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 . One step at a time, a piecewise polynomial, of degree and class , is constructed, which yields an approximation of order <!-- MATH $O({h^{2n}})$ --> at the mesh points and <!-- MATH $O({h^{n + 1}})$ --> between mesh points. In addition, the th derivatives of the approximation on each subinterval have errors of order <!-- MATH $O({h^{n - j + 1}}),1 \leqq j \leqq n$ --> . 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 -stable.