A Family of Three Stages Embedded Explicit Six-Step Methods with Eliminated Phase-Lag and its Derivatives for the Numerical Solution of Second Order Problems

In the present paper we investigate a family of three stages high algebraic order embedded explicit six-step methods with eliminated phase-lag and its derivatives for the numerical solution of second order periodic initial or boundary-value problems. The basis of the construction of the new proposed family of embedded methods is (1) the elimination phase-lag and (2) the elimination of the phase-lag's derivatives. The produced methods with the above mentioned methodology are studied on: their local truncation error, the asymptotic form of the local truncation error which is produced applying them to the radial Schrodinger equation, he comparison of the asymptotic forms of the local truncation errors which leads to conclusions on the effectiveness of each method of the family, the stability and the interval of periodicity of the developed methods of the new family of embedded pairs, the application the new obtained family of embedded methods to the numerical solution of several second order problems like the radial Schrodinger equation, astronomical problems etc in order to show their effectiveness.