Projection method II: Godunov-Ryabenki analysis

Weinan E School of Mathematics, Institute for Advanced Study Jian-Guo Liu Temple University, P

Numerical Analysis and Scientific Computing mathscidoc:1702.25067

SIAM Journal on Numerical Analysis, 33, (4), 1996.8
This is the second of a series of papers on the subject of projection methods for viscous incompressible flow calculations. The purpose of the present paper is to explain why the accuracy of the velocity approximation is not affected by (1) the numerical boundary layers in the approximation of pressure and the intermediate velocity field and (2) the noncommutativity of the projection operator and the laplacian. This is done by using a Godunov–Ryabenki type of analysis in a rigorous fashion. By doing so, we hope to be able to convey the message that normal mode analysis is basically sufficient for understanding the stability and accuracy of a finite-difference method for the Navier–Stokes equation even in the presence of boundaries. As an example, we analyze the second-order projection method based on pressure increment formulations used by van Kan and Bell, Colella, and Glaz. The leading order error term in this case is of $O(\Delta t)$ and behaves as high frequency oscillations over the whole domain, compared with the $O(\Delta t^{{1 / 2}} )$ numerical boundary layers found in the second-order Kim–Momn method.
, viscous incompressible flows, projection method, convergence, numerical boundary layers, GodunovRyabenki analysis
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  title={Projection method II: Godunov-Ryabenki analysis},
  author={Weinan E, and Jian-Guo Liu},
  booktitle={SIAM Journal on Numerical Analysis},
Weinan E, and Jian-Guo Liu. Projection method II: Godunov-Ryabenki analysis. 1996. Vol. 33. In SIAM Journal on Numerical Analysis.
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