Numerical-Analytical Evidence of Convergence of the Semi-Discrete Lagrangian-Eulerian Numerical Method for the Korteweg-de Vries Equation

In this work, we present a new extended formulation of the semi-discrete Lagrangian-Eulerian numerical method applied to the Korteweg-de Vries equation, which has a smooth convex flux function. This new scheme is applied to one-dimensional scalar problems incorporating a linear dispersive term with a constant dispersive coefficient. We present here some numerical experiments that provide strong evidence of the method’s convergence. Whenever possible, a comparison is made between the numerical results and exact solutions or highly accurate approximations.