We propose a numerical approach, of the BGK kinetic type, that is able to approximate with a given, but arbitrary, order of accuracy the solution of linear and non-linear convection-diffusion type problems: scalar advection-diffusion, non-linear scalar problems of this type and the compressible Navier-Stokes equations. Our kinetic model can use \emph{finite} advection speeds that are independent of the relaxation parameter, and the time step does not suffer from a parabolic constraint. Having finite speeds is in contrast with many of the previous works about this kind of approach, and we explain why this is possible: paraphrasing more or less \cite{golse:hal-00859451}, the convection-diffusion like PDE is not a limit of the BGK equation, but a correction of the same PDE without the parabolic term at the second order in the relaxation parameter that is interpreted as Knudsen number. We then show that introducing a matrix collision instead of the well-known BGK relaxation makes it possible to target a desired convection-diffusion system. Several numerical examples, ranging from a simple pure diffusion model to the compressible Navier-Stokes equations illustrate our approach

翻译：我们提出了一种BGK动力学类型的数值方法，能够以给定但任意阶精度逼近线性和非线性对流扩散型问题的解：包括标量对流扩散、此类非线性标量问题以及可压缩Navier-Stokes方程。该动力学模型采用与松弛参数无关的有限对流速度，且时间步长不受抛物线型约束限制。采用有限速度与以往同类研究的多数工作形成鲜明对比，我们阐释了其可行性：大致转述文献[golse:hal-00859451]的思想，即对流扩散型偏微分方程并非BGK方程的极限，而是在以努森数为松弛参数的二阶项上对不含抛物线项的同种偏微分方程进行修正。进一步证明，通过引入矩阵碰撞替代经典的BGK松弛，能够针对特定对流扩散系统实现精准求解。从简单纯扩散模型到可压缩Navier-Stokes方程等多个数值算例验证了本方法的有效性。