使我们能够利用离散梯度时间积分器，并证明在完全离散层面上能量可被耗散且质量可同时守恒。对于朗道方程，我们证明了该方法在完全离散层面还能守恒动量和动能。文中给出了若干数值算例，以展示所提方法的耗散与守恒特性。