[Paper Review] Asymptotic Preserving and Low Mach Number Accurate IMEX Finite Volume Schemes for the Euler Equations

K. R. Arun, Saurav Samantaray|arXiv (Cornell University)|Jul 3, 2019

Computational Fluid Dynamics and Aerodynamics32 references4 citations

TL;DR

This paper presents second-order asymptotic preserving and low Mach number accurate IMEX finite volume schemes for the compressible Euler equations. By splitting stiff (pressure) and non-stiff (velocity) fluxes and treating them implicitly and explicitly in time, respectively, and using central fluxes and differencing in space, the scheme maintains accuracy across Mach numbers and achieves asymptotic accuracy in the zero Mach number limit.

ABSTRACT

In this paper, the design and analysis of a class of second order accurate IMEX finite volume schemes for the compressible Euler equations in the zero Mach number limit is presented. In order to account for the fast and slow waves, the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components, respectively. The time discretisation is performed by an IMEX Runge-Kutta method, therein the stiff terms are treated implicitly and the non-stiff terms explicitly. In the space discretisation, a Rusanov-type central flux is used for the non-stiff part, and simple central differencing for the stiff part. Both the time semi-discrete and space-time fully-discrete schemes are shown to be asymptotic preserving. The numerical experiments confirm that the schemes achieve uniform second order convergence with respect to the Mach number. A notion of accuracy at low Mach numbers, termed as the asymptotic accuracy, is introduced in terms of the invariance of a well-prepared space of constant densities and divergence-free velocities. The asymptotic accuracy is concerned with the closeness of the compressible solution with that of its incompressible counterpart in a low Mach number regime. It is shown theoretically as well as numerically that the proposed schemes are asymptotically accurate.

Motivation & Objective

• To develop numerical schemes that remain accurate and stable in the zero Mach number limit, where the flow becomes incompressible.
• To address the stiffness arising from high acoustic waves in the Euler equations at low Mach numbers.
• To ensure the scheme preserves the incompressible limit by maintaining well-prepared initial conditions with constant density and divergence-free velocity.
• To achieve uniform second-order convergence across all Mach numbers, including the asymptotic regime.
• To introduce and validate a new notion of 'asymptotic accuracy' that measures closeness to the incompressible solution in the low Mach number regime.

Proposed method

• The Euler equations are split into stiff (pressure-related) and non-stiff (velocity-related) components to separate fast and slow waves.
• An IMEX Runge-Kutta time discretization is employed, treating stiff terms implicitly and non-stiff terms explicitly to handle stiffness and maintain stability.
• A Rusanov-type central flux is used for the non-stiff fluxes to ensure robustness and consistency in the hyperbolic part.
• Simple central differencing is applied to the stiff fluxes in space, ensuring second-order accuracy and consistency with the asymptotic limit.
• The time semi-discrete and fully-discrete schemes are proven to be asymptotic preserving, meaning they converge to the incompressible limit as Mach number approaches zero.
• The scheme is designed to preserve the well-prepared manifold of constant density and divergence-free velocity, ensuring asymptotic accuracy.

Experimental results

Research questions

• RQ1 Can a second-order accurate finite volume scheme maintain stability and accuracy in the zero Mach number limit?
• RQ2 How can stiffness from acoustic waves be effectively handled in low Mach number flows using IMEX time integration?
• RQ3 What conditions ensure that the compressible solution remains close to the incompressible solution in the asymptotic regime?
• RQ4 Can a notion of 'asymptotic accuracy' be formally defined and numerically verified for finite volume schemes?
• RQ5 Does the proposed scheme achieve uniform second-order convergence across all Mach numbers, including very low values?

Key findings

• The proposed IMEX finite volume schemes are asymptotic preserving, meaning they converge to the incompressible Euler equations as the Mach number tends to zero.
• The schemes achieve uniform second-order convergence in both space and time across all Mach numbers, including the low Mach number regime.
• The notion of asymptotic accuracy is successfully defined and validated, ensuring the solution remains close to the incompressible counterpart in the low Mach number limit.
• Numerical experiments confirm that the scheme preserves the well-prepared manifold of constant density and divergence-free velocity, which is essential for asymptotic accuracy.
• The combination of flux splitting, IMEX time integration, and central spatial discretization ensures robustness and accuracy in both compressible and incompressible flow regimes.
• The theoretical analysis and numerical results together demonstrate that the scheme maintains second-order accuracy even as the Mach number approaches zero.