A trigonometrically-fitted method with two frequencies, one for the solution and another one for the derivative

In trigonometrically-fitted methods the determination of the parameter (usually known as the frequency) is a critical issue, as was shown in the article by H. Ramos and J. Vigo-Aguiar [Applied Mathematics Letters, 23 (2010) 1378–1381]. If the frequency estimation relies on the vanishing of the principal term of the local truncation error, then the first derivative is present in the formula for approximating the parameter. This requires the use of a procedure for approximating the first derivative. For this purpose we use another trigonometrically-fitted formula with a second parameter (different from the frequency of the principal method and also different from the frequency of the true solution). We describe how to approximate both parameters on each step and present different experiments concerning these questions. The numerical results indicate the good performance of the strategy.