Abstract
A family of six-stages embedded explicit six-step methods of high algebraic order is studied in this paper. The applicability of the new proposed family of schemes is the numerical solution of second order initial and/or boundary-value problems for which their solution has periodic and/or oscillating behavior.
Requesting the zeroing of the phase-lag and its derivatives for the methods of the above mentioned family, we obtain a system of of equations. Based on the solution of the above described system of equations, we determine the free parameters of the new proposed methods.
The development of the new family of methods is followed by a theoretical and computational analysis which consists the following stages:
Computation of the local truncation error (LTE) of the resulting methods.
Investigation of the asymptotic form of the LTE which are developed using as model the radial Schrodinger equation.
Comparison of the asymptotic forms of LTEs for several methods of the new proposed family. Based on this comparison, we obtain conclusions on the effectiveness of the methods of the new proposed family.
Investigation of the stability and the interval of periodicity of the methods of the new proposed family.
Investigation of the effectiveness of the methods of the new proposed family, by applying them to the numerical solution of several second order problems like the radial Schrodinger equation, astronomical problems etc. Based on the above described tests, we arrive to conclusions on the efficiency of the methods.