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

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

doi:10.1002/fld.4177

Cited by 47 publications

(58 citation statements)

References 42 publications

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“…This section is devoted to the derivation of the numerical flux function for the finite volume scheme. We decide to adapt a modified Suliciu relaxation technique on the modified system , where we use the approach for a Suliciu relaxation technique for the original system . For simplicity, we consider only one spatial dimension, and we suggest the following relaxation system:

\begin{matrix} right \partial_{t} ρ + \partial_{x_{1}} ρ u_{1} & left = 0, & right \\ right \partial_{t} ρ u_{1} + \partial_{x_{1}} (ρ u_{1}^{2} + π) & left = K \partial_{x_{1}} Z, \\ right \partial_{t} ρ u_{2} + \partial_{x_{1}} (ρ u_{1} u_{2}) & left = 0, \\ right \partial_{t} ρ u_{3} + \partial_{x_{1}} (ρ u_{1} u_{3}) & left = 0, \\ right \partial_{t} E + \partial_{x_{1}} (E + π) u_{1} & left = K u_{1} \partial_{x_{1}} Z, \\ right \partial_{t} ρ π + \partial_{x_{1}} (ρ π + a^{2}) u_{1} & left = \frac{ρ}{ϵ} (p (τ, e) - π), \\ right \partial_{t} ρ Z + \partial_{x_{1}} ρ Z u_{1} & left = \frac{ρ}{ϵ} (β - Z) . \end{matrix}

…”

Section: The Suliciu Relaxation Solvermentioning

confidence: 99%

“…Then, the approximate Riemann solver is consistent with the entropy inequality. This follows directly from Theorem 8 in the work of Desveaux et al, since the given proof there is independent of the source term and, thus, can be directly applied here.…”

Section: The Suliciu Relaxation Solvermentioning

confidence: 99%

“…In addition, we assume that the pressure obeys the second law of thermodynamics. This means that there exists a specific entropy

η false(τ, e false) : R^{+} \times R^{+} \to R^{+}

, with which one can define the Lax entropy pair for system . The specific entropy η ( τ , e ) satisfies for a temperature T ( τ , e ) > 0 the relations

- T d η = d e + p d τ,

η {false(τ, e false)}_{τ} = - \frac{p false(τ, e false)}{T false(τ, e false)} < 0, η {false(τ, e false)}_{e} = - \frac{1}{T false(τ, e false)} < 0 .

We assume, in addition, that the specific entropy is strictly convex.…”

Section: Introductionmentioning

confidence: 99%

“…Also challenging is how the source term is treated. It can be included into the flux function or left outside …”

Section: Introductionmentioning

confidence: 99%

“…Common to the previous mentioned works is that they are algebraically consistent with a certain class of equilibria, for example, hydrostatic equilibria with constant entropy or isothermal and polytropic equilibria . An interesting approach is found in the work of Käppeli and Mishra, where a discrete approximation of the hydrostatic equilibrium is used to achieve second‐order accuracy to general equilibria.…”

Section: Introductionmentioning

confidence: 99%

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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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\begin{matrix} right \partial_{t} ρ + \partial_{x_{1}} ρ u_{1} & left = 0, & right \\ right \partial_{t} ρ u_{1} + \partial_{x_{1}} (ρ u_{1}^{2} + π) & left = K \partial_{x_{1}} Z, \\ right \partial_{t} ρ u_{2} + \partial_{x_{1}} (ρ u_{1} u_{2}) & left = 0, \\ right \partial_{t} ρ u_{3} + \partial_{x_{1}} (ρ u_{1} u_{3}) & left = 0, \\ right \partial_{t} E + \partial_{x_{1}} (E + π) u_{1} & left = K u_{1} \partial_{x_{1}} Z, \\ right \partial_{t} ρ π + \partial_{x_{1}} (ρ π + a^{2}) u_{1} & left = \frac{ρ}{ϵ} (p (τ, e) - π), \\ right \partial_{t} ρ Z + \partial_{x_{1}} ρ Z u_{1} & left = \frac{ρ}{ϵ} (β - Z) . \end{matrix}

…”

Section: The Suliciu Relaxation Solvermentioning

confidence: 99%

Section: The Suliciu Relaxation Solvermentioning

confidence: 99%

“…In addition, we assume that the pressure obeys the second law of thermodynamics. This means that there exists a specific entropy

η false(τ, e false) : R^{+} \times R^{+} \to R^{+}

, with which one can define the Lax entropy pair for system . The specific entropy η ( τ , e ) satisfies for a temperature T ( τ , e ) > 0 the relations