A Well-Balanced Scheme for the Euler Equation with a Gravitational Potential

Desveaux, Vivien; Zenk, Markus; Berthon, Christophe; Klingenberg, Christian

doi:10.1007/978-3-319-05684-5_20

Cited by 15 publications

(15 citation statements)

References 9 publications

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“…Well-balanced schemes for certain classes of hydrostatic equilibrium have been presented by LeVeque (1998LeVeque ( , 2011, Botta et al (2004), Fuchs et al (2010), Xing & Shu (2013), Desveaux et al (2014), Käppeli & Mishra (2014), Chandrashekar & Klingenberg (2015), Li & Xing (2015). However, Eq.…”

Section: Introductionmentioning

confidence: 99%

A well-balanced finite volume scheme for the Euler equations with gravitation

Käppeli

Mishra

2016

A&A

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Context. Many problems in astrophysics feature flows which are close to hydrostatic equilibrium. However, standard numerical schemes for compressible hydrodynamics may be deficient in approximating this stationary state, where the pressure gradient is nearly balanced by gravitational forces. Aims. We aim to develop a second-order well-balanced scheme for the Euler equations. The scheme is designed to mimic a discrete version of the hydrostatic balance. It therefore can resolve a discrete hydrostatic equilibrium exactly (up to machine precision) and propagate perturbations, on top of this equilibrium, very accurately. Methods. A local second-order hydrostatic equilibrium preserving pressure reconstruction is developed. Combined with a standard central gravitational source term discretization and numerical fluxes that resolve stationary contact discontinuities exactly, the wellbalanced property is achieved. Results. The resulting well-balanced scheme is robust and simple enough to be very easily implemented within any existing computer code that solves time explicitly or implicitly the compressible hydrodynamics equations. We demonstrate the performance of the well-balanced scheme for several astrophysically relevant applications: wave propagation in stellar atmospheres, a toy model for core-collapse supernovae, convection in carbon shell burning, and a realistic proto-neutron star.

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Section: Introductionmentioning

confidence: 99%

A well-balanced finite volume scheme for the Euler equations with gravitation

Käppeli

Mishra

2016

A&A

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“…Cells located on a solid wall have only half the size of interior cells. For example, if the bottom boundary is a solid wall, then the finite volume scheme is given by (13) and the flux on the wall is given bŷ…”

Section: Solid Wall Boundary Conditionmentioning

confidence: 99%

A second-order, discretely well-balanced finite volume scheme for euler equations with gravity

Varma

Chandrashekar

2019

Computers & Fluids

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We present a well-balanced, second order, Godunov-type finite volume scheme for compressible Euler equations with gravity. By construction, the scheme admits a discrete stationary solution which is a second order accurate approximation to the exact stationary solution. Such a scheme is useful for problems involving complex equations of state and/or hydrostatic solutions which are not known in closed form expression. Noá priori knowledge of the hydrostatic solution is required to achieve the well-balanced property. The performance of the scheme is demonstrated on several test cases in terms of preservation of hydrostatic solution and computation of small perturbations around a hydrostatic solution.

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“…An improper treatment of the gravitational force and thus the steady states may lead to a solution, which either oscillates around the equilibrium or deviates from the equilibrium in time. This problem is well known in the numerical community and received a great attention in the literature; see, e.g., [1,7,9,17,36,37,39,55,56,60,61]. For the compressible Euler equations that describe the gas phase, we use a well-balanced version of the second-order semi-discrete CU scheme introduced in [9].…”

Section: Three-dimensional Dusty Gas Flow Model With Axial Symmetrymentioning

confidence: 99%

“…For the gas phase, the energy is updated by 17) and for the dust phase, the energy weight of the i th particle is evolved by…”

Section: : Interaction Time Step For Heat Transfermentioning

confidence: 99%

Hybrid Finite-Volume-Particle Method for Dusty Gas Flows

Chertock¹,

Cui²,

Kurganov³

2017

The SMAI Journal of Computational Mathematics

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Abstract. We first study the one-dimensional dusty gas flow modeled by the two-phase system composed of a gaseous carrier (gas phase) and a particulate suspended phase (dust phase). The gas phase is modeled by the compressible Euler equations of gas dynamics and the dust phase is modeled by the pressureless gas dynamics equations. These two sets of conservation laws are coupled through source terms that model momentum and heat transfers between the phases. When an Eulerian method is adopted for this model, one can notice the obtained numerical results are typically significantly affected by numerical diffusion. This phenomenon occurs since the pressureless gas equations are nonstrictly hyperbolic and have degenerate structure in which singular delta shocks are formed, and these strong singularities are vulnerable to the numerical diffusion.We introduce a low dissipative hybrid finite-volume-particle method in which the compressible Euler equations for the gas phase are solved by a central-upwind scheme, while the pressureless gas dynamics equations for the dust phase are solved by a sticky particle method. The obtained numerical results demonstrate that our hybrid method provides a sharp resolution even when a relatively small number of particle is used.We then extend the hybrid finite-volume-particle method to the three-dimensional dusty gas flows with axial symmetry. In the studied model, gravitational effects are taken into account. This brings an additional level of complexity to the development of the finite-volume-particle method since a delicate balance between the flux and gravitational source terms should be respected at the discrete level. We test the proposed method on a number of numerical examples including the one that models volcanic eruptions.Math. classification. 65M08, 76M12, 76M28, 86-08, 76M25, 35L65.

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A Well-Balanced Scheme for the Euler Equation with a Gravitational Potential

Cited by 15 publications

References 9 publications

A well-balanced finite volume scheme for the Euler equations with gravitation

A well-balanced finite volume scheme for the Euler equations with gravitation

A second-order, discretely well-balanced finite volume scheme for euler equations with gravity

Hybrid Finite-Volume-Particle Method for Dusty Gas Flows

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