# A low Mach two-speed relaxation scheme for the compressible Euler equations with gravity

@article{Birke2021ALM, title={A low Mach two-speed relaxation scheme for the compressible Euler equations with gravity}, author={Claudius Birke and Christophe Chalons and Christian Klingenberg}, journal={ArXiv}, year={2021}, volume={abs/2112.02986} }

We present a numerical approximation of the solutions of the Euler equations with a gravitational source term. On the basis of a Suliciu type relaxation model with two relaxation speeds, we construct an approximate Riemann solver, which is used in a ﬁrst order Godunov-type ﬁnite volume scheme. This scheme can preserve both stationary solutions and the low Mach limit to the corresponding incompressible equations. In addition, we prove that our scheme preserves the positivity of density and…

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## References

SHOWING 1-10 OF 35 REFERENCES

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

- MathematicsCommunications in Computational Physics
- 2020

The proposed scheme is positivity preserving with respect to the density and internal energy and asymptotic preserving towards the incompressible Euler equations and gives a second order extension which maintains the positivity property.

### An entropy satisfying two-speed relaxation system for the barotropic Euler equations: application to the numerical approximation of low Mach number flows

- Computer Science, MathematicsNumerische Mathematik
- 2020

It is proved that the underlying scheme satisfies the well-known asymptotic-preserving property in the sense that it is uniformly (first-order) accurate with respect to the Mach number, and at the same time it satisfies a fully discrete entropy inequality.

### An all speed second order well-balanced IMEX relaxation scheme for the Euler equations with gravity

- MathematicsJ. Comput. Phys.
- 2020

### A low-Mach Roe-type solver for the Euler equations allowing for gravity source terms

- Physics
- 2017

In order to perform simulations of low Mach number flow in presence of gravity the technique from [23] is found insufficient as it is unable to cope with the presence of a hydrostatic equilibrium.…

### A Second Order Well-Balanced Finite Volume Scheme for Euler Equations with Gravity

- MathematicsSIAM J. Sci. Comput.
- 2015

A well-balanced, second order, Godunov-type finite volume scheme for compressible Euler equations with gravity, which admits a discrete stationary solution which is a second order accurate approximation to the exact stationary solution.

### An all Mach number relaxation upwind scheme

- Mathematics
- 2020

The present paper concerns the derivation of finite volume methods to approximate the weak solutions of the Euler equations within all Mach number regimes. To address such an issue, we develop a…

### A Numerical Scheme for the Compressible Low-Mach Number Regime of Ideal Fluid Dynamics

- PhysicsJ. Sci. Comput.
- 2017

Property analysis and performance study of a new technique that allows to correctly represent low Mach number flows with a discretization of the compressible Euler equations through the evolution of its condition number.

### Entropy Stable Numerical Fluxes for Compressible Euler Equations which are Suitable for All Mach Numbers

- Computer ScienceArXiv
- 2020

Two novel two-state approximate Riemann solvers for the compressible Euler equations are proposed which are provably entropy dissipative and suitable for the simulation of low Mach numbers and one is provably kinetic energy stable.

### A well‐balanced scheme to capture non‐explicit steady states in the Euler equations with gravity

- Mathematics
- 2016

This paper describes a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the…