Multiscale Modeling &Simulation, Volume 19, Issue 1, Page 478-505, January 2021.
We design a variational asymptotic preserving scheme for the Vlasov--Poisson--Fokker--Planck system with the high field scaling, which describes the Brownian motion of a large system of particles in a surrounding bath. Our scheme builds on an implicit-explicit framework, wherein the stiff terms coming from the collision and field effects are solved implicitly while the convection terms are solved explicitly. To treat the implicit part, we propose a variational approach by viewing it as a Wasserstein gradient flow of the relative entropy, and solve it via a proximal quasi-Newton method. In so doing we get positivity and asymptotic preservation for free. The method is also massively parallelizable and thus suitable for high dimensional problems. We further show that the convergence of our implicit solver is uniform across different scales. A suite of numerical examples are presented at the end to validate the performance of the proposed scheme.

中文翻译：

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Vlasov--Poisson--Fokker--Planck系统的变分渐近保持方案

多尺度建模与仿真，第 19 卷，第 1 期，第 478-505 页，2021 年 1 月。
我们为具有高场标度的 Vlasov--Poisson--Fokker--Planck 系统设计了一种变分渐近保持方案，该方案描述了周围浴槽中大型粒子系统的布朗运动。我们的方案建立在隐式-显式框架上，其中来自碰撞和场效应的刚性项被隐式求解，而对流项被显式求解。为了处理隐含部分，我们提出了一种变分方法，将其视为相对熵的 Wasserstein 梯度流，并通过近端拟牛顿法求解。通过这样做，我们可以免费获得正性和渐近保存。该方法还可以大规模并行化，因此适用于高维问题。我们进一步表明，我们的隐式求解器的收敛性在不同的尺度上是一致的。