3 - THEORY OF ONE-STEP METHODS

Abstract

This chapter discusses the theory of one-step methods. The conventional one-step numerical integrator for the IVP can be described as yn+1 = yn + hnф (xn, yn; hn), where ф(x, y; h) is the increment function and hn is the mesh size adopted in the subinterval [xn, xn +1]. For the sake of convenience and easy analysis, hn shall be considered fixed. The consistency of a one-step numerical integrator ensures that the scheme is at least of order one. The consistency of a formula ensures that the method approximates the ODE in some sense. The chapter discusses the Euler scheme, the inverse Euler scheme, and Richardson's extrapolation. Euler proposed the simplest and the most analyzed numerical integration algorithm. It is a one-step scheme. The fact that the explicit Euler scheme has a small interval of absolute stability renders it unsuitable for stiff systems. A significant improvement can be attained by adopting the implicit Euler scheme. The inverse Euler method is component applicable to systems of differential equations. The stability property of Euler method is the same as that of the implicit Euler for scalar ODEs, while it eliminates the need for the solution of nonlinear systems. The main drawback of both the Euler and inverse Euler schemes is that they are both of order one, which is rather low. However, this situation can be significantly improved by the application of the Richardson extrapolation to the limit of either of the two schemes.