An Energy Stable, Well-Balanced, and Asymptotic-Preserving Finite Volume Scheme for the Barotropic Euler System with Gravity under the Anelastic Scaling

Conceitos Básicos

The authors design and analyze an energy stable, structure preserving, well-balanced and asymptotic preserving (AP) finite volume scheme for the barotropic Euler system with gravity in the anelastic limit.

Resumo

The key aspects of the proposed numerical scheme are:

Energy Stability: The introduction of appropriate velocity shifts in the convective fluxes of mass and momenta ensures the dissipation of mechanical energy.

Well-Balancing: The scheme exactly satisfies the discrete counterparts of the hydrostatic steady states of the Euler system. The stability of solutions with respect to the relative energy leads to well-balancing.

Asymptotic Preserving (AP) Property: The scheme's stability and consistency with the anelastic Euler system is rigorously established as the Mach and Froude numbers vanish. The scheme automatically transitions between the compressible, weakly compressible and the anelastic regimes.

Positivity Preserving: The scheme supports the positivity of density and yields consistency with the weak solutions of the Euler system upon mesh refinement.

The semi-implicit in time and finite volume in space fully-discrete scheme is resolved in two steps: by solving a non-linear elliptic problem for the density and a subsequent explicit computation of the velocity. The authors present results from several benchmark case studies to corroborate the proposed claims.

Estatísticas

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Principais Insights Extraídos De

An Energy Stable Well-balanced Scheme for the Barotropic Euler System with Gravity under the Anelastic Scaling

by K. R. Arun,M... às arxiv.org 05-02-2024

https://arxiv.org/pdf/2405.00559.pdf

Perguntas Mais Profundas

What are the potential extensions or generalizations of the proposed numerical scheme to other types of fluid flow models with gravity, such as the full compressible Navier-Stokes equations or magnetohydrodynamics

The proposed numerical scheme for the barotropic Euler system with gravity can be extended or generalized to other types of fluid flow models with gravity, such as the full compressible Navier-Stokes equations or magnetohydrodynamics.
For the full compressible Navier-Stokes equations, the key idea would be to incorporate the additional terms representing viscosity and thermal conduction into the discretization scheme. This would involve modifying the momentum balance equation to include the viscous stress tensor and the heat conduction term. The discretization of these terms would need to be carefully designed to ensure stability and accuracy of the scheme.
In the case of magnetohydrodynamics (MHD), which describes the behavior of electrically conducting fluids in the presence of magnetic fields, the numerical scheme would need to account for the additional magnetic field equations. This would involve introducing the magnetic induction equation and coupling it with the momentum and energy equations. The discretization of the magnetic field terms would require special consideration to maintain the well-balanced and energy stability properties of the scheme.
Overall, the extension of the proposed numerical scheme to these more complex fluid flow models would involve adapting the discretization to include the additional physical phenomena while ensuring the stability, accuracy, and conservation properties of the scheme.

How can the well-balancing and asymptotic preserving properties of the scheme be leveraged to develop efficient numerical methods for multiscale atmospheric flow simulations

The well-balancing and asymptotic preserving properties of the scheme can be leveraged to develop efficient numerical methods for multiscale atmospheric flow simulations.
In multiscale atmospheric flow simulations, different physical processes occur at varying spatial and temporal scales. The well-balanced property of the scheme ensures that the numerical solutions accurately capture the balance between gravitational and internal forces, particularly in scenarios where the flow is close to hydrostatic equilibrium. This is crucial for simulating atmospheric phenomena such as gravity waves and large-scale circulation patterns.
The asymptotic preserving property of the scheme allows it to accurately capture the behavior of the flow in different regimes, such as the anelastic limit where the Mach and Froude numbers approach zero. This is important for simulating atmospheric flows that transition between compressible and incompressible regimes.
By leveraging these properties, the numerical scheme can efficiently handle the complex interactions between different scales in atmospheric flow simulations, leading to more accurate and reliable results. Additionally, the scheme's energy stability ensures that the numerical solutions remain physically meaningful over long simulation times, making it well-suited for studying multiscale atmospheric phenomena.

Can the ideas behind the velocity stabilization and the specific interface discretization used in this work be adapted to develop high-order accurate schemes for the Euler system with gravity

The ideas behind the velocity stabilization and the specific interface discretization used in this work can be adapted to develop high-order accurate schemes for the Euler system with gravity.
To develop high-order accurate schemes, one approach would be to incorporate higher-order spatial discretization schemes, such as finite difference or spectral methods, into the numerical scheme. This would involve using higher-order interpolation techniques for the density and velocity variables on the grid points, leading to more accurate approximations of the solution.
Additionally, the velocity stabilization technique can be enhanced by introducing higher-order terms in the discretization of the convective fluxes. This can help improve the accuracy of the scheme, especially in capturing sharp gradients and discontinuities in the flow field.
Furthermore, the specific interface discretization method, such as the 'γ-mean' approach used in the scheme, can be extended to higher-order accuracy by incorporating more sophisticated interpolation schemes at the interfaces. This can help reduce numerical diffusion and improve the resolution of the scheme, particularly in regions where the flow undergoes rapid changes.
Overall, by combining higher-order spatial discretization techniques with the velocity stabilization and interface discretization methods used in this work, it is possible to develop high-order accurate schemes for the Euler system with gravity, leading to more precise and reliable numerical simulations of fluid flow phenomena.