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UBC Theses and Dissertations

On optimum Runge-Kutta methods for the numorical solution of ordianry differential equations Johnston, Robert Laurence

Abstract

After a brief discussion of numerical methods for the solution of the ordinary differential equation x'= f(t, x) the problem of finding optimum methods is considered. The thesis then deals with this problem for the family of Runge-Kutta methods. Criteria for optimum methods are discussed and then the derivation of third-order methods is examined in detail. The next part of the thesis deals with possible approaches to finding optimum methods. The first approach consists of finding some sort of estimate for f and its derivatives contained in the truncation error T[subscript n]. The resulting expression is then dependent on some free parameter or parameters (depending on the order of the method) which are chosen in order to minimize this expression. The second approach assumes the independence of terms in the truncation error and minimizes, in some sense, the coefficients of these terms. Several procedures based on these approaches, are used to predict optimum second-order and third-order methods and comparisons are made with experimental results. While no definite conclusions could be drawn it was seen that one particular procedure gave a good prediction. This result encourages further studies in this area. I hereby certify that this abstract is satisfactory.

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