Numerical boundary control of multi-dimensional discrete-velocity kinetic models

TL;DR

This paper develops numerical control methods for multi-dimensional discrete-velocity kinetic models, ensuring exponential decay of solutions through operator splitting, Lyapunov functions, and implicit schemes, validated by simulations.

Contribution

It introduces a novel numerical control approach with stability analysis for multi-dimensional kinetic models, extending previous theoretical results to practical algorithms.

Findings

Numerical control laws guarantee exponential decay of solutions.

Implicit schemes ensure stability for stiff source terms.

Validated effectiveness through three 2D numerical simulations.

Abstract

This paper extends our recent results on multi-dimensional discrete-velocity models to the numerical level. By adopting an operator splitting scheme and introducing a suitable discrete Lyapunov function, we derive numerical control laws that ensure the corresponding numerical solutions decay exponentially in time. To handle the stiff source term, we also use an implicit scheme for the collision part and prove the stability of the resulting schemes. The theoretical results are validated through three numerical simulations for the two-dimensional coplanar model.