A Second Order Well-Balanced Finite Volume Scheme for Euler Equations with Gravity

Varma, Deepak; Chandrashekar, Praveen

doi:10.1137/140984373

Cited by 90 publications

(72 citation statements)

References 15 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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“…This results in a jump in density by an amount of Δ ρ at the interface defined by r = r i , whereas the pressure is continuous. Following the work of Chandrashekar and Klingenberg, we take Δ ρ = 0.1, η = 0.02, and k = 20. For computation, we use a mesh of 240 × 240 cells.…”

Section: Numerical Resultsmentioning

confidence: 99%

A second‐order positivity‐preserving well‐balanced finite volume scheme for Euler equations with gravity for arbitrary hydrostatic equilibria

Thomann

Zenk

Klingenberg

2019

Numerical Methods in Fluids

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Summary We present a well‐balanced finite volume scheme for the compressible Euler equations with gravity, where the approximate Riemann solver is derived using a Suliciu relaxation approach. Besides the well‐balanced property, the scheme is robust with respect to the physical admissible states. General hydrostatic solutions are captured up to machine precision by deriving, for a given initial value problem, suitable time‐independent functions and using them in the discretization of the source term. The first‐order scheme is extended to a second‐order scheme by reconstructing in equilibrium variables while preserving the well‐balanced and robustness properties. Numerical examples are performed to demonstrate the accuracy, well‐balanced, and robustness properties of the presented scheme for up to three space dimensions.

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“…This kind of simple model is of interest not only in hydrodynamics [126,127,128,129,130], but also in the astrophysical community [1,60,131].…”

Section: Euler Equations With Source Termmentioning

confidence: 99%

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

Gaburro

Boscheri

Chiocchetti

et al. 2020

Journal of Computational Physics

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We present a new family of very high order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on moving Voronoi meshes that are regenerated at each time step and which explicitly allow topology changes in time. The Voronoi tessellations are obtained from a set of generator points that move with the local fluid velocity. We employ an AREPOtype approach [1], which rapidly rebuilds a new high quality mesh exploiting the previous one, but rearranging the element shapes and neighbors in order to guarantee that the mesh evolution is robust even for vortex flows and for very long computational times. The old and new Voronoi elements associated to the same generator point are connected in space-time to construct closed space-time control volumes, whose bottom and top faces may be polygons with a different number of sides. We also need to incorporate some degenerate space-time sliver elements, which are needed in order to fill the space-time holes that arise because of the topology changes in the mesh between time t n and time t n+1 . The final ALE FV-DG scheme is obtained by a novel redesign of the high order accurate fully discrete direct ALE schemes of Boscheri and Dumbser [2,3], which have been extended here to general moving Voronoi meshes and space-time sliver elements. Our new numerical scheme is based on the integration over arbitrary shaped closed spacetime control volumes combined with a fully-discrete space-time conservation formulation of the governing hyperbolic PDE system. In this way the discrete solution is conservative and satisfies the geometric conservation law (GCL) by construction. Numerical convergence studies as well as a large set of benchmark problems for hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and robustness of the proposed method. Our numerical results clearly show that the new combination of very high order schemes with regenerated meshes that allow topology changes in each time step lead to substantial improvements over the existing state of the art in direct ALE methods.Keywords: Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes, arbitrary high order in space and time, moving Voronoi tessellations with topology change, a posteriori sub-cell finite volume limiter, fully-discrete one-step ADER approach for hyperbolic PDE, compressible Euler and MHD equations directly integrated by means of a high order fully discrete one-step ADER method. To the best knowledge of the authors, this is the first time that arbitrary high order accurate direct ALE FV and DG schemes are developed with an embedded mesh generator that builds a new mesh with a different topology at each time step. State of the artLagrangian algorithms [4,5,6,7,8,9,10,11,12] are characterized by a moving computational mesh displaced with a velocity chosen as close as possible to the local fluid velocity. In the Lagrangian description of the fluid, the ...

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A Second Order Well-Balanced Finite Volume Scheme for Euler Equations with Gravity

Cited by 90 publications

References 15 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 positivity‐preserving well‐balanced finite volume scheme for Euler equations with gravity for arbitrary hydrostatic equilibria

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

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