Asymptotic analysis of high order IMEX-RK methods for ES-BGK model at Navier-Stokes level

¹ Department of Mathematics and Computer Science, University of Catania, 95125 Catania, Italy
² Department of Mathematics, Gyeongsang National University, 52828 Jinju, Republic of Korea

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 22 August 2025
Accepted: 6 January 2026

Abstract

Implicit explicit Runge–Kutta (IMEX-RK) time discretization methods are very popular when solving stiff kinetic equations. In the work of Hu and Zhang in [J. Sci. Comput. 73 (2017) 797–818], an asymptotic analysis shows that a specific class of high-order IMEX-RK schemes can accurately capture the Navier-Stokes limit without needing to resolve the small scales dictated by the Knudsen number. In this work, we extend the asymptotic analysis to general IMEX-RK schemes, known in literature as Type I and Type II. We further introduce some IMEX-RK methods developed by Boscarino and Pareschi [J. Comput. Appl. Math. 316 (2017) 60–73 to attain uniform accuracy in the wide range of Knudsen numbers. Several numerical examples are presented to verify the validity of the obtained theoretical results and the effectiveness of the methods.