DERIVATION OF CONTINUOUS LINEAR MULTISTEP METHODS FOR INITIAL VALUE PROBLEMS OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATION

Abstract

In this work, a class of numerical methods with step numbers k > 1 is studied to solve initial value problems of first order ordinary differential equations. Four new linear multistep collocation methods were developed. Collocations were taken at even grid points for even k, while for odd k, collocations were taken at odd grid points. For maximal order method k = 4 collocation were taken at all grid points. In both cases interpolations were taken at all the grid points except the last grid point. At X=Xn+k the discrete methods obtained are symmetric for even k and of order p = 3, 4, 6 and 8 respectively. The methods are consistent but not zero stable. This is so because the interpolation points were not restricted to a single point Xn+k-1 or Xn as it is in Adams methods. The accuracy of the methods was tested, using linear and non-linear first order sample problems. The results show a Significant improvement over the existing methods in terms of accuracy. It is also striking to note that orders of accuracy of the correctors and predictor are equal for corresponding step numbers.