  Computationally efficient, structure-preserving reduced-order methods are
developed for the Korteweg de Vries (KdV) equations in Hamiltonian form. The
KdV equation is discretized in space by finite differences. The resulting
skew-gradient system of ordinary differential equations (ODEs) is integrated
with the linearly implicit Kahan's method, which preserves the Hamiltonian
approximately. We have shown, using proper orthogonal decomposition (POD), the
Hamiltonian structure of the full-order model (FOM) is preserved by the
reduced-order model (ROM). The quadratic nonlinear terms of the KdV equation
are evaluated efficiently by the use of tensorial methods, clearly separating
the offline-online cost of the FOMs and ROMs. The accuracy of the reduced
solutions, preservation of the Hamiltonian, momentum and mass, and
computational speed-up gained by ROMs are demonstrated for the one-dimensional
KdV equation, coupled KdV equations and two-dimensional Zakharov-Kuznetzov
equation with soliton solutions
