Numerical simulation of three-dimensional nonlinear water waves

  • Authors:

  • Liwei Xu;Philippe Guyenne

  • Affiliations:

  • Department of Mathematical Sciences, University of Delaware, 501 Ewing Hall, Newark, DE 19716-2553, USA and College of Mathematics and Physics, Chongqing University, Chongqing 400044, China;Department of Mathematical Sciences, University of Delaware, 501 Ewing Hall, Newark, DE 19716-2553, USA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2009

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Abstract

We present an accurate and efficient numerical model for the simulation of fully nonlinear (non-breaking), three-dimensional surface water waves on infinite or finite depth. As an extension of the work of Craig and Sulem [19], the numerical method is based on the reduction of the problem to a lower-dimensional Hamiltonian system involving surface quantities alone. This is accomplished by introducing the Dirichlet-Neumann operator which is described in terms of its Taylor series expansion in homogeneous powers of the surface elevation. Each term in this Taylor series can be computed efficiently using the fast Fourier transform. An important contribution of this paper is the development and implementation of a symplectic implicit scheme for the time integration of the Hamiltonian equations of motion, as well as detailed numerical tests on the convergence of the Dirichlet-Neumann operator. The performance of the model is illustrated by simulating the long-time evolution of two-dimensional steadily progressing waves, as well as the development of three-dimensional (short-crested) nonlinear waves, both in deep and shallow water.