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Séminaire de Calcul Scientifique et Modélisation
[Séminaire CSM] Nonlinear Boundary Conditions for Initial Boundary Value Problems with Applications in Computational Fluid Dynamics
Jan Nordström
( Linköping University )Salle 2
le 15 juin 2023 à 14:00
We derive new boundary conditions and implementation procedures for nonlinear initial boundary
value problems (IBVPs) that lead to energy and entropy bounded solutions. The new boundary
procedure is applied to nonlinear IBVPs on skew-symmetric form. For easy of presentation, the
analysis focus on the nonlinear IBVPs part involving first derivatives. However, the boundary
procedure is general in the sense that it can be used to also bound dissipative IBVPs involving
second derivatives, present for example in the Navier-Stokes equations, and we show how that is
done. The complete procedure has two main ingredients. In the first part (published in [1, 2]),
the energy and entropy rate in terms of a surface integral with boundary terms was produced. In
this second part we shortly reiterate the previous analysis for completeness and complement it by
adding second derivative dissipative terms.
This main part of this paper deals with the boundary terms, which are controlled using a new
nonlinear boundary procedure which generalise the well known characteristic boundary procedure
for linear problems. Both strong and weak imposition of the nonlinear boundary conditions are
discussed. We stress that the second part in itself does not depend on the first part. It only
requires that an energy rate in terms a surface integral with boundary terms exist. The new
boundary procedure is exemplified on three important IBVPs in computational fluid dynamics: the
incompressible Euler equations, the shallow water equations and the compressible Euler equations
(all on skew-symmetric form). We also discuss how to formally extend the analysis to the NavierStokes equations. Finally we show that stable semi-discrete approximations follow promptly if
summation-by-parts operators in combination with weak boundary conditions are used.
References
[1] J. Nordström (2022). Nonlinear and linearised primal and dual initial boundary value problems:
When are they bounded? How are they connected?. Journal of Computational Physics, vol 455,
no 111001.
[2] J. Nordström (2022). A skew-symmetric energy and entropy stable formulation of the compressible Euler equations. Journal of Computational Physics, vol 470, no 111573.
value problems (IBVPs) that lead to energy and entropy bounded solutions. The new boundary
procedure is applied to nonlinear IBVPs on skew-symmetric form. For easy of presentation, the
analysis focus on the nonlinear IBVPs part involving first derivatives. However, the boundary
procedure is general in the sense that it can be used to also bound dissipative IBVPs involving
second derivatives, present for example in the Navier-Stokes equations, and we show how that is
done. The complete procedure has two main ingredients. In the first part (published in [1, 2]),
the energy and entropy rate in terms of a surface integral with boundary terms was produced. In
this second part we shortly reiterate the previous analysis for completeness and complement it by
adding second derivative dissipative terms.
This main part of this paper deals with the boundary terms, which are controlled using a new
nonlinear boundary procedure which generalise the well known characteristic boundary procedure
for linear problems. Both strong and weak imposition of the nonlinear boundary conditions are
discussed. We stress that the second part in itself does not depend on the first part. It only
requires that an energy rate in terms a surface integral with boundary terms exist. The new
boundary procedure is exemplified on three important IBVPs in computational fluid dynamics: the
incompressible Euler equations, the shallow water equations and the compressible Euler equations
(all on skew-symmetric form). We also discuss how to formally extend the analysis to the NavierStokes equations. Finally we show that stable semi-discrete approximations follow promptly if
summation-by-parts operators in combination with weak boundary conditions are used.
References
[1] J. Nordström (2022). Nonlinear and linearised primal and dual initial boundary value problems:
When are they bounded? How are they connected?. Journal of Computational Physics, vol 455,
no 111001.
[2] J. Nordström (2022). A skew-symmetric energy and entropy stable formulation of the compressible Euler equations. Journal of Computational Physics, vol 470, no 111573.
