Stability analysis of the Crank-Nicolson scheme on a staggered grid
Speaker:Prof. Jin-Yun Yuan
Professor
Federal University of Paraná
Date & Time:13 Nov 2013 (Wednesday) 10:30
Venue:WLG113
Organized by:Department of Mathematics

Abstract

This talk considers the stability of explicit, implicit and Crank-Nicolson schemes for the one-dimensional heat equation on a staggered grid. Furthermore, we consider the cases when both explicit and implicit approximations of the boundary conditions are employed. Why we choose to do this is clearly motivated and arises from solving fluid flow equations with free surfaces when the Reynolds number can be very small, in at least parts of the spatial domain. A comprehensive stability analysis is supplied: a novel result is the precise stability restriction on the Crank-Nicolson method when the boundary conditions are approximated explicitly, that is, at t =n(delta)t rather than t =(n+1)(delta)t. The two-dimensional Navier-Stokes equations were then solved by a marker and cell approach for two simple problems that had analytic solutions. It was found that the stability results r were qualitatively very similar, thereby providing insight as to why a Crank-Nicolson approximation of the momentum equations is only conditionally stable.

Biography

Prof. Jin-Yun Yuan is currently the Full Professor of Computational Mathematics at Federal University of Paraná, Brazil, and member of Brazilian Academy of Science. Prof. Yuan obtained his D.Sc. degree in Applied Mathematics in Instituto Nacional de Matemática Pura e Aplicada, Brazil. He received the Brazilian National Scientific Merit Prize awarded by President Lula Silva in 2008. Prof. Yuan is interested in Numerical Linear Algebra, Optimization, Numerical Analysis, Industrial Mathematics, and Environment Mathematics.