# Algorithm for the development of families of numerical methods based on phase-lag Taylor series

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## Abstract

In the present research paper we propose a new generator of families of numerical methods with increasing number of internal layers in an attempt to achieve higher order accuracy. The intermediate stages consist of predictor-corrector methods. The final layer is a symmetric two step method with constant coefficients and also free parameters. Those parameters define each family of methods. At first the method is constructed with unknown parameters and subsequently their value is estimated in order to fulfill the requirement of maximum phase-lag order. The stability of the new numerical algorithm is analyzed and the local truncation error is computed. The generator of the new families is applied to well known problems and is found to be more efficient compared to other methods and numerical methods generators with similar characteristics, which attempt to numerically solve such problems.

## Keywords

Two-step methods Explicit methods Hybrid methods Phase-lag Phase-fitted Schrödinger equation## Abbreviation

- LTE
Local Truncation Error

## Notes

### Acknowledgements

This work was supported by Special Account for Research Funds(SARF) of Democritus University of Thrace-project no. 81896.

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