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Accurate projection methods for the incompressible Navier-Stokes equations. (English) Zbl 1153.76339
Summary: This paper considers the accuracy of projection method approximations to the initial-boundary-value problem for the incompressible Navier-Stokes equations. The issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second-order methodology a decade and a half ago. It has been observed that while the velocity can be reliably computed to second-order accuracy in time and space, the pressure is typically only first-order accurate in the \(L_\infty\)-norm. This paper identifies the source of this problem in the interplay of the global pressure-update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second-order accurate, as demonstrated by a normal mode analysis and numerical experiments. In addition, a numerical method based on a gauge variable formulation of the incompressible Navier-Stokes equations, which provides another option for obtaining fully second-order convergence in both velocity and pressure, is discussed. The connection between the boundary conditions for projection methods and the gauge method is explained in detail.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
76M25 Other numerical methods (fluid mechanics) (MSC2010)
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