Numerical solution of the Korteweg-de Vries equation on a moving grid using two-layer difference schemes

This paper presents the results of a numerical and analytical study of 2-layer explicit and implicit difference schemes for the KdV equation. On Eulerian computational grids, a satisfactory numerical solution was obtained only when using an explicit-implicit difference scheme of the Crank-Nichols type of the second order of approximation in time t and spatial x variables. A completely implicit 2-layer scheme of the 1st order in time t and 2nd in the space x, although it is absolutely stable, but the presence of a high schematic viscosity leads to a significant distortion of the solution. The use of moving grids with dynamic adaptation made it possible to obtain high-precision numerical solutions not only for Crank-Nichols-type schemes, but also for a family of completely implicit 2-layer schemes of the 1st order in time t and 2nd in space x. An important advantage of the considered schemes is their simplicity and transparency of the basic mathematical constructions.