Well-Balanced Formulation of Gravitational Source Terms for Conservative Finite-Difference Atmospheric Flow Solvers

Ghosh, Debojyoti; Constantinescu, Emil M.

doi:10.2514/6.2015-2889

Cited by 8 publications

(12 citation statements)

References 50 publications

(85 reference statements)

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“…In case of WB-Exact, we see that the errors are not as large as in the case of NWB-Exact scheme. We run this case for a long time of 750 units and plot the error norm as a function of time as shown in figure (4). We can observe that while the velocity error tends towards machine epsilon (10 −16 ), the errors for density and pressure reaches steady state with a value ≈ 10 −7 , but importantly, they do not grow with time.…”

Section: Polytropic Hydrostatic Solutionmentioning

confidence: 98%

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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Section: Polytropic Hydrostatic Solutionmentioning

confidence: 98%

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

Varma

Chandrashekar

2019

Computers & Fluids

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“…Therefore, we seek for a parameterization of the hydrostatic equilibrium relation . Since

\overset{ρ}{true}

and

\overset{p}{true}

are time independent, we choose two time‐independent functions α and β such that they coincide with a given hydrostatic solution as

\overset{ρ}{true} false(boldx false) = α false(boldx false) and \overset{p}{true} false(boldx false) = β false(boldx false) .

…”

Section: The Hydrostatic Steady Statesmentioning

confidence: 99%

“…Analogously to (1), we define the state vector W = ( , u, E, , Z), which belongs to Ω W = { W ∈ R 4+d , > 0, e > 0 } . For a given function , a relaxation equilibrium state for model (11) is defined by…”

Section: The Suliciu Relaxation Solvermentioning

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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“…After that they proposed a more general pressure reconstruction using a local analytical integration of hydrostatic equation, and demonstrated the efficiency of their well-balanced schemes for a broad set of astrophysical scenarios with several types of equation of state 10 . Ghosh and Constantinescu 11,12 extended Xing and Shu's work to more general flows encountered in atmospheric simulations. Besides an isothermal equilibrium, the proposed well-balanced scheme can hold for many other hydrostatic equilibrium states.…”

Section: Introductionmentioning

confidence: 99%

A well-balanced gas kinetic scheme for Navier-Stokes equations with gravitational potential

Chen,

Guo,

2019

Preprint

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The hydrostatic equilibrium state is the consequence of the exact hydrostatic balance between hydrostatic pressure and external force. Standard finite volume or finite difference schemes cannot keep this balance exactly due to their unbalanced truncation errors. In this study, we introduce an auxiliary variable which becomes constant at isothermal hydrostatic equilibrium state and propose a well-balanced gas kinetic scheme for the Navier-Stokes equations with a global reconstruction. Through reformulating the convection term and the force term via the auxiliary variable, zero numerical flux and zero numerical source term are enforced at the hydrostatic equilibrium state instead of the balance between hydrostatic pressure and external force.Several problems are tested numerically to demonstrate the accuracy and the stability of the new scheme, and the results confirm that, the new scheme can preserve the exact hydrostatic solution. The small perturbation riding on hydrostatic equilibria can be calculated accurately. The viscous effect is also illustrated through the propagation of small perturbation and the Rayleigh-Taylor instability. More importantly, the new scheme is capable of simulating the process of converging towards hydrostatic equilibrium state from a highly non-balanced initial condition. The ultimate state of zero velocity and constant temperature is achieved up to machine accuracy. As demonstrated by the numerical experiments, the current scheme is very suitable for small amplitude perturbation and long time running under gravitational potential.

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Well-Balanced Formulation of Gravitational Source Terms for Conservative Finite-Difference Atmospheric Flow Solvers

Cited by 8 publications

References 50 publications

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

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

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

A well-balanced gas kinetic scheme for Navier-Stokes equations with gravitational potential

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