Arbitrarily High-order Accurate Entropy Stable Essentially Nonoscillatory Schemes for Systems of Conservation Laws

Fjordholm, Ulrik Skre; Mishra, Siddhartha; Tadmor, Eitan

doi:10.1137/110836961

Cited by 265 publications

(302 citation statements)

References 32 publications

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“…The viscous terms in 2.8 can be rewritten as 9) and because an entropy dissipative regularization [12] requires that…”

Section: Nonlinear Conservation Laws and The Entropy Conditionmentioning

confidence: 99%

“…These schemes have been made computationally tractable for the Navier-Stokes equations through the work of Ismail and Roe [8]. A methodology for constructing entropy stable schemes satisfying a cell entropy inequality and capable of simulating flows with shocks in periodic domains has been developed by Fjordholm et al [9] Herein, an alternative approach is developed based on a finitedomain entropy stability proof, which yields entropy stable methods with formal boundary closures.…”

Section: Introductionmentioning

confidence: 99%

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High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

Fisher

Carpenter

2013

Journal of Computational Physics

288

265

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“…The viscous terms in 2.8 can be rewritten as 9) and because an entropy dissipative regularization [12] requires that…”

Section: Nonlinear Conservation Laws and The Entropy Conditionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

Fisher

Carpenter

2013

Journal of Computational Physics

288

265

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“…Some explicit examples of (2.16) are described in [6]. We remark that the arbitrarily high-order entropy conservative fluxes are linear combinations of the two point entropy conservative flux (2.7).…”

Section: High-order Entropy Conservative Fluxesmentioning

confidence: 95%

“…The conservative finite difference (finite volume) method updates point values (cell averages in I i ) of the solution U resulting in the semi-discrete scheme: 6) with numerical flux F i+1/2 = F(U i (t), U i+1 (t)) computed from the (approximate) solution of the Riemann problem for (2.1) at the interface x i+1/2 , [13]. Second order spatial accuracy can be obtained with nonoscillatory TVD [11] and even higher order of accuracy can be obtained with ENO [8] and WENO [15] piecewise polynomial reconstructions.…”

Section: Conservative Finite Difference Schemesmentioning

confidence: 99%

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Entropy Stable ENO Scheme

Fjordholm¹,

Mishra²,

Tadmor³

2012

Series in Contemporary Applied Mathematics

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We prove the first stability estimates for the ENO reconstruction procedure. They take the form of a sign property: we show that the jump in the reconstructed pointvalues at each cell interface has the same sign as the jump in underlying cell averages (cell centered values) across the interface. Moreover their ratio is upper bounded. These estimates hold for arbitrary orders of accuracy of the reconstruction as well as for non-uniform meshes. We then combine the ENO reconstruction together with entropy conservative fluxes to construct new entropy stable ENO schemes of arbitrary order.

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Numerics and subgrid‐scale modeling in large eddy simulations of stratocumulus clouds

Pressel

Mishra

Schneider

et al. 2017

J Adv Model Earth Syst

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Stratocumulus clouds are the most common type of boundary layer cloud; their radiative effects strongly modulate climate. Large eddy simulations (LES) of stratocumulus clouds often struggle to maintain fidelity to observations because of the sharp gradients occurring at the entrainment interfacial layer at the cloud top. The challenge posed to LES by stratocumulus clouds is evident in the wide range of solutions found in the LES intercomparison based on the DYCOMS‐II field campaign, where simulated liquid water paths for identical initial and boundary conditions varied by a factor of nearly 12. Here we revisit the DYCOMS‐II RF01 case and show that the wide range of previous LES results can be realized in a single LES code by varying only the numerical treatment of the equations of motion and the nature of subgrid‐scale (SGS) closures. The simulations that maintain the greatest fidelity to DYCOMS‐II observations are identified. The results show that using weighted essentially non‐oscillatory (WENO) numerics for all resolved advective terms and no explicit SGS closure consistently produces the highest‐fidelity simulations. This suggests that the numerical dissipation inherent in WENO schemes functions as a high‐quality, implicit SGS closure for this stratocumulus case. Conversely, using oscillatory centered difference numerical schemes for momentum advection, WENO numerics for scalars, and explicitly modeled SGS fluxes consistently produces the lowest‐fidelity simulations. We attribute this to the production of anomalously large SGS fluxes near the cloud tops through the interaction of numerical error in the momentum field with the scalar SGS model.

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Arbitrarily High-order Accurate Entropy Stable Essentially Nonoscillatory Schemes for Systems of Conservation Laws

Cited by 265 publications

References 32 publications

High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

Entropy Stable ENO Scheme

Numerics and subgrid‐scale modeling in large eddy simulations of stratocumulus clouds

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