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 $n$ is constructed and the arcs are joined together continuously, but not smoothly, to form a piecewise polynomial of degree $n$ and class ${C^0}$. If the $n$-point quadrature rule used for the innerproducts is of order $\nu + 1,\nu \geqq n$, then the Galerkin method is of order $\nu$ at the mesh points. In between the mesh points, the $j$th derivatives have accuracy of order $O({h^{\min (\nu ,n + 1)}})$, for $j = 0$ and $O({h^{n - j + 1}})$ for $1 \leqq j \leqq n$.