Symmetric five–stages ten–step complete in phase embedded methods (S5S10SEMB)

The development of a family of symmetric five–stages ten–step embedded (S5S10SEMB) methods which are complete in phase is presented in this paper. In order to produce efficient numerical methods from the above family, we chose to request the vanishing of the phase–lag and its derivatives for each of the methods of the family. For the methods of the family we investigate the phase-lag, the local truncation error, the stability and the interval of periodicity. An important notice is that the construction and the analysis of the new family of embedded algorithms is based on the whole method and no on each stage of each method. The applicability of the new developed methods is the initial and/or boundary-value problems with periodical and/or oscillating solution.