Numerical methods for stiff systems

Ashi, Hala (2008) Numerical methods for stiff systems. PhD thesis, University of Nottingham.

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Some real-world applications involve situations where different physical phenomena acting on very different time scales occur simultaneously. The partial differential equations (PDEs) governing such situations are categorized as "stiff" PDEs. Stiffness is a challenging property of differential equations (DEs) that prevents conventional explicit numerical integrators from handling a problem efficiently. For such cases, stability (rather than accuracy) requirements dictate the choice of time step size to be very small. Considerable effort in coping with stiffness has gone into developing time-discretization methods to overcome many of the constraints of the conventional methods. Recently, there has been a renewed interest in exponential integrators that have emerged as a viable alternative for dealing effectively with stiffness of DEs.

Our attention has been focused on the explicit Exponential Time Differencing (ETD) integrators that are designed to solve stiff semi-linear problems. Semi-linear PDEs can be split into a linear part, which contains the stiffest part of the dynamics of the problem, and a nonlinear part, which varies more slowly than the linear part. The ETD methods solve the linear part exactly, and then explicitly approximate the remaining part by polynomial approximations.

The first aspect of this project involves an analytical examination of the methods' stability properties in order to present the advantage of these methods in overcoming the stability constraints. Furthermore, we discuss the numerical difficulties in approximating the ETD cofficients, which are functions of the linear term of the PDE. We address ourselves to describing various algorithms for approximating the cofficients, analyze their performance and their computational cost, and weigh their advantages for an efficient implementation of the ETD methods.

The second aspect is to perform a variety of numerical experiments to evaluate the usefulness of the ETD methods, compared to other competing stiff integrators, for integrating real application problems. The problems considered include the Kuramoto-Sivashinsky equation, the nonlinear Schr�odinger equation and the nonlinear Thin Film equation, all in one space dimension. The main properties tested are accuracy, start-up overhead cost and overall computation cost, since these parameters play key roles in the overall eciency of the methods.

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Matthews, P.C.
Cummings, L.J.
Subjects: Q Science > QA Mathematics > QA299 Analysis
Faculties/Schools: UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID: 10663
Depositing User: EP, Services
Date Deposited: 17 Feb 2009 09:29
Last Modified: 14 Oct 2017 06:10

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