A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes

Abgrall, Rémi

doi:10.1016/j.jcp.2018.06.031

Cited by 60 publications

(42 citation statements)

References 35 publications

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“…We denote with the Galerkin method using Bernstein polynomials with polynomial order 1, 2 or 3 similar denoting by applying a Lagrange basis. The basic implementation is done in the RD framework, see [ 38 ]. The two approaches only differ slightly.…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…Here, again nothing has changed from the calculations before except that the quadrature rules are changed which leads to an error in the interior of the spatial matrix , which cannot be stabilized with the SAT boundary treatment. We will focus on this test again in the second part of the paper series [ 42 ] to demonstrate the entropy correction term as presented in [ 38 ] and applied in [ 43 , 44 ] can also be seen as a stabilization factor for linear problems.…”

Section: Numerical Simulationsmentioning

confidence: 99%

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Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

et al. 2020

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In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.

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Section: Numerical Simulationsmentioning

confidence: 99%

Section: Numerical Simulationsmentioning

confidence: 99%

Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

et al. 2020

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“…While most of the aforementioned schemes are thermodynamically compatible only at the semi-discrete level, a fully discrete entropy-stable scheme has been recently presented in [95]. We also would like to point out that a very general framework for the construction of numerical schemes satisfying additional extra conservation laws has been recently forwarded by Abgrall in [2].…”

Section: Introductionmentioning

confidence: 99%

On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

et al. 2021

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In this paper we propose a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows recently proposed in Gavrilyuk et al. (J Comput Phys 366:252–280, 2018). The novelty of the formulation forwarded here is the use of a new evolution variable that guarantees the trace of the discrete Reynolds stress tensor to be always non-negative. The mathematical model is particularly challenging because one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. Therefore, in this paper we first consider a thermodynamically compatible viscous extension of the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamically compatible numerical scheme. We then introduce two different, but related, families of numerical methods for its solution. The first scheme is a provably thermodynamically compatible semi-discrete finite volume scheme that makes direct use of the Godunov form of the equations and can therefore be called a discrete Godunov formalism. The new method mimics the underlying continuous viscous system exactly at the semi-discrete level and is thus consistent with the conservation of total energy, with the entropy inequality and with the vanishing viscosity limit of the model. The second scheme is a general purpose high order path-conservative ADER discontinuous Galerkin finite element method with a posteriori subcell finite volume limiter that can be applied to the inviscid as well as to the viscous form of the model. Both schemes have in common that they make use of path integrals to define the jump terms at the element interfaces. The different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in Gavrilyuk et al. (2018) on the example of three Riemann problems. Moreover, we make the comparison with a fully resolved solution of the underlying viscous system with small viscosity parameter (vanishing viscosity limit). In all cases an excellent agreement between the different schemes is achieved. We furthermore show numerical convergence rates of ADER-DG schemes up to sixth order in space and time and also present two challenging test problems for the model where we also compare with available experimental data.

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“…Over the last decade, there have been rapid developments in spatially entropy stable high order methods, which are crucial for the construction of provably high order nonlinearly stable robust solvers for the compressible Euler and Navier-Stokes equations; see, for instance, [10][11][12][13][14][15][16][17][18][19][20][21][22][23]. High order entropy stable schemes are often based on the well known matrix-vector nodal formulation collocated at quadrature points; see, for instance, [24] for a clear introduction of nodal discontinuous Galerkin methods.…”

Section: Introductionmentioning

confidence: 99%

Simulation of Turbulent Flows Using a Fully Discrete Explicit hp-nonconforming Entropy Stable Solver of Any Order on Unstructured Grids

Parsani

Boukharfane²,

Nolasco

et al. 2021

AIAA Scitech 2021 Forum

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We report the numerical solution of two challenging turbulent flow test cases simulated with the SSDC framework, a compressible, fully discrete hp-nonconforming entropy stable solver based on the summation-by-parts discontinuous collocation Galerkin discretizations and the relaxation Runge-Kutta methods. The algorithms at the core of the solver are systematically designed with mimetic and structure-preserving techniques that transfer fundamental properties from the continuous level to the discrete one. We aim at providing numerical evidence of the robustness and maturity of these entropy stable scale-resolving methods for the new generation of adaptive unstructured computational fluid dynamics tools. The two selected turbulent flows are i) the flow past two spheres in tandem at a Reynolds number based on the sphere diameter of Re D = 3.9 × 10 3 and 10 4 , and a Mach number of Ma ∞ = 0.1, and ii) the NASA junction flow experiment at a Reynolds number based on the crank chord length of Re = 2.4 × 10 6 and Ma ∞ = 0.189.

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A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes

Cited by 60 publications

References 35 publications

Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

Simulation of Turbulent Flows Using a Fully Discrete Explicit hp-nonconforming Entropy Stable Solver of Any Order on Unstructured Grids

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