CAIMS Prize Award 426

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Department of Computer Science, University of Toronto

Supervisor:  Wayne Enright

Thesis title: Implicit Runge-Kutta Methods for Two-Point Boundary Value Problems.


We develop efficient classes of implicit Runge-Kutta methods for the solution of two-point boundary value problems. We call these new methods TPERK and ATPERK methods. They are shown to be considerably more efficient than the standard implicit Runge-Kutta methods, while at the same time retaining the good stability properties of these methods. By using this new class of methods we are able to avoid having to solve a nonlinear system of equations on each subinterval as is usually the case for an implicit Runge-Kutta method. The operation counts for the solution of two-point boundary value problem based on an implicit Runge-Kutta method are about (n3 s3)/24 per subinterval; when one of the new methods is used we show that the operation counts become about n3 s + n2 s2 per subinterval, where n is the number of differential equations and s is the number of stages of the Runge-Kutta method. We present a detailed theoretical investigation of the new methods which involves an extension of the work of [Stetter 1973]. This investigation includes results concerning the stability functions of the new methods and also results establishing a symmetry property for the ATPERK methods. Also included is a study of desirable stability properties for Runge-Kutta methods applied to singular perturbation problems using coarse meshes, and an investigation of several error measures for use in mesh selection.