Solving first-order initial-value problems by using an explicit non-standard A-stable one-step method in variable step-size formulation

This paper presents the construction of a new family of explicit schemes for the numerical solution of initial-value problems of ordinary differential equations (ODEs). The one-parameter family is constructed by considering a suitable rational approximation to the theoretical solution, resulting a family with second-order convergence and A-stable. Imposing that the principal term in the local truncation error vanishes, we obtain an expression for the parameter value in terms of the point (xn, yn) on each step. With this approach, the resulting method has third order convergence maintaining the characteristic of A-stability. Finally, combining this last method with other of order two in order to get an estimation for the local truncation error, an implementation in variable step-size has been considered. The proposed method can be used in a wide range of problems, for solving numerically a scalar ODE or a system of first order ODEs. Several numerical examples are given to illustrate the efficiency and performance of the proposed method in comparison with some existing methods in the literature.