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 h . One step at a time, a piecewise polynomial, of degree n n and class C 0 {C^0} , is constructed, which yields an approximation of order O ( h 2 n ) O({h²ⁿ}) at the mesh points and O ( h n + 1 ) O({hn + 1}) between mesh points. In addition, the j j th derivatives of the approximation on each subinterval have errors of order O ( h n − j + 1 ) , 1 ≦ j ≦ n 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 A -stable.