An efficient flexible-order model for coastal and ocean water waves

Current work are directed toward the development of an improved 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. The mathemathical 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. The improved 3D numerical model is based on a finite difference method as in the original works \cite{LiFleming1997,BinghamZhang2007}. The new and improved approach 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. A robust method is achieved through a special treatment of the boundary conditions along solid boundaries, and is necessary for a robust multigrid preconditioning strategy. Full details and other aspects of the 3D solution will appear in \cite{EngsigKarupBinghamLindberg2008}. At the symposium, we will present examples demonstrating the fundamental properties of the numerical model together with the latests achievements.