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

Abstract

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.