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.
