High-Order Kinetic Relaxation Schemes as High-Accuracy Poisson Solvers

TL;DR

This paper introduces high-order kinetic relaxation schemes derived from Boltzmann's theory to solve the Poisson equation with high accuracy, significantly reducing computational time compared to standard methods.

Contribution

The authors develop a novel high-order expansion technique to improve the accuracy of kinetic schemes for Poisson problems, achieving substantial computational efficiency gains.

Findings

Achieves high-accuracy solutions for the Poisson equation using kinetic schemes.

Reduces computational time by up to six orders of magnitude compared to standard lattice Boltzmann methods.

Demonstrates effectiveness for problems with sizeable Knudsen numbers.

Abstract

We present a new approach to find accurate solutions to the Poisson equation, as obtained from the steady-state limit of a diffusion equation with strong source terms. For this purpose, we start from Boltzmann's kinetic theory and investigate the influence of higher order terms on the resulting macroscopic equations. By performing an appropriate expansion of the equilibrium distribution, we provide a method to remove the unnecessary terms up to a desired order and show that it is possible to find, with high level of accuracy, the steady-state solution of the diffusion equation for sizeable Knudsen numbers. In order to test our kinetic approach, we discretise the Boltzmann equation and solve the Poisson equation, spending up to six order of magnitude less computational time for a given precision than standard lattice Boltzmann methods.