On the Properties of Difference Schemes for Solving Nonlinear Dispersive Equations of Increased Accuracy. I. The Case of One Spatial Variable

A difference scheme of the predictor–corrector type is constructed for solving nonlinear dispersive equations of wave hydrodynamics with an increased 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.