### Abstract

The incompressible Euler equations are solved with a free surface, the position of which is captured by applying an Eulerian kinematic boundary condition. The solution strategy follows that of [1, 2], applying a coordinate-transformation to obtain a time-constant spatial computational domain which is discretized using arbitrary-order finite difference schemes on a staggered grid with one optional stretching in each coordinate direction. The momentum equations and kinematic free surface condition are integrated in time using the classic fourth-order Runge-Kutta scheme. Mass conservation is satisfied implicitly, at the end of each time stage, by constructing the pressure from a discrete Poisson equation, derived from the discrete continuity and momentum equations and taking the time-dependent physical domain into account. An efficient preconditionedDefect Correction (DC) solution of the discrete Poisson equation for the pressure is presented, in which the preconditioning step is based on an order-multigrid formulation with a direct solution on the lowest order-level. This ensures fast convergence of the DC method with a computational effort which scales linearly with the problem size. Results obtained with a two-dimensional implementation of the model are compared with highly accurate stream function solutions to the nonlinear wave problem, which show the approximately expected convergence rates and a clear advantage of using high-order finite difference schemes in combination with the Euler equations.

Original language | English |
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Title of host publication | Proceedings of the ASME 2012 31st International Conference on Ocean, Offshore and Arctic |

Number of pages | 10 |

Publisher | American Society of Mechanical Engineers |

Publication date | 2012 |

Publication status | Published - 2012 |

Event | 31st ASME International Conference on Ocean, Offshore and Arctic Engineering - Rio de Janeiro, Brazil Duration: 1 Jul 2012 → 6 Jul 2012 Conference number: 31 https://www.asmeconferences.org/OMAE2012/ |

### Conference

Conference | 31st ASME International Conference on Ocean, Offshore and Arctic Engineering |
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Number | 31 |

Country | Brazil |

City | Rio de Janeiro |

Period | 01/07/2012 → 06/07/2012 |

Internet address |

## Cite this

Christiansen, T. R. B., Bingham, H. B., & Engsig-Karup, A. P. (2012). High-order Finite Difference Solution of Euler Equations for Nonlinear Water Waves. In

*Proceedings of the ASME 2012 31st International Conference on Ocean, Offshore and Arctic*American Society of Mechanical Engineers.