A Flexible Velocity Boltzmann Scheme for Convection-Diffusion Equations

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.