A high-order nonconservative approach for hyperbolic equations in fluid dynamics

Abgrall, Rémi; Bacigaluppi, Paola; Tokareva, Svetlana

doi:10.1016/j.compfluid.2017.08.019

Cited by 47 publications

(55 citation statements)

References 21 publications

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“…We have developed a general technique to guaranty that the discretisation of a conservative hyperbolic problem is consistent with an additional conservation relation. This is a generalisation of the work [34] and a sequel of [32]. This construction uses in depth a reinterpretation of conservative schemes as Residual distribution schemes.…”

Section: Resultsmentioning

confidence: 92%

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

Abgrall

2018

Journal of Computational Physics

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We are interested in the approximation of a steady hyperbolic problem. In some cases, the solution can satisfy an additional conservation relation, at least when it is smooth. This is the case of an entropy. In this paper, we show, starting from the discretisation of the original PDE, how to construct a scheme that is consistent with the original PDE and the additional conservation relation. Since one interesting example is given by the systems endowed by an entropy, we provide one explicit solution, and show that the accuracy of the new scheme is at most degraded by one order. In the case of a discontinuous Galerkin scheme and a Residual distribution scheme, we show how not to degrade the accuracy. This improves the recent results obtained in [1,2,3,4] in the sense that no particular constraints are set on quadrature formula and that a priori maximum accuracy can still be achieved. We study the behaviour of the method on a non linear scalar problem. However, the method is not restricted to scalar problems.

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Section: Resultsmentioning

confidence: 92%

“…The equality, together with the remark of the appendix B shows that we get indeed local conservation of the entropy, too. Following [33,34],the idea is to write:…”

Section: Construction Of Entropy Conservative Schemesmentioning

confidence: 99%

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

Abgrall

2018

Journal of Computational Physics

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“…There is no ambiguity in the definition of the last integral in (8c) because u is continuous across ∂K and the numerical fluxτ n is well defined. 4 Hereafter, we use the notation K i to indicate that the summation is done over all elements K containing a degree of freedom i 4. Residual distribution scheme.…”

Section: Staggered Grid Formulationmentioning

confidence: 99%

Multidimensional Staggered Grid Residual Distribution Scheme for Lagrangian Hydrodynamics

Abgrall

Lipnikov²,

Morgan³

et al. 2020

SIAM J. Sci. Comput.

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We present the second-order multidimensional Staggered Grid Hydrodynamics Residual Distribution (SGH RD) scheme for Lagrangian hydrodynamics. The SGH RD scheme is based on the staggered finite element discretizations as in [Dobrev et al., SISC, 2012]. However, the advantage of the residual formulation over classical FEM approaches consists in the natural mass matrix diagonalization which allows one to avoid the solution of the linear system with the global sparse mass matrix while retaining the desired order of accuracy. This is achieved by using Bernstein polynomials as finite element shape functions and coupling the space discretization with the deferred correction type timestepping method. Moreover, it can be shown that for the Lagrangian formulation written in non-conservative form, our residual distribution scheme ensures the exact conservation of the mass, momentum and total energy. In this paper we also discuss construction of numerical viscosity approximations for the SGH RD scheme allowing to reduce the dissipation of the numerical solution. Thanks to the generic formulation of the staggered grid residual distribution scheme, it can be directly applied to both single-and multimaterial and multiphase models. Finally, we demonstrate computational results obtained with the proposed residual distribution scheme for several challenging test problems.

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“…This modified form of NS equations written in Equation (4) only fulfills rigorous conservation at infinite resolution, and it has been shown that the numerical solutions of such a system might fail in converging to the correct weak solutions in the vicinity of shock discontinuities . Recent progress in solving this system was made by using a residual distribution formulation . Nonetheless, the applications of our particular interests would be smooth flows but might involve multiphysics and complex nonlinear dynamics.…”

Section: Governing Equationsmentioning

confidence: 99%

Development of a nonconservative discontinuous Galerkin formulation for simulations of unsteady and turbulent flows

2019

Numerical Methods in Fluids

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Summary In this paper, discontinuous Galerkin (DG) discretization schemes for Navier‐Stokes equations of a nonconservative form was proposed, in which the constitutional equation for energy conservation is written in terms of pressure. The primary focus is to address the treatment of nonconservative products in the pressure equation, for which we formulate four different scheme variants by using the path‐conservative scheme or solely regarding the nonconservative products as source terms. In addition to the complete description of the discretization formulations, we highlight the positivity‐preserving properties of the proposed nonconservative DG schemes. The performance of the four schemes are thoroughly assessed and tested with the consideration of a series of classical flow configurations that involve steady/unsteady, inviscid/viscous, and laminar/turbulence flow physics. It is found that two out of the four schemes are able to provide the optimal convergence and similar accuracies as the fully conservative formulation. The nonconservative formulations provide an alternative way to model fluid flows with substantial thermodynamic and compositional complexities. The present work aims to lay the theoretical foundation for further algorithmic extensions to simulations of such fluid flows of practical relevance.

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A high-order nonconservative approach for hyperbolic equations in fluid dynamics

Cited by 47 publications

References 21 publications

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

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

Multidimensional Staggered Grid Residual Distribution Scheme for Lagrangian Hydrodynamics

Development of a nonconservative discontinuous Galerkin formulation for simulations of unsteady and turbulent flows

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