One-Step Piecewise Polynomial Galerkin Methods for Initial Value Problems*

A new approach to the numerical solution of systems of first-order ordinary differential equations is given by finding local Galerkin approximations on each subinterval of a given mesh of size h. One step at a time, a piecewise polynomial, of degree n and class C°, is constructed, which yields an approximation of order 0(A*") at the mesh points and 0(A"+1) between mesh points. In addition, the y'th derivatives of the approximation on each subinterval have errors of order 0(An_'+1), 1 £ j £ n. The methods are related to collocation schemes and to implicit Runge-Kutta schemes based on Gauss-Legendre quadrature, from which it follows that the Galerkin methods are /4-stable.