论文标题
Vlasov-Poisson-Fokker-Planck系统的变分渐近保护方案
Variational Asymptotic Preserving Scheme for the Vlasov-Poisson-Fokker-Planck System
论文作者
论文摘要
我们设计了一个具有高场缩放的Vlasov-Poisson-Fokker-Planck系统的变异渐近保存方案，该系统描述了周围浴室中大型颗粒系统的布朗运动。我们的方案建立在一个隐式解释框架上，其中，从碰撞和野外效应的僵硬术语被隐式求解，而对流术语则明确解决。为了治疗隐式部分，我们通过将其视为相对熵的Wasserstein梯度流来提出一种变分方法，并通过近端准Newton方法求解它。这样，我们可以免费获得积极性和渐近保护。该方法也非常平行，因此适用于高维问题。我们进一步表明，我们隐式求解器的收敛性在不同尺度上是统一的。在末尾提供了一套数值示例，以验证拟议方案的性能。
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.
