Numerical modelling of water-wave evolution based on the Zakharov equation

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.