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Application of the rosenbrock methods to the solution of unsteady 3D incompressible Navier-Stokes equations. (English) Zbl 1411.76059
Summary: We consider the Rosenbrock methods, namely a family of methods for differential algebraic equations, for the solution of the unsteady three-dimensional Navier-Stokes equations. These multistage schemes are attractive for non-linear problems because they achieve high order in time, ensuring stability properties and linearizing the system to be solved at each timestep. Moreover, as they provide inexpensive ways to estimate the local truncation error, adaptive timestep strategies can be easily devised. In this work, we test the Rosenbrock methods for the solution of three-dimensional unsteady incompressible flows. We derive the correct essential boundary conditions to impose at each stage in order to retain the convergence order of the schemes. Then, we consider two benchmark tests: a flow problem with imposed oscillatory pressure gradient whose analytical solution is known and the classical flow past a cylinder. In the latter case, we especially focus on the accuracy in the approximation of the drag and lift coefficients. In both benchmarks we test the performance of a time adaptivity scheme.

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
65L80 Numerical methods for differential-algebraic equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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