Analysis of the error of approximation of two-layer difference schemes for the Korteweg de Vries equation

A family of weighted two-layer finite-difference schemes is presented. Using the example of the numerical solution of model problems on the propagation of a single soliton and the interaction of two solitons, the high quality of explicit-implicit schemes of the Crank-Nichols type with a weight parameter of $0.5$ and a second order of approximation in the time and space variables is shown. Absolute stability with a low accuracy of the solution due to a large approximation error is characteristic of completely implicit two-layer difference schemes with a weight parameter of $l$, first order in time and second in space. A family of explicitly implicit difference schemes is absolutely unstable if the explicitness parameter less than $0.5$ prevails. Analysis of the structure of the approximation error, performed using the modified equation method, confirmed the results of numerical simulation.