Projection Method II: Godunov-Ryabenki Analysis

• Published in: SIAM journal on numerical analysis Vol. 33; no. 4; pp. 1597 - 1621
• Main Authors: Weinan E, Liu, Jian-Guo
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.08.1996
• Subjects: Accuracy; Airy equation; Applied mathematics; Approximation; Boundary conditions; Boundary layers; Commutators; Computational methods in fluid dynamics; Computational techniques; Convergent boundaries; Exact sciences and technology; Finite-difference methods; Fluid dynamics; Fundamental areas of phenomenology (including applications); Mathematical methods in physics; Mathematics; Navier Stokes equation; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation and analysis; Ordinary and partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Physics; Sciences and techniques of general use; Sine function; Velocity; Velocity distribution; Numerical approximation; Projection method; Stability; Fluid dynamics; Error estimation; Navier Stokes equation; Viscous fluid; Boundary layer approximation; Incompressible fluid; Partial differential equation; Boundary layer; Finite difference method
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/s003614299426450x

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 (Δ t) and behaves as high frequency oscillations over the whole domain, compared with the O (Δ t1/2) numerical boundary layers found in the second-order Kim-Moin method.