Four-step exponential-fitted methods for nonlinear physical problems

We describe a four-step exponential-fitted method for systems of second-order differential equations of the form y″ = ƒ(x,y). This is a sixth-order method depending on five parameters which are automatically adjusted in terms of the equations to be solved. Some other relevant features are as follows: (i) it requires only two solution values to start; (ii) it allows modification of the stepsize during the integration process; (iii) it works in the predictor-corrector mode with only one function evaluation per step; (iv) the whole integration process is controlled in terms of the requested value for the local truncation error. Our method was tested on a representative set of problems taken from physics and found to behave particularly well on the problems involving oscillatory phenomena. A selection of experimental results is given in which our method is compared with a widely used code.