The accurate element method: A novel method for integrating ordinary differential equations

This paper presents the development of a new methodology for numerically solving ordinary differential equations. This methodology, which we call the accurate element method (AEM), breaks the connection between the solution accuracy and the number of unknowns. The differential equations are discretized by using finite elements, like in the finite element method (FEM). A key feature of the AEM is the methodology developed for eliminating unknowns inside the element by using the relations provided by the governing equations. The results of the AEM applied to a second-order ordinary differential equation (ODE) show that the degree of the local approximation function does not affect the number of unknowns used to discretize the differential equation. For a second-order ODE it is shown that the discretized solution has the same number of unknowns, whether the local approximation (or interpolation) function is a third-degree or a nineteen-degree polynomial. The implication of this result is that higher-degree interpolation functions can be used without increasing the number of unknowns. For a specified accuracy, the usage of elements with high-degree interpolation functions leads to a reduction in the necessary number of elements. Numerical results using the AEM, the shooting method and the relaxation method are presented and compared for several ODE boundary value problems.

The accurate element method: A novel method for integrating ordinary differential equations Conference Paper