Toward a scalable flexible-order model for 3D nonlinear water waves

For marine and coastal applications, current work are directed toward the development of a scalable numerical 3D model for fully nonlinear potential water waves over arbitrary depths. The model is high-order accurate, robust and efficient for large-scale problems, and support will be included for flexibility in the description of structures by the use of curvilinear boundary-fitted meshes.

The mathematical equations for potential waves in the physical domain is transformed through $\sigma$-mapping(s) to a time-invariant boundary-fitted domain which then becomes a basis for an efficient solution strategy on a time-invariant mesh. The 3D numerical model is based on a finite difference method as in the original works \cite{LiFleming1997,BinghamZhang2007}.

Full details and other aspects of an improved 3D solution can be found in \cite{EBL08}. The new and improved approach for three-dimensional problems employs a GMRES solver with multigrid preconditioning to achieve optimal scaling of the overall solution effort, i.e., directly with $n$ the total number of grid points.

Grid independent iteration count and optimal scaling has been demonstrated to be independent of the mesh and the physics. A robust method is achieved through a special treatment of the boundary conditions along solid boundaries using a fictitious a ghost point technique, and is necessary for a robust multigrid preconditioning strategy.

The solution strategy is found to be both robust for general nonlinear wave problems, with no need for additional smoothing or filtering over that imposed naturally by the finite difference scheme. By the adjusting the numerical discretization parameters, the accuracy in dispersion and flow kinematics (accuracy) together with the solution effort (efficiency) can be optimized for the model to be nearly competitive with dedicated models based on simplified equations, e.g.

Boussinesq-type equations. At the symposium, we will present examples demonstrating the fundamental properties of the numerical model (OceanWave3D) together with the latests achievements.