John Chrispell, Clemson University

A Fractional Step $\theta$-Method for Time Dependent PDEs

Abstract:  The accurate numerical approximation of viscoelastic fluid flow poses two difficulties: the large number of unknowns in the approximating algebraic system (corresponding to velocity, pressure, and stress), and the different mathematical types of the modeling equations. An appealing approximation approach is to use an operator splitting method which decouples the conservation of momentum equation from the constitutive equation. This split reduces the size of the linear systems that need to be solved and separates the parabolic and hyperbolic equations into different substeps. Motivated by the viscoelastic fluid flow problem, we analyze an operator splitting fractional step $\theta$-scheme for the numerical approximation of the convection-diffusion problem and Burger' equation. In this presentation, we describe the approximation schemes for both equations, present numerical simulations, and give theoretical results.
 
Advisors:  Eleanor W. (Lea) Jenkins, and Vincent J. Ervin. (Clemson)