This research study has undertaken the further development of a non-iterative method for predicting the fluid temperature variations in a counterflow regenerator operating at cyclic equilibrium. The method utilises the closed-form solutions derived by earlier workers from the equations of the regenerator, finite-conductivity thermal model. The numerical techniques required to compute the fluid temperature histories from these 'closed' solutions are detailed, and it is noted that in a previous attempt to apply this method, many computations had serious numerical errors. Also, the theory developed to cater for heating and cooling periods of unequal duration was incomplete and virtually untested. As a result of the present investigation, all of the previously reported computational difficulties have been resolved satisfactorily. In addition, techniques have been developed which have enabled regenerator performance to be predicted for a wide range of dimensionless regenerator parameters, including cases of unequal heating and cooling periods. The successful application of this method has been shown to depend on the ability to control, to suitably small values, the numerical errors which can arise during the summation of infinite series and in the application of methods for numerical quadrature and for the numerical integration of differential equations. The application of the techniques developed to achieve the necessary error control has enabled thermally well-balanced solutions to be attained consistently although, in many cases, the need to resort to small step sizes in the time dimension has reflected the relatively poor accuracy of the method adopted for numerical quadrature. This, in turn, has necessitated the development of an extrapolation method for predicting the regenerator effectiveness. The use of a more accurate quadrature formula is suggested in order to realise the full potential of this computational approach.