Comparison of Stability of selected Numerical Methods in Solving Stiff Semi-Linear Differential Equations
AbstractMany 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 part of this project involves a general study of the stiff semi-linear differential equations. The second part of this project involves an analytical examination of the asymptotic stability properties of the Exponential Time Differencing Schemes in order to present the advantage of these methods in overcoming the stability constraints.
A Thesis submitted to the Department f Mathematics, Kwame Nkrumah University of Science and Technology in partial fulfillment of the requirement for degree of Master of Science Industrial Mathematics Institute of Distance Learning.2013