A new class of high order semi-Lagrangian schemes for rarefied gas dynamics

TL;DR

This paper introduces a high-order semi-Lagrangian scheme for rarefied gas dynamics that improves accuracy and long-term solution preservation by reconstructing the distribution function at extreme points, applicable across different regimes.

Contribution

The paper generalizes a semi-Lagrangian scheme to high order accuracy using polynomial reconstructions and extreme point updates, enhancing performance over existing methods.

Findings

Achieves high-order accuracy for all gas regimes

Preserves solution structure over long simulation times

Demonstrates improved results through numerical tests

Abstract

In this paper we genealize the fast semi-Lagrangian scheme developed in [J. Comput. Phys., Vol. 255, 2013, pp 680-698] to the case of high order reconstructions of the distribution function. The original first order accurate semi-Lagrangian scheme is supplemented with polynomial reconstructions of the distribution function and of the collisional operator leading to an effective high order accurate numerical scheme for all regimes, from extremely rarefied gas to highly collisional siuation. The main idea relies on updating at each time step the extreme points of the distribution function for each velocity of the lattice instead of updating the solution in the cell centers, these extremes points being located at different positions for any fixed velocity of the lattice. The result is a class of scheme which permits to preserve the structure of the solution over very long times compared to existing schemes from the literature. We propose a proof of concept of this new approach along with numerical tests and comparisons with classical numerical methods.