Numerical methods for kinetic equations

Abstract

In this survey we consider the development and the mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity.\ud \ud Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic preserving methods and the construction of hybrid schemes

Numerical simulation of flows from free molecular regime to continuum regime by a DVM with streaming and collision processes

“…By applying the above approaches, a number of flow problems from free molecular regime to continuum regime have been well resolved. Different from the above mentioned kinetic schemes [2][3][4][5][6][17][18][19][20][21][22][23][24], the semi-Lagrangian method [33][34][35] and lattice Boltzmann method (LBM) [36][37][38][39][40] have been devised and applied to simulate rarefied flows with streaming and collision processes. The semi-Lagrangian method is indeed a DVM.…”

“…Otherwise, the distribution functions at the surrounding points should be computed by suitable reconstruction from those at the mesh points. On the other hand, to the best of our knowledge, the BGK model is usually utilized in the conventional semi-Lagrangian method [33][34][35]. In addition, the application of the semi-Lagrangian method for simulation of near continuum flows and hypersonic rarefied flows is still limited.…”

“…As indicated in [36], LBM can also be considered as a special form of DVM. However, different from the conventional DVM [2][3][4][5][6][17][18][19][20][21][22][23][24][33][34][35], the LBM is intended to seek a minimal set of velocities in the phase velocity space rather than directly discretizes the whole velocity space. This is an important feature of LBM for effective reduction of the computational cost.…”

Tensor-Product Discretization for the Spatially Inhomogeneous and Transient Boltzmann Equation in Two Dimensions

“…The classical Fourier spectral method [26,11] is known to converge to a constant distribution in velocity for long times [7,Rem. 5.13].…”

Spectral method for a kinetic swarming model

“…Spectral methods exploit the weighted convolution structure of the Fourier transform [20] of the interaction terms for high accuracy. Spectral approximations for Boltzmann collision operators were first proposed by Pareschi and Perthame [21], and many other authors developed numerical methods for Boltzmann and Fokker-Planck type collisions [22][23][24][25][26][27][28][29][30].…”