Multisymplectic box schemes and the Korteweg–de Vries equation

We develop and compare some geometric integrators for the Korteweg-de Vries equation, especially with regard to their robustness for large steps in space and time, ∆x and ∆t, and over long times. A standard, semi-explicit, symplectic finite difference scheme is found to be fast and robust. However, in some parameter regimes such schemes are susceptible to developing small wiggles. At the same instances the fully implicit and multisymplectic Preissmann scheme, written as a 12-point box scheme, stays smooth. This is accounted for by the ability of the box scheme to preserve the shape of the dispersion relation of any hyperbolic system for all ∆x and ∆t. We also develop a simplified 8-point version of this box scheme which maintains its advantageous features.