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Convergence of a vector penalty projection scheme for the Navier Stokes equations with moving body. (English) Zbl 1406.35220

Summary: In this paper, we analyse a Vector Penalty Projection Scheme (see [1]) to treat the displacement of a moving body in incompressible viscous flows in the case where the interaction of the fluid on the body can be neglected. The presence of the obstacle inside the computational domain is treated with a penalization method introducing a parameter \({\eta}\) to enforce the velocity on the solid boundary. The incompressibility constraint is approached using a Vector Projection method which introduces a relaxation parameter \({\varepsilon}\). We show the stability of the scheme and that the pressure and velocity converge towards a limit when the relaxation parameter \({\epsilon}\) and the time step \({\delta}t\) tend to zero with a proportionality constraint \({\varepsilon} = {\lambda}\delta t\). Finally, when \({\eta}\) goes to 0, we show that the problem admits a weak limit which is a weak solution of the Navier-Stokes equations with no-slip condition on the solid boundary.

MSC:

35Q30 Navier-Stokes equations
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76D05 Navier-Stokes equations for incompressible viscous fluids
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35B35 Stability in context of PDEs
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