一种用于对流-扩散方程的灵活速度Boltzmann格式
A Flexible Velocity Boltzmann Scheme for Convection-Diffusion Equations
AI总结：
本文提出一种速度可调的有限速度Boltzmann框架，结合通量差分裂与广义动力学Lax-Wendroff格式及限制器，有效控制数值扩散并求解一维、二维非线性对流-扩散方程。
摘要：
针对对流-扩散方程，建立了一个基于有限速度模型的Boltzmann方程框架。这些速度保持灵活并可调节，以控制数值扩散。随后引入一种基于通量差分裂的动力学格式，用于数值求解多种非线性对流-扩散方程。在此框架基础上，还推导出一种广义动力学Lax-Wendroff格式，经典的Lax-Wendroff方法可作为其中的一种选择得到恢复。进一步提出了该动力学通量差分裂格式的全变差衰减版本，并利用限制器函数将其与动力学Lax-Wendroff格式相结合。该数值格式经过了广泛测试，文中给出了一维和二维非线性对流方程及对流-扩散方程基准测试算例的结果。
英文摘要：
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.
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