Date of Award

1-1-2013

Document Type

Open Access Thesis

Department

Mathematics

First Advisor

Lili Ju

Abstract

In this thesis we apply the compact implicit integration factor (cIIF) scheme towards solving the Allen-Cahn equations with zero-flux or periodic boundary conditions. The Allen-Cahn equation is a second-order nonlinear PDE which has been the focus of many applications spanning a wide range of fields, such as in material science where it was first introduced to model the phase separation of two metallic alloys, and in biology to study population dynamics, just to name a few. The compact implicit integration method works by first transforming the PDE into a system of ODEs by discretizing the spatial derivatives using the central differencing scheme. This yields a semi-discretized form which produces a nice compact representation to the original PDE. The resulting system is then integrated with respect to time, thereby treating the linear component of the PDE exactly. The nonlinear portion which represents the integrand is then approximated by a Lagrange interpolation polynomial of order r and then integrated exactly, with r = 0,1,2 in our study . Altogether, this a fully discrete scheme which is second-order accurate in space and (r + 1)-order accurate in time. Experiments are also performed to numerically demonstrate the stability and convergence properties of the proposed scheme.

Rights

© 2013, Meshack K. Kiplagat

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