Well-balanced schemes for the Euler equations with gravitation

Käppeli, Roger; Mishra, Siddhartha

doi:10.1016/j.jcp.2013.11.028

Cited by 121 publications

(113 citation statements)

References 19 publications

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“…Therefore, our reconstruction preserves discrete hydrostatic equilibria with arbitrary temperature or entropy stratification. This represents a substantial generalization of the reconstruction advocated by Käppeli & Mishra (2014).…”

Section: One-dimensional Cartesian Casementioning

confidence: 73%

“…However, a standard piecewise linear reconstruction of the pressure will not preserve the discrete hydrostatic equilibrium (18). To design an equilibrium preserving pressure reconstruction within the ith cell p i (x), we decompose it into an equilibrium p 0,i (x) and a perturbation p 1,i (x) part (Botta et al 2004;Käppeli & Mishra 2014):…”

Section: Second-order Schemementioning

confidence: 99%

“…Note that we use the scheme proposed in this paper. This has to be contrasted with the scheme of Käppeli & Mishra (2014), which could handle either the isentropic or the isothermal case at a time. Arbitrary entropy/temperature stratifications are considered in the astrophysical applications below.…”

Section: Hydrostatic Atmospheresmentioning

confidence: 99%

“…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%

“…All the afore mentioned well-balanced schemes need to assume a certain thermal equilibrium in order to uniquely determine the stationary state. For instance, Käppeli & Mishra (2014) have designed a well-balanced scheme under the further assumption of isentropic (isothermal) conditions. Then Eq.…”

Section: Introductionmentioning

confidence: 99%

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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: One-dimensional Cartesian Casementioning

confidence: 73%

Section: Second-order Schemementioning

confidence: 99%

Section: Hydrostatic Atmospheresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Käppeli

Mishra

2016

A&A

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

Desveaux

Zenk

Berthon

et al. 2015

Numerical Methods in Fluids

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Summary 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 special case of stationary solutions with zero velocity, described by a nonlinear partial differential equation, whose solutions are called hydrostatic equilibria. We present a well‐balanced method, meaning that besides discretizing the complete equations, the method is also able to maintain all hydrostatic equilibria. The method is a finite volume method, whose Riemann solver is approximated by a so‐called relaxation Riemann solution that takes all hydrostatic equilibria into account. Relaxation ensures robustness, accuracy, and stability of our method, because it satisfies discrete entropy inequalities. We will present numerical examples, illustrating that our method works as promised. Copyright © 2015 John Wiley & Sons, Ltd.

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Well‐balanced finite difference weighted essentially non‐oscillatory schemes for the blood flow model

Wang

Delestre

2016

Numerical Methods in Fluids

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SUMMARYThe blood flow model maintains the steady-state solutions, in which the flux gradients are non-zero but exactly balanced by the source term. In this paper, we design high order finite difference weighted essentially non-oscillatory (WENO) schemes to this model with such well-balanced property and at the same time keeping genuine high order accuracy. Rigorous theoretical analysis as well as extensive numerical results all indicate that the resulting schemes verify high order accuracy, maintain the well-balanced property, and keep good resolution for smooth and discontinuous solutions.

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Well-balanced schemes for the Euler equations with gravitation

Cited by 121 publications

References 19 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 well‐balanced scheme to capture non‐explicit steady states in the Euler equations with gravity

Well‐balanced finite difference weighted essentially non‐oscillatory schemes for the blood flow model

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