Projection Method I: Convergence and Numerical Boundary Layers

• Published in: SIAM journal on numerical analysis Vol. 32; no. 4; pp. 1017 - 1057
• Main Authors: Weinan E, Liu, Jian-Guo
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.08.1995
• Subjects: A priori knowledge; Accuracy; Approximation; Boundary conditions; Boundary layers; Computational methods in fluid dynamics; Convergent boundaries; Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Incompressibility; Incompressible flow; Lagrange multiplier; Mathematical methods in physics; Mathematics; Methods; 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; Reynolds number; Sciences and techniques of general use; Velocity; Velocity distribution; Projection method; Fluid dynamics; Partial differential equations; Error estimation; Viscous fluid; Boundary layer approximation; Boundary conditions; Dirichlet problem; Incompressible fluid; Convergence
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/0732047

This is the first of a series of papers on the subject of projection methods for viscous incompressible flow calculations. The purpose of these papers is to provide a thorough understanding of the numerical phenomena involved in the projection methods, particularly when boundaries are present, and point to ways of designing more efficient, robust, and accurate numerical methods based on the primitive variable formulation. This paper contains the following topics: 1. convergence and optimal error estimates for both velocity and pressure up to the boundary; 2. explicit characterization of the numerical boundary layers in the pressure approximations and the intermediate velocity fields; 3. the effect of choosing different numerical boundary conditions at the projection step. We will show that a different choice of boundary conditions gives rise to different boundary layer structures. In particular, the straightforward Dirichlet boundary condition for the pressure leads to O(1) numerical boundary layers in the pressure and deteriorates the accuracy in the interior; and 4. postprocessing the numerical solutions to get more accurate approximations for the pressure.