一种多尺度粒子方法用于均场方程：一般情况
A multi-scale particle method for mean field equations: the general case
AI总结：
本文提出一种多尺度无网格粒子方法，用于求解广义相互作用粒子模型的宏观均场近似，适用于大和小相互作用半径，并扩展到更一般的情况，包括超几何极限。
摘要：
一种用于宏观均场近似的多尺度无网格粒子方法被开发并研究。该方法在大和小相互作用半径的情况下均有效。本文将[39]中针对多孔介质类型极限方程的方法扩展到更一般的情况，包括超几何极限。该方法可视为一种从微观相互作用粒子系统（DEM型方法）到宏观方程（无网格粒子方法）的数值过渡。我们详细讨论了该方案在各种示例中的数值性能以及计算时间的潜在节省。后者在接近宏观极限的情况下尤为显著。该方法可应用于涉及均场近似的群体行为、交通、行人或颗粒流模拟等问题。
英文摘要：
A multi-scale meshfree particle method for macroscopic mean field approximations of generalized interacting particle models is developed and investigated. The method is working in a uniform way for large and small interaction radii. The well resolved case for large interaction radius is treated, as well as underresolved situations with small values of the interaction radius. In the present work we extend the approach from [39] for porous media type limit equations to a more general case, including in particular hyperbolic limits. The method can be viewed as a numerical transition between a DEM-type method for microscopic interacting particle systems and a meshfree particle method for macroscopic equations. We discuss in detail the numerical performance of the scheme for various examples and the potential gain in computation time. The latter is shown to be particularly high for situations near the macroscopic limit. There are various applications of the method to problems involving mean field approximations in swarming, tra?c, pedestrian or granular fow simulation.
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