A Flexible Velocity Boltzmann Scheme for Convection-Diffusion Equations

TL;DR

This paper introduces a flexible finite-velocity Boltzmann-based framework and kinetic schemes for solving nonlinear convection-diffusion equations, allowing control over numerical diffusion and improving numerical stability.

Contribution

It develops a novel kinetic framework with adjustable velocities and introduces a flux difference splitting scheme, including a TVD version, for enhanced numerical solutions of convection-diffusion problems.

Findings

Effective control of numerical diffusion through velocity adjustment

Successful application to 1D and 2D nonlinear convection-diffusion equations

Demonstrated stability and accuracy in benchmark tests

Abstract

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.