A three-point formula for numerical quadrature of oscillatory integrals with variable frequency

Abstract A simple three-point formula is constructed for the evaluation of general oscillatory integrals. A rigorous derivation of the local error term is presented, and the implications to high frequency oscillations are discussed. Simple examples given include integrals with variable frequency for which the usual Filon formula would be inappropriate. For cases where Filon's formula is appropriate, the new formula appears to be computationally more efficient. The main application of the formula is to an example chosen from a class of integrals arising in the theory of water waves on a sloping beach. Comparison with exact results is possible from the work of Stoker [16] for a case which, whilst special in the physical sense, does not simplify the integral involved. In all cases the implementation of the formula is as straightforward as the implementation of the ordinary Simpson Rule.

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