On the properties of difference schemes for solving nonlinear dispersion equations of ingreased precision. I. The case of one spatial variable

A difference scheme of the predictor-corrector type is constructed for solving nonlinear dispersion equations of wave hydrodynamics with a high order of approximation of the dispersion relation, based on splitting of the original system of equations into a hyperbolic system and a scalar equation of the elliptic type. A dissipation and dispersion analysis of the new scheme is performed, a condition for its stability is obtained, and a formula for the phase error is written and analyzed. Parameters are found at which the phase characteristics of the difference scheme, the nonlinear-dispersive model approximated by it, and the full model of potential flows have the same order of accuracy.