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 C⁰, is constructed, which yields an approximation of order O(h²ⁿ) at the mesh points and O(hn + 1) between mesh points. In addition, the jth derivatives of the approximation on each subinterval have errors of order O(hn - j + 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 A-stable.