initial value problems; convergence; stability; error constants; theta-method; Euler formula; implicit methods; explicit method
The Euler methods are the most popular, simplest and widely used methods for the solution of the Cauchy problem for the first order ODE. The simplest and usual generalization of these methods are the so called theta-methods (notated also as $\theta$-methods), which are, in fact, the convex linear combination of the two basic variants of the Euler methods, namely of the explicit Euler method (EEM) and of the implicit Euler method (IEM). This family of the methods is well-known and it is introduced almost in any arbitrary textbook of the numerical analysis, and their consistency is given. However, in its qualitative investigation the convergence is proven for the EEM, only, almost everywhere. At the same time, for the rest of the methods it is usually missed (e.g. [1,2,7,8]). While the consistency is investigated, the stability (and hence, the convergence) property is usually shown as a consequence of some more general theory. In this communication we will present an easy and elementary prove for the convergence of the general methods for the scalar ODE problem. This proof is direct and it is available for the non-specialists, too.