A Level Set Discontinuous Galerkin Method for Free Surface Flows - and Water-Wave Modeling

We present a discontinuous Galerkin method on a fully unstructured grid for the modeling of unsteady incompressible fluid flows with free surfaces. The surface is modeled by a level set technique. We describe the discontinuous Galerkin method in general, and its application to the flow equations. The discontinuous Galerkin method is based on triangular elements, giving maximum flexibility in the handling of complex domains.

A high order nodal basis is used to enable local high order and is well suited for problems with many scales and wave lengths. The use of high-order methods for advancing the unsteady equations in time are discussed. We investigate theory of differential algebraic equations, and connect the theory to current methods for solving the unsteady fluid flow equations.

We explore the use of a semi-implicit spectral deferred correction method having potential to achieve high temporal order. The deferred correction method is applied on the fluid flow equations and show good results in periodic domains. We describe the design of a level set method for the free surface modeling.

The level set utilize the high order accurate discontinuous Galerkin method fully and represent smooth surfaces very accurately. We present techniques for reinitialization, and outline the strengths and weaknesses of the level set method. Through a few numerical tests, the robustness and versatility of the proposed scheme is confirmed.