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 and class . If the -point quadrature rule used for the innerproducts is of order <!-- MATH $\nu + 1,\nu \geqq n$ --> , then the Galerkin method is of order at the mesh points. In between the mesh points, the th derivatives have accuracy of order <!-- MATH $O({h^{\min (\nu ,n + 1)}})$ --> , for and <!-- MATH $O({h^{n - j + 1}})$ --> for <!-- MATH $1 \leqq j \leqq n$ --> .