99-17   Berichtsreihe des Mathematischen Seminars der Universität Kiel

Andreas Prohl:

On Pressure Approximation via Projection Methods in Computational Fluid Dynamics

Over the last three decades, projection-/time-splitting methods turned out to be an efficient tool for simulating fluid flow problems. These time-discretization methods compute iterates for velocity field and pressure independently at each time step. As the major draw-back of Chorin's projection method which is considered to be the most popular projection method, this method often suffers from induced boundary layers that pollute the approximation of the pressure close to the boundary. A thorough error analysis of Chorin's method is given in \cite{P1}, together with the new Chorin-Uzawa method that avoids this deficiency. The first goal of this work is to survey the state of the art as well as to show exponential decay of the pollution effect to the interior of the domain for the Chorin method. Secondly, we propose a modification of the Chorin-Uzawa method called Chorin-Penalty method that allows for the error bound $\max_{0\leq m \leq M} \tau_m\, \norm{p(t_{m}) - p^{m}}{L^2/\R} \leq C\, k$. In recent works of the author, projection-based time-splitting methods are analyzed for (chemically-) reacting flows and magneto-hydrodynamical flows, where a severe (global) pollution effect is stated onto the additional quantities being temperature, concentrations or magnetic field, respectively, when Chorin's method is used. This drawback can be avoided if Chorin-Uzawa or the Chorin-Penalty method are employed in a time-splitting framework. The verification of this assertion is the third goal of the present work.

Mathematics Subject Classification (1991): 35Q30, 65M12, 65N15, 76D05, 76V05, 76W05

Keywords: Navier-Stokes equations, LBB condition, projection method, splitting method, Chorin method, Chorin-Uzawa method, pressure stabilization, artificial compressibility, penalty method, time grids, boundary layers, chemically reacting flows, magnetohydrodynamics

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[Thu Feb 19 18:56:34 2009]
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