A framework of finite-velocity model based Boltzmann equation has been developed for convection-diffusion equations. These velocities are kept flexible and adjusted to control numerical diffusion. A flux difference splitting based kinetic scheme is then introduced for solving a wide variety of nonlinear convection-diffusion equations numerically. Based on this framework, a generalized kinetic Lax-Wendroff scheme is also derived, recovering the classical Lax-Wendroff method as one of the choices. Further, a total variation diminishing version of this kinetic flux difference splitting scheme is presented, combining it with the kinetic Lax-Wendroff scheme using a limiter function. The numerical scheme has been extensively tested and the results for benchmark test cases, for 1D and 2D nonlinear convection and convection-diffusion equations, are presented.

翻译：本文发展了一种基于有限速度模型玻尔兹曼方程的框架，用于求解对流-扩散方程。这些速度保持灵活并可调节，以控制数值耗散。随后，引入了一种基于通量差分分裂的动理学格式，用于数值求解多种非线性对流-扩散方程。基于此框架，还推导出一种广义的动理学Lax-Wendroff格式，经典Lax-Wendroff方法可作为其特例之一。此外，通过结合该动理学通量差分分裂格式与动理学Lax-Wendroff格式，并利用限制器函数，提出了该格式的总变差递减版本。该数值格式经过了广泛测试，文中展示了一维和二维非线性对流方程及对流-扩散方程基准测试案例的结果。