Studies of one-leg collocation methods for ordinary differential equations

Abstract: "One-leg collocation methods are studied for initial value problems of systems of ordinary differential equations. The methods are investigated experimentally and theoretically. Implementations have been made with variable step size and variable order. An implementation of a method based on the optimal (corrector) one-leg collocation formula is described and analysed. An algorithm is given for selecting the number of steps (k) used for interpolation. k varies with time and is chosen individually for each equation in the system. The model problem y' = [lambda]y, [lambda] [member of] C, is used to analyse, [sic] the stability of the predictor-corrector one-leg collocation methods. An attempt to use, in certain cases, one evaluation of the right hand side of the differential system (f) in every step, rather than two or more, leads to an analysis of different predictors. The stability of a new predictor is analysed and it is concluded that such an approach cannot be recommended, mainly due to too small stability region. The same technique is also applied to the Adams methods although they are not one-leg collocation methods. Stability of multistep methods, for a vibrating system with a finite number of degrees of freedom, is studied in terms of the previous model problem for imaginary values [lambda] mainly. Both corrector and predictor-corrector methods are examined. Quantitative results are given for the one-leg collocation methods."