A conservative, second order, unconditionally stable artificial compression method. (English) Zbl 1439.76059
Summary: This report presents a new artificial compression method for incompressible, viscous flows. The method has second order consistency error and is unconditionally, long time, energy stable for the velocity and, weighted by the timestep, for the pressure. It uncouples the pressure and velocity and requires no artificial pressure boundary conditions. When the viscosity \(\nu = 0\) the method also exactly conserves a system energy. The method is based on a Crank-Nicolson Leapfrog time discretization of the slightly compressible model \[\begin{aligned}(1 - \varepsilon_1 \operatorname{grad} \operatorname{div}) u_t + u \cdot \nabla u + \frac{1}{2}(\operatorname{div} u) u - \nu \Delta u + \nabla p = f \\ \text{and } \varepsilon_2 p_t + \operatorname{div} u = 0 .\end{aligned}\] This report presents the method, gives a stability analysis, presents numerical tests and gives a preliminary analysis with tests of the non-physical acoustic waves generated. Consideration of the physical fidelity of the artificial compression method leads to a related method.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35Q30 | Navier-Stokes equations |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
Software:
FreeFem++References:
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