Entropy stable wall boundary conditions for the three-dimensional compressible Navier–Stokes equations

Parsani, Matteo; Carpenter, Mark H.; Nielsen, Eric J.

doi:10.1016/j.jcp.2015.03.026

Cited by 94 publications

(222 citation statements)

References 47 publications

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“…The fully nonlinear theory, necessary for problems with discontinuities, is incomplete. Entropy estimates can be used to bound the solution, see for example [12,17,18], but neither uniqueness nor existence follows. In this paper we do not consider problems with discontinuities.…”

Section: Remark 33mentioning

confidence: 99%

“…a roadmap for the whole computational chain. In order not to be tangled up in technical and yet unresolved theoretical difficulties, we do not deal with discontinuous problems, such as for example in [12,17,18], only sufficiently smooth solutions with related smooth and compatible data are considered. The subscripts t, x, y, z denotes partial differentiation with respect to time and space.…”

Section: Introductionmentioning

confidence: 99%

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A Roadmap to Well Posed and Stable Problems in Computational Physics

Nordström

2016

J Sci Comput

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All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact, and exemplify by discussing well-posedness of a prototype problem: the time-dependent compressible Navier-Stokes equations. We do not deal with discontinuous problems, smooth solutions with smooth and compatible data are considered. In particular, we will discuss how many boundary conditions are required, where to impose them and which form they should have in order to obtain a well posed problem. Once the boundary conditions are known, one issue remains; they can be imposed weakly or strongly. It is shown that the weak and strong boundary procedures produce similar continuous energy estimates. We conclude by relating the well-posedness results to energy-stability of a numerical approximation on summation-by-parts form. It is shown that the results obtained for weak boundary conditions in the well-posedness analysis lead directly to corresponding stability results for the discrete problem, if schemes on summation-by-parts form and weak boundary conditions are used. The analysis in this paper is general and can without difficulty be extended to any coupled system of partial differential equations posed as an initial boundary value problem coupled with a numerical method on summation-by parts form with weak boundary conditions. Our ambition in this paper is to give a general roadmap for how to construct a well posed continuous problem and a stable numerical approximation, not to give exact answers to specific problems.

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Section: Remark 33mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Roadmap to Well Posed and Stable Problems in Computational Physics

Nordström

2016

J Sci Comput

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“…Such split forms can be used to prove nonlinear stability, for example of the Burgers and Euler equations, and more generally are related to the entropy stability theory in Refs. [1][2][3]21,22]. Moreover, such split-forms can be advantageous in improving robustness [23].…”

Section: Linear Variable-coefficient Pdesmentioning

confidence: 99%

“…The SBP property mimics integration by parts, and it greatly facilitates the construction of high-order schemes that are conservative and provably stable (linearly and nonlinearly) [1][2][3]. We are, in addition, interested in the flexibility provided by SBP operators that do not have a standard polynomial basis representation.…”

Section: Introductionmentioning

confidence: 99%

Simultaneous Approximation Terms for Multidimensional Summation-by-parts Operators

Hicken

Fernández

Zingg

2016

46th AIAA Fluid Dynamics Conference

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This paper is concerned with the accurate, conservative, and stable imposition of boundary conditions and inter-element coupling for multi-dimensional summation-byparts (SBP) finite-difference operators. More precisely, the focus is on diagonal-norm SBP operators that are not based on tensor products and are applicable to unstructured grids composed of arbitrary elements. We show how penalty terms -simultaneous approximation terms (SATs) -can be adapted to discretizations based on multi-dimensional SBP operators to enforce boundary and interface conditions. A general SAT framework is presented that leads to conservative and stable discretizations of the variable-coefficient advection equation. This framework includes the case where there are no nodes on the boundary of the SBP element at which to apply penalties directly. This is an important generalization, because elements analogous to Legendre-Gauss collocation, i.e. without boundary nodes, typically have higher accuracy for the same number of degrees of freedom. Symmetric and upwind examples of the general SAT framework are created using a decomposition of the symmetric part of an SBP operator; these particular SATs enable the pointwise imposition of boundary and inter-element conditions. We illustrate the proposed SATs using triangular-element SBP operators with and without nodes that lie on the boundary. The accuracy, conservation, and stability properties of the resulting SBP-SAT discretizations are verified using linear advection problems with spatially varying divergence-free velocity fields.

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“…A lot of physical problems are described by using the general structures of entropy and partial differential equations. In this sense, Parsani et al have investigated the entropy concept for the three-dimensional compressible Navier-Stokes equations [22]. Zhao et al studied crowd macro state detection using entropy model [23].…”

Section: Open Accessmentioning

confidence: 99%

New Hyperbolic Function Solutions for Some Nonlinear Partial Differential Equation Arising in Mathematical Physics

Baskonus

2015

Entropy

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Abstract:In this study, we investigate some new analytical solutions to the (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin-Bona-Mahony equation and the (2 + 1)-dimensional cubic Klein-Gordon equation by using the generalized Kudryashov method. After we submitted the general properties of the generalized Kudryashov method in Section 2, we applied this method to these problems to obtain some new analytical solutions, such as rational function solutions, exponential function solutions and hyperbolic function solutions in Section 3. Afterwards, we draw two-and three-dimensional surfaces of analytical solutions by using Wolfram Mathematica 9.

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Entropy stable wall boundary conditions for the three-dimensional compressible Navier–Stokes equations

Cited by 94 publications

References 47 publications

A Roadmap to Well Posed and Stable Problems in Computational Physics

A Roadmap to Well Posed and Stable Problems in Computational Physics

Simultaneous Approximation Terms for Multidimensional Summation-by-parts Operators

New Hyperbolic Function Solutions for Some Nonlinear Partial Differential Equation Arising in Mathematical Physics

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