Block Third Derivative Trigonometrically-Fitted Methods for Stiff and Periodic Problems

This article constructed and implemented a family of a third derivative trigonometric fitted method of order k+3 whose coefficients are functions of frequency and step size for the integration of systems of first-order stiff and periodic Initial Value Problems. The Block Third Derivative Trigonometric Fitted methods (BTDTFMs) are constructed via multistep collocation technique and applied in block form as simultaneous numerical integrators which make them self-starting. The convergence, accuracy, and efficiency of the methods are established through some standard numerical examples.

Trigonometrically-Fitted Methods: A Review

Numerical methods for the solution of ordinary differential equations are based on polynomial interpolation. In 1952, Brock and Murray have suggested exponentials for the case that the solution is known to be of exponential type. In 1961, Gautschi came up with the idea of using information on the frequency of a solution to modify linear multistep methods by allowing the coefficients to depend on the frequency. Thus the methods integrate exactly appropriate trigonometric polynomials. This was done for both first order systems and second order initial value problems. Gautschi concluded that “the error reduction is not very substantial unless” the frequency estimate is close enough. As a result, no other work was done in this direction until 1984 when Neta and Ford showed that “Nyström’s and Milne-Simpson’s type methods for systems of first order initial value problems are not sensitive to changes in frequency”. This opened the flood gates and since then there have been many papers on the subject.