Conservative semi-Lagrangian schemes for kinetic equations Part I: Reconstruction

TL;DR

This paper introduces a high-order conservative semi-Lagrangian reconstruction technique for kinetic equations, enabling accurate and stable numerical solutions, with analysis and demonstrations on models like Xin-Jin and Broadwell.

Contribution

It presents a novel reconstruction method based on sliding averages of polynomial reconstructions, enhancing semi-Lagrangian schemes for kinetic equations.

Findings

High-order conservative schemes achieved with the new reconstruction.

Effective handling of shock problems in kinetic models.

Mathematical properties of the reconstruction are rigorously analyzed.

Abstract

In this paper, we propose and analyse a reconstruction technique which enables one to design high-order conservative semi-Lagrangian schemes for kinetic equations. The proposed reconstruction can be obtained by taking the sliding average of a given polynomial reconstruction of the numerical solution. A compact representation of the high order conservative reconstruction in one and two space dimension is provided, and its mathematical properties are analyzed. To demonstrate the performance of proposed technique, we consider implicit semi-Lagrangian schemes for kinetic-like equations such as the Xin-Jin model and the Broadwell model, and then solve related shock problems which arise in the relaxation limit. Applications to BGK and Vlasov-Poisson equations will be presented in the second part of the paper.