Giselle Sosa Jones
Applied Mathematics, University of Waterloo
Space-Time Hybridizable Discontinuous Galerkin Methods for Free Surface Flow Problems
Free surface problems are of great interest in many fields such as naval and maritime engineering. For example, one may be interested in how water waves will interact and affect ships and offshore structures so that they can be designed properly.
Mathematically, free surface problems are modeled by systems of partial differential equations that describe the motion of a fluid, and certain nonlinear boundary conditions that define the free surface. These problems are particularly hard to solve, because the free surface that defines the shape of the domain is part of the solution of the problem and has to be determined at each time step. This is why we require a numerical method that easily handles time dependent domains, that is higher order accurate in both space and time, and that satisfies desirable conservation and stability properties.
Space-time discontinuous Galerkin (DG) methods are known for being suitable for problems where the domain changes in time, which makes them appropriate for free surface problems. Also, by making no distinction between space and time, it is easy to make them higher order accurate. Moreover, Hybridizable/Embedded Discontinuous Galerkin (HDG/EDG) methods have been proved to have several advantages over standard DG methods. In the case of free surface problems, HDG methods are of great interest since, for incompressible flows, they are H(div)-conforming, are locally momentum conserving and result in pointwise divergence free velocity fields.
In this talk we discuss the challenges associated to solving free surface problems as well as the work that has been developed so far in this field. We will also explain space-time HDG/EDG methods and what advantages they have over other numerical methods. Previous work has resulted in issues with stability due to the discontinuous approximation of the free-surface. For this reason we propose using space-time HDG to discretize the system of partial differential equations and space-time EDG to discretize the free surface boundary conditions to overcome these issues. Moreover, we propose to do convergence, stability and error analysis for different water wave models with nonlinear free surface conditions.