# An all speed second order IMEX relaxation scheme for the Euler equations

## TL;DR

This paper introduces a second-order IMEX relaxation scheme for the Euler equations that is positivity-preserving, asymptotic-preserving, and applicable in 1D and 2D, improving numerical stability and accuracy in fluid simulations.

## Contribution

The paper develops a novel all-speed second-order IMEX relaxation scheme for Euler equations that maintains positivity and asymptotic properties, with demonstrated convergence and applicability.

## Key findings

- Scheme is positivity preserving for density and internal energy.
- Scheme is asymptotic preserving towards incompressible Euler equations.
- Numerical experiments confirm convergence and effectiveness in 1D and 2D.

## Abstract

We present an implicit-explicit finite volume scheme for the Euler equations. We start from the non-dimensionalised Euler equations where we split the pressure in a slow and a fast acoustic part. We use a Suliciu type relaxation model which we split in an explicit part, solved using a Godunov-type scheme based on an approximate Riemann solver, and an implicit part where we solve an elliptic equation for the fast pressure. The relaxation source terms are treated projecting the solution on the equilibrium manifold. The proposed scheme is positivity preserving with respect to the density and internal energy and asymptotic preserving towards the incompressible Euler equations. For this first order scheme we give a second order extension which maintains the positivity property. We perform numerical experiments in 1D and 2D to show the applicability of the proposed splitting and give convergence results for the second order extension.