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Schémas de discrétisation Compatible Discrete Operator pour les équations de Navier–Stokes d’un fluide incompressible en régime instationnaire

Abstract : We develop face-based Compatible Discrete Operator (CDO-Fb) schemes for the unsteady, incompressible Stokes and Navier–Stokes equations. We introduce operators discretizing the gradient, the divergence, and the convection term. It is proved that the discrete divergence operator allows one to recover a discrete inf-sup condition. Moreover, the discrete convection operator is dissipative, a paramount property for the energy balance. The scheme is first tested in the steady case on general and deformed meshes in order to highlight the flexibility and the robustness of the CDO-Fb discretization. The focus is then moved onto the time-stepping techniques. In particular, we analyze the classical monolithic approach, consisting in solving saddle-point problems, and the Artificial Compressibility (AC) method, which allows one to avoid such saddle-point systems at the cost of relaxing the mass balance. Three classic techniques for the treatment of the convection term are investigated: Picard iterations, the linearized convection and the explicit convection. Numerical results stemming from first-order and then from second-order time-schemes show that the AC method is an accurate and efficient alternative to the classical monolithic approach
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Riccardo Milani. Schémas de discrétisation Compatible Discrete Operator pour les équations de Navier–Stokes d’un fluide incompressible en régime instationnaire. Mathématiques générales [math.GM]. Université Paris-Est, 2020. Français. ⟨NNT : 2020PESC1037⟩. ⟨tel-03413250⟩

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