Time behaviour of the error when simulating finite-band periodic waves. The case of the KdV equation

This paper is devoted to study the error growth of numerical time integrators for N-phase or N-band quasi-periodic (in time) solutions of the periodic Korteweg–de Vries equation. It is shown that the preservation, through numerical time integration, of conserved quantities of the periodic problem of the equation, may be an element to take into account in the selection of the numerical method. We explain why the inclusion of these properties of conservation provides a better error propagation. In particular, we emphasize how the preservation of invariants makes influence in the simulation of some physical parameters of the waves.