A Family of Four Stages Embedded Explicit six-step Methods with Eliminated phase-lag and its Derivatives for the Numerical Solution of the Second Order Problems

A family of four stages high algebraic order embedded explicit six-step methods, for the numerical solution of second order initial or boundary-value problems with periodical and/or oscillating solutions, are studied in this paper. The free parameters of the new proposed methods are calculated solving the linear system of equations which is produced by requesting the vanishing of the phase-lag of the methods and the vanishing of the phase-lag's derivatives of the schemes. For the new obtained methods we investigate: Its local truncation error (LTE) of the methods. The asymptotic form of the LTE obtained using as model problem the radial Schrodinger equation. The comparison of the asymptotic forms of LTEs for several methods of the same family. This comparison leads to conclusions on the efficiency of each method of the family. The stability and the interval of periodicity of the obtained methods of the new family of embedded finite difference pairs. The applications of the new obtained family of embedded finite difference pairs to the numerical solution of several second order problems like the radial Schrodinger equation, astronomical problems etc. The above applications lead to conclusion on the efficiency of the methods of the new family of embedded finite difference pairs.