Families of backward differentiation methods based on trigonometric polynomials

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Abstract
Backward differentiation methods based on trigonometric polynomials for the initial value problems whose solutions are known to be periodic are constructed. It is assumed that the frequency wcan be estimated in advance. The resulting methods depend on a parameter v = hw, where his the step size, and reduce to classical backward methods if v→0. Neta and Ford [6] constructed Nyström and generalized Milne-Simpson type methods. Those methods require the Jacobian matrix to have purely imaginary eigenvalues. The methods we construct here will not suffer of this deficiency.