Discrete Galerkin and Related One - Step Methods for Ordinary Differential Equations

New techniques for numerically solving systems of first-order ordinary differential equations are obtained by finding local Galerkin approximations on each subinterval of a given mesh. Different classes of methods correspond to different quadrature rules used to evaluate the innerproducts involved. At each step, a polynomial of degree/; is constructed and the arcs are joined together continuously, but not smoothly, to form a piecewise polynomial of degree n and class C°. If the n-point quadrature rule used for the innerproducts is of order r -+• I, r sE ft, then the Galerkin method is of order » at the mesh points. In between the mesh points, the y'th derivatives have accuracy of order oUim'n<-",n+1)), for j 0 and 0(A»->'+1) for 1 g j g n.