We develop a new approach to numerical modelling of water-wave evolution based
on the Zakharov integrodifferential equation and outline its areas of application.
The Zakharov equation is known to follow from the exact equations of potential
water waves by the symmetry-preserving truncation at a certain order in wave
steepness. This equation, being formulated in terms of nonlinear normal variables, has
long been recognized as an indispensable tool for theoretical analysis of surface wave
dynamics. However, its potential as the basis for the numerical modelling of wave
evolution has not been adequately explored. We partly fill this gap by presenting
a new algorithm for the numerical simulation of the evolution of surface waves,
based on the Hamiltonian form of the Zakharov equation taking account of quintet
interactions. Time integration is performed either by a symplectic scheme, devised as
a canonical transformation of a given order on a timestep, or by the conventional
Runge–Kutta algorithm. In the latter case, non-conservative effects, small enough to
preserve the Hamiltonian structure of the equation to the required order, can be
taken into account. The bulky coefficients of the equation are computed only once,
by a preprocessing routine, and stored in a convenient way in order to make the
subsequent operations vectorized.
The advantages of the present method over conventional numerical models are
most apparent when the triplet interactions are not important. Then, due to the
removal of non-resonant interactions by means of a canonical transformation, there
are incomparably fewer interactions to consider and the integration can be carried out
on the slow time scale (O(ε2), where ε is a small parameter characterizing wave slope),
leading to a substantial gain in computational efficiency. For instance, a simulation of
the long-term evolution of 103 normal modes requires only moderate computational
resources; a corresponding simulation in physical space would involve millions of
degrees of freedom and much smaller integration timestep.
A number of examples aimed at problems of independent physical interest, where
the use of other existing methods would have been difficult or impossible, illustrates
various aspects of the implementation of the approach. The specific problems include
establishing the range of validity of the deterministic description of water wave
evolution, the emergence of sporadic horseshoe patterns on the water surface, and the
study of the coupled evolution of a steep wave and low-intensity broad-band noise.