Projection Method II : Godunov-Ryabenki Analysis

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 non-commutativity of the projection operator and the laplacian. This is done by using 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 et. al.. The leading order error term in this case is of O(∆t) and behaves as high frequency oscillations over the whole domain, compared to the O(∆t) numerical boundary layers found in the second order Kim-Moin’s method. School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540. Email: [email protected] Department of Mathematics, Temple University, Philadelphia, PA 19122, and School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540. Email: [email protected] §