Wave Propagation Methods for Conservation Laws with Source Terms

LeVeque, Randall J.; Bale, Derek S.

doi:10.1007/978-3-0348-8724-3_12

Cited by 45 publications

(67 citation statements)

References 1 publication

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“…Let us stress out that as δ tends to zero, π = p and φ = ψ, and the evolution equations for (ρ, ρu, ρE) T are thus equivalently to those found in system (8). Moreover, defining W δ = (ρ, ρu, ρE, ρπ, ψ)…”

Section: Derivation Of the One-dimensional Relaxation Modelmentioning

confidence: 94%

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A relaxation scheme for inviscid flows under gravitational influence

Vides¹,

Audit²,

Nkonga³

2014

ESAIM: ProcS

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Abstract. Many astrophysical flows are modeled by the Euler equations with gravity source terms derived from a potential, the evolution of which is described by a Poisson equation. Several gravitational flows reach equilibrium states that are necessary to preserve in the numerical formulation. In this paper, we present the derivation of the relaxation model [17], in which the pressure is a supplementary variable and the Poisson equation is transformed into a hyperbolic equation with a penalty parameter. The corresponding scheme is obtained in the limit as the parameter tends to zero. The proposed Riemann solver, implemented in the software HERACLES [10], provides better robustness compared to other approaches available in the same software and is capable of preserving gravitational equilibria when required. Several numerical tests and results are presented, as well. Résumé. De nombreux écoulements en astrophysique sont modélisés par les équations d'Euler avecdes termes sources de gravité dérivant d'un potentiel dont l'évolution est décrite par une équation de Poisson. Certains écoulements gravitationnels développent des états d'équilibre qu'il est nécessaire de préserver dans la formulation numérique. Nous présentons ici la dérivation du modèle de relaxation [17], dans lequel la pression est une variable complémentaire et l'équation de Poisson est transformée en une équation hyperbolique avec un paramètre de pénalisation. Le schéma est obtenu à la limite quand ce paramètre tend vers zéro. Le solveur de Riemann proposé, mis en oeuvre dans la plate-forme de calcul HERACLES [10], offre plus de robustesse numérique par rapport aux précédentes approches disponibles dans la plate-forme et permet de préserver les équilibres gravitationnels lorsque le problème l'exige. Des tests et des résultats numériques sont ainsi présentés.

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Section: Derivation Of the One-dimensional Relaxation Modelmentioning

confidence: 94%

“…LeVeque and Bale [8] first proposed this test to determine a method's ability to capture perturbed nearequilibrium solutions. The problem consists of an ideal gas in isothermal equilibrium.…”

Section: Perturbed One-dimensional Isothermal Equilibriummentioning

confidence: 99%

A relaxation scheme for inviscid flows under gravitational influence

Vides¹,

Audit²,

Nkonga³

2014

ESAIM: ProcS

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“…Since the f 0 is totally determined, taking the limit t → 0 in equation (8) and substituting its solution into conservation constraint (g − f ) ψdΞ = 0 at the cell interface, we have…”

Section: Gas Evolutionmentioning

confidence: 99%

“…This test case is similar to LeVeque and Bale's Rayleigh-Taylor instability case [8], which was used to validate their quasi-steady wave propagation scheme. Consider an isothermal equilibrium ideal gas Figure 9 shows a scatter plot of the density as a function of the radius.…”

Section: Rayleigh-taylor Instabilitymentioning

confidence: 99%

“…The solution will either oscillate around the equilibrium solution, or simply deviate from equilibrium due to artificial heating, which triggers numerical gravitation-thermal instability, i.e., the collapse of the gas core. There have been many attempts to construct such a well-balanced gas dynamic code that preserves the hydrostatic solution accurately [8,20,2].…”

Section: Introductionmentioning

confidence: 99%

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A three-dimensional multidimensional gas-kinetic scheme for the Navier–Stokes equations under gravitational fields

Tian

Chan

et al. 2007

Journal of Computational Physics

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This paper extends the gas-kinetic scheme for one-dimensional inviscid shallow water equations (J. Comput. Phys. 178 (2002), pp. 533-562) to multidimensional gas dynamic equations under gravitational fields. Four important issues in the construction of a well-balanced scheme for gas dynamic equations are addressed. First, the inclusion of the gravitational source term into the flux function is necessary. Second, to achieve second-order accuracy of a well-balanced scheme, the Chapman-Enskog expansion of the Boltzmann equation with the inclusion of the external force term is used. Third, to avoid artificial heating in an isolated system under a gravitational field, the source term treatment inside each cell has to be evaluated consistently with the flux evaluation at the cell interface. Fourth, the multidimensional approach with the inclusion of tangential gradients in two-dimensional and three-dimensional cases becomes important in order to maintain the accuracy of the scheme. Many numerical examples are used to validate the above issues, which include the comparison between the solutions from the current scheme and the Strang splitting method. The methodology developed in this paper can also be applied to other systems, such as semi-conductor device simulations under electric fields.

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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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Wave Propagation Methods for Conservation Laws with Source Terms

Cited by 45 publications

References 1 publication

A relaxation scheme for inviscid flows under gravitational influence

A relaxation scheme for inviscid flows under gravitational influence

A three-dimensional multidimensional gas-kinetic scheme for the Navier–Stokes equations under gravitational fields

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

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