High-order well-balanced finite volume schemes for the Euler equations with gravitation
Introduction
A multitude of interesting physical phenomena are modeled by the Euler equations with gravitational source terms. Applications range from the study of atmospheric phenomena, such as numerical weather prediction and climate modeling, to the numerical simulation of the climate of exoplanets, convection in stars and core-collapse supernova explosions. The Euler equations with gravitational source terms express the conservation of mass, momentum and energy: Here ρ is the mass density, v the velocity and the total fluid energy density being the sum of internal and kinetic energy densities. The pressure p is related to the density and specific internal energy through an equation of state .
The source terms on the right-hand side of the momentum and energy equations model the effect of the gravitational forces on the fluid. They are dictated by the variation of the gravitational potential ϕ, which can either be a given function or, in the case of self-gravity, be determined by the Poisson equation where G is the gravitational constant.
In many physically relevant applications, such as the ones named above, (parts of) the flow of interest may be realized close to hydrostatic equilibrium As a matter of fact, the numerical simulation of near equilibrium flows is challenging for standard finite volume methods. The reason for this is that these methods may in general not satisfy a discrete equivalent of the equilibrium. Thus such states are not preserved exactly but are solely approximated with an error proportional to the truncation error of the scheme. So if the interest relies in the simulation of small perturbations on top of a hydrostatic equilibrium, the numerical resolution has to be increased to the point that the truncation errors do not obscure these small perturbations. This may result in prohibitively high computational costs, especially in several space dimensions.
A design principle to overcome the challenge was introduced by Greenberg and Leroux [1] leading to the concept of so-called well-balanced schemes. In these schemes, a discrete equivalent of the equilibrium is exactly satisfied. Therefore, they possess the ability to maintain discrete equilibrium states down to machine precision and are capable of resolving small equilibrium perturbations effectively. Many well-balanced schemes have been designed, especially for the shallow water equations with non-trivial bottom topography, see e.g. [2], [3], [4] and references therein. An extensive review on well-balanced schemes for many different applications is also given in the book by Gosse [5].
Well-balanced schemes for the Euler equations with gravitation have received a considerable amount of attention in the recent literature. First, LeVeque and Bale [6] have applied the quasi-steady wave-propagation algorithm [2] to the Euler equations with gravity. Few years later, Botta et al. [7] designed a well-balanced finite volume scheme for numerical weather prediction applications. More recently, several well-balanced finite volume [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], finite difference [18], [19] and discontinuous Galerkin [20], [21], [22] schemes have been presented. Magnetohydrostatic steady state preserving well-balanced finite volume schemes were devised in [23]. To the best of our knowledge, many of the mentioned schemes are at most second-order accurate and only [18], [20], [21], [19], [22], [24] go to higher orders.
In fact, equation (1.6) only specifies a mechanical equilibrium. In order to fully characterize the equilibrium a thermal variable, such as the specific entropy s or the temperature T, needs to be supplemented. As a concrete astrophysically relevant example of a stationary state we consider the case of constant entropy. The relevant thermodynamic relation for isentropic hydrostatic equilibrium is where h is the specific enthalpy T the temperature and s the specific entropy. Then we can write (1.6) for the isentropic case () as The last equation can then be trivially integrated to obtain In [9] this equilibrium was used to build a second-order accurate well-balanced finite volume scheme. Along the same lines, well-balanced schemes for isothermal hydrostatic equilibrium can be constructed [15]. In the latter case, the relevant thermodynamic potential is the Gibbs free energy. More generally, the presented approach can be applied to barotropic fluids characterized by the fact that the density is a funcion of pressure only. In that case, the expression is integrable and the hydrostatic equilibrium takes the form We note that the isentropic and the isothermal as well as the polytropic equilibrium [11], [22] fall into this category. However, in the following we focus on the isentropic case. Many interesting physical phenomena involve steady convection that takes place near the isentropic state. For example athmospheric convection on Earth and exoplanets [25], convection in stars and neutron stars [26], [27].
In this paper, we extend the well-balanced finite volume schemes [9] beyond second-order accuracy. The scheme possesses the following novel features:
- •
An arbitrarily high-order accurate local hydrostatic profile is constructed based on the equilibrium (1.10).
- •
An arbitrarily high-order equilibrium preserving reconstruction is designed on the basis of any standard high-order reconstruction procedure.
- •
A well-balanced source term discretization is built from the equilibrium preserving reconstruction.
- •
It is well-balanced for any consistent numerical flux, which allows a straightforward implementation within any standard finite volume method.
- •
It is well-balanced for multi-dimensional hydrostatic equilibria.
- •
It is not tied to any particular equation of state such as the ideal gas law. This is important, especially for astrophysical applications.
The rest of the paper is structured as follows: the well-balanced finite volume scheme is presented in section 2. Extensive numerical results are presented in section 3 and conclusions are provided in section 4.
Section snippets
One-dimensional scheme
We first consider the Euler equations with gravitation (1.1), (1.2), (1.3) in one space dimension and write them in the following compact form with where u, f and s are the vectors of conserved variables, fluxes and source terms. An equation of state (EoS) relates the pressure to the density ρ and specific internal energy e (or any other thermodynamic quantity such as specific entropy s or temperature T). For example, a simple
Numerical experiments
In this section we assess the performance of our well-balanced scheme on a series of numerical experiments. For comparison, we also present results obtained with a standard (unbalanced) base scheme. The fully-discrete finite volume base scheme consists of
- •
the temporally third-order accurate SSP-RK scheme for time integration (see [36]),
- •
the spatially third-order accurate CWENO3 [41] reconstruction procedure ,
- •
the spatially fourth-order accurate two-point Gauss–Legendre quadrature rule for .
Conclusion
We presented a novel well-balanced, high-order finite volume scheme for Euler equations with gravity. We are able to well-balance a large class of astrophysically relevant hydrostatic equilibria without imposing the exact equilibrium apriori. Rather, we only assume some thermodynamic information about the equilibrium, e.g. constant entropy, and solve for the equilibrium in every time step. Since the equilibrium defined by (1.6) is only a mechanical equilibrium, it seems natural that some
Acknowledgements
The work was supported by the Swiss National Science Foundation (SNSF) under grant 169631. The authors also acknowledge the use of computational resources provided by the Swiss SuperComputing Center (CSCS), under the allocation grant s661, s665, s667 and s744. We acknowledge the computational resources provided by the EULER cluster of ETHZ.
We happily acknowledge the use of the following C++ software libraries: OpenMPI [46], HDF5 [47] and Boost. As well as the Python packages NumPy and SciPy [48]
References (50)
Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm
J. Comput. Phys.
(1998)- et al.
Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows
J. Comput. Phys.
(2006) - et al.
Well balanced finite volume methods for nearly hydrostatic flows
J. Comput. Phys.
(2004) - et al.
Well-balanced schemes for the Euler equations with gravitation
J. Comput. Phys.
(2014) - et al.
High order finite volume WENO schemes for the Euler equations under gravitational fields
J. Comput. Phys.
(2016) - et al.
Well-balanced schemes for the Euler equations with gravitation: conservative formulation using global fluxes
J. Comput. Phys.
(2018) - et al.
Well-balanced finite difference weighted essentially non-oscillatory schemes for the Euler equations with static gravitational fields
Comput. Math. Appl.
(2018) - et al.
Well-balanced discontinuous Galerkin methods with hydrostatic reconstruction for the Euler equations with gravitation
J. Comput. Phys.
(2018) - et al.
High order well-balanced finite volume schemes for simulating wave propagation in stratified magnetic atmospheres
J. Comput. Phys.
(2010) Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method
J. Comput. Phys.
(1979)
High resolution schemes for hyperbolic conservation laws
J. Comput. Phys.
The piecewise parabolic method (ppm) for gas-dynamical simulations
J. Comput. Phys.
Uniformly high order accurate essentially non-oscillatory schemes, III
J. Comput. Phys.
A well-balanced scheme for the numerical processing of source terms in hyperbolic equations
SIAM J. Numer. Anal.
A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows
SIAM J. Sci. Comput.
Computing Qualitatively Correct Approximations of Balance Laws
Wave propagation methods for conservation laws with source terms
A well-balanced path-integral f-wave method for hyperbolic problems with source terms
J. Sci. Comput.
A well-balanced scheme for the Euler equation with a gravitational potential
Springer Proc. Math. Stat.
A second order well-balanced finite volume scheme for Euler equations with gravity
SIAM J. Sci. Comput.
A well-balanced finite volume scheme for the Euler equations with gravitation. the exact preservation of hydrostatic equilibrium with arbitrary entropy stratification
Astron. Astrophys.
Well-balanced unstaggered central schemes for the Euler equations with gravitation
SIAM J. Sci. Comput.
A well-balanced scheme for the Euler equations with gravitation
Well balanced arbitrary-Lagrangian–Eulerian finite volume schemes on moving nonconforming meshes for the Euler equations of gasdynamics with gravity
Mon. Not. R. Astron. Soc.
High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields
J. Sci. Comput.
Cited by (41)
Well-balanced positivity-preserving high-order discontinuous Galerkin methods for Euler equations with gravitation
2024, Journal of Computational PhysicsA well-balanced discontinuous Galerkin method for the first–order Z4 formulation of the Einstein–Euler system
2024, Journal of Computational PhysicsOn high order positivity-preserving well-balanced finite volume methods for the Euler equations with gravitation
2023, Journal of Computational PhysicsHigh order well-balanced positivity-preserving scale-invariant AWENO scheme for Euler systems with gravitational field
2023, Journal of Computational PhysicsWell-balanced adaptive compact approximate Taylor methods for systems of balance laws
2023, Journal of Computational PhysicsWell balanced finite volume schemes for shallow water equations on manifolds
2023, Applied Mathematics and Computation