Numerical approximation of kinetic Fokker–Planck equations with specular reflection boundary conditions
Numerical approximation of kinetic Fokker–Planck equations with specular reflection boundary conditions
复制标题
具有镜面反射边界条件的动力学福克普朗克方程的数值近似
DOI：
10.1016/j.jcp.2024.112841
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发表时间：
2024
影响因子：
4.1
通讯作者：
Borzì, A.
中科院分区：
文献类型：
--
作者：
Roy, S.;Borzì, A.
本文研究了高维Fokker-Planck(FP)动力学方程在镜面反射边界条件下的数值逼近问题。这种数值近似是基于将动力学FP模型分裂为空间中的输运方程和速度坐标中的FP扩散模型。前者用Kurganov-TAdmor有限体积格式离散，后者用广义Chang&Cooper有限体积法近似。时间积分由强保持稳定性的Runge-Kutta方法进行，其中反应项和源项通过Strang分裂技术和Magnus积分器进行调节。证明了在连续模型具有这些性质的情况下，所得到的数值解方法是守恒和正保的，并且满足CFL条件，在时间和相空间上具有L 1范数的二阶精度。数值实验结果验证了这些理论结果。
This work is devoted to the analysis of a numerical approximation to a general multi-dimensional kinetic Fokker–Planck (FP) equation with reaction and source terms and subject to specular reflection boundary conditions. This numerical approximation is based on splitting the kinetic FP model into a transport equation in space and a FP diffusive model in the velocity coordinates. The former is discretized by a Kurganov-Tadmor finite-volume scheme, while the latter is approximated by a generalized Chang & Cooper finite-volume method. Time integration is performed by a strong stability-preserving Runge-Kutta method where the reaction and source terms are accommodated with a Strang splitting technique and the use of a Magnus integrator. It is proved that the resulting numerical solution method is conservative and positive preserving, in the case where the continuous model has these properties, and it is second-order accurate in time and in phase space in the L 1-norm, subject to a CFL condition. Results of numerical experiments are reported that validate these theoretical results.