Second-Order Convergence of a Projection Scheme for the Incompressible Navier–Stokes Equations with Boundaries

A rigorous convergence result is given for a projection scheme for the Navies–Stokes equations in the presence of boundaries. The numerical scheme is based on a finite-difference approximation, and the pressure is chosen so that the computed velocity satisfies a discrete divergence-free condition. This choice for the pressure and the particular way that the discrete divergence is calculated near the boundary permit the error in the pressure to be controlled and the second-order convergence in the velocity and the pressure to the exact solution to be shown. Some simplifications in the calculation of the pressure in the case without boundaries are also discussed

A Numerical Method for Tracking Curve Networks Moving with Curvature Motion.

A finite difference method is proposed to track curves whose normal velocity is given by their curvature and which meet at different types of junctions. The prototypical example is that of phase interfaces that meet at prescribed angles, although eutectic junctions and interactions through nonlocal effects are also considered. The method is based on a direct discretization of the underlying parabolic problem and boundary conditions. A linear stability analysis is presented for our scheme as well as computational studies that confirm the second order convergence to smooth solutions. After a singularity in the curve network where the solution is no longer smooth, we demonstrate "almost" second order convergence. A numerical study of singularity types is done for the case of networks that meet at prescribed angles at triple junctions. Finally, different discretizations and methods for implicit time stepping are presented and compared. Key Words: grain growth, interface tracking, finite di..

Stability Analysis for the Immersed Fiber Problem

A linear stability analysis is performed on a two-dimensional version of the "immersed fiber problem", formulated by C. Peskin to model the flow of fluid in the presence of a mesh of moving, elastic fibers. The purpose of the analysis is to isolate the modes in the solution which are associated with the fiber, and thereby determine the effect of the presence of a fiber on the fluid. The results are used not only to make conclusions about the stability of the problem, but also to suggest guidelines for developing numerical methods for flows with immersed fibers. 1 Introduction This paper is concerned with the stability of incompressible, viscous fluid flows in the presence of moving, elastic fibers. A mesh of such fibers was used by Peskin in [7] to model muscle tissue immersed in blood, leading to the development of a numerical scheme for computing the flow of blood within the heart. His "Immersed Boundary Method" facilitated realistic computations of flows with complex, elastic struc..