Applied Analysis

Place: Loyola University - Monroe Hall, Room 222
Day: Wednesday
Time: 4:00pm - 5:00pm

Prof. Dongming Wei

Department of Mathematics, University of New Orleans

Modeling of Nonlinear Waves in Strain Hardening Structures

Prof. Ralph Saxton

Department of Mathematics, University of New Orleans

Flow in an Infinite Channel

Dr. Alexei Medovikov

Department of Mathematics, Tulane University

Increasing Asymptotic Stability of the Crank-Nicolson Method

Abstract:
We study the optimization of the Crank-Nicolson method for stiff ordinary differential equations. The Crank-Nicolson method for numerical integration of first order ordinary differential equations is absolutely stable, but the stability function |R(z)| tends to 1 rather than zero as Re z approaches infinity. This causes unexpected oscillatory behavior of the numerical solution of stiff differential equations. In order to avoid this problem, we optimize the stability property of the stability function. Variable steps within the sequence of steps by the Crank-Nicolson method allows us to obtain different stability functions and formulate an optimization problem for roots and poles of the stability function. The optimal solution of this problem is the classical rational Zolotarev function. An appropriate selection of the sequence of step-sizes eliminates oscillatory behavior of the numerical solution.

The talk is based on a joint work with Vyacheslav I. Lebedev (Institute of Numerical Mathematics of The Russian Academy of Science 117334 Russia, Moscow, Gubkina Street 8.)

Abstract:
When a Lie group acts on some fiber bundle of a manifold it induces what is called a prolonged action on the jet bundle of the manifold. This in turn induces an action (transformation) on local functions and local functionals which are defined on the jet bundle. The transformations that leave the Poisson bracket of two functionals unchanged are called canonical transformations. We find the necessary and sufficient conditions for a transformation to be canonical.