A class of momentum-preserving finite difference schemes for the Korteweg–de Vries equation

To preserve some invariant properties of the original differential equation is an important criterion to judge the success of a numerical simulation. In this paper, we construct, analyze and numerically validate a class of momentum-preserving finite difference methods for the Korteweg–de Vries equation. The proposed schemes can conserve the discrete momentum to machine precision. Numerical experiments reveal that the phase and amplitude errors, after long time simulation, are well controlled due to the momentum-preserving property. Besides, the numerical results show that the numerical errors grow only linearly as a function of time.