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