A Finite Element Method for Fully Nonlinear Water Waves

  title={A Finite Element Method for Fully Nonlinear Water Waves},
  author={Xing Cai and Hans Petter Langtangen and Bj{\o}rn Fredrik Nielsen and Aslak Tveito},
  journal={Journal of Computational Physics},
We introduce a numerical method for fully nonlinear, three-dimensional water surface waves, described by standard potential theory. The method is based on a transformation of the dynamic water volume onto a fixed domain. Regridding at each time step is thereby avoided. The transformation introduces an elliptic boundary value problem which is solved by a preconditioned conjugate gradient method. Moreover, a simple domain imbedding precedure is introduced to solve problems with an obstacle in the… 

Figures from this paper

Boundary Element Simulation of Linear Water Waves in a Model Basin

This work considers the linearized model of this time-dependent three-dimensional problem, and uses the Galerkin boundary element method (BEM) for the approximate evaluation of the Dirichlet-to-Neumann map.

Space-time discontinuous Galerkin method for nonlinear water waves

A Spectral Element Method for Nonlinear and Dispersive Water Waves

A high-order general-purpose three-dimensional numerical model solving fully nonlinear and dispersive potential flow equations with a free surface is presented.

Quasi ALE finite element method for nonlinear water waves

This paper presents a newly developed quasi arbitrary Lagrangian–Eulerian finite element method (QALE-FEM) for simulating water waves based on fully nonlinear potential theory. The main difference of

A Mixed Eulerian–Lagrangian Spectral Element Method for Nonlinear Wave Interaction with Fixed Structures

We present a high-order nodal spectral element method for the two-dimensional simulation of nonlinear water waves. The model is based on the mixed Eulerian–Lagrangian (MEL) method. Wave interaction

Variational space-time (dis)continuous Galerkin method for linear free surface waves

A new variational finite element method is developed for nonlinear free surface gravity water waves that handles waves generated by a wave maker together with a space-time finite element discretization that is continuous in space and discontinuous in time.

Finite element simulation of fully non‐linear interaction between vertical cylinders and steep waves. Part 1: methodology and numerical procedure

A methodology for computing three‐dimensional interaction between waves and fixed bodies is developed based on a fully non‐linear potential flow theory. The associated boundary value problem is



An analysis of a preconditioner for the discretized pressure equation arising in reservoir simulation

The use of fast solvers as preconditioners for the discretized pressure equation arising in reservoir simulation is analyzed and the number of iterations for the conjugate gradient method is bounded independently of both the lower bound δ of the permeability and theDiscretization parameter h.

An iterative method for elliptic problems on regions partitioned into substructures

Analytic estimates are given which guarantee that under appropriate hypotheses, the preconditioned iterative procedure converges to the solution of the discrete equations with a rate per iteration that is independent of the number of unknowns.

Stability of periodic waves of finite amplitude on the surface of a deep fluid

We study the stability of steady nonlinear waves on the surface of an infinitely deep fluid [1, 2]. In section 1, the equations of hydrodynamics for an ideal fluid with a free surface are transformed

On the Approximation of the Solution of the Pressure Equation by Changing the Domain

Using the theory of Sobolev spaces, it is proved that the approximations converge toward the correct solution as the permeability tends to zero in the proper regions as well as the estimated rate of convergence is sharp.

Splines and Ocean Wave Modelling

The chapter introduces spline functions as approximants in the approximate solution of differential equations. These solutions are continuous and accurate such that results can be evaluated anywhere

Modified linear multistep methods for a class of stiff ordinary differential equations

Implicit and explicit Adams-like multistep formulas are derived for equations of the typeP(d/dt)y=f(t,y) whereP is a polynomial with constant coefficients and where ∣∂f/∂y∣ is considered small

Numerical solutions of partial differential equations by the finite element method , by C. Johnson. Pp 278. £40 (hardback), £15 (paperback). 1988. ISBN 0-521-34514-6, 34758-0 (Cambridge University Press)

Boundary value problems are considered in chapters five and six. The shooting method is used for ordinary differential equations and the eigenvalue problem associated with the twopoint boundary value

Multi-grid methods and applications

  • W. Hackbusch
  • Mathematics
    Springer series in computational mathematics
  • 1985
This paper presents the Multi-Grid Method of the Second Kind, a method for solving Singular Perturbation Problems and Eigenvalue Problems and Singular Equations of the Two-Grid Iteration.

The Direct Solution of the Discrete Poisson Equation on a Rectangle

where G is a rectangle, Au = 82u/8x2 + 82u/8y2, and v, w are known functions. For computational purposes, this partial differential equation is frequently replaced by a finite difference analogue.

The direct solution of the discrete Poisson equation on irregular regions

There are several very fast direct methods which can be used to solve the discrete Poisson equation on rectangular domains. We show that these methods can also be used to treat problems on irregular