Optimization of explicit two-step hybrid methods for solving orbital and oscillatory problems

The construction of optimized explicit two-step hybrid methods for solving orbital problems and oscillatory second order IVPs is analyzed. These methods have variable coefficients depending on the parameter ν=ωh, where hh is the integration step-size and ωω represents an approximation of the main frequency of the problem. Some optimized explicit two-step hybrid schemes with orders four and six are derived and their stability and phase properties are analyzed. The new methods have the property of being zero dissipative for all the values of the parameter ωω whereas their dispersion errors (phase-lag) are optimized in terms of the relative error committed in the approximation of the main frequency of the problem. The numerical experiments carried out with several orbital and oscillatory problems show that the new optimized two-step schemes are more efficient than other methods recently proposed in the scientific literature for solving this class of problems.