Giselle Sosa Jones | Applied Math, University of Waterloo
Space-Time Hybridizable Discontinuous Galerkin Methods for Free Surface Waves 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 proven to have several advantages over standard DG methods. In HDG/EDG methods, a variable that lives only on the facets is introduced. This allows to couple the degrees of freedom on elements only to the degrees of freedom on facets. Therefore, the element degrees of freedom can be eliminated and a linear system only for the facet variables has to be solved. In EDG methods, the facet variables are approximated by continuous functions. In the case of incompressible flows, for example, HDG methods are H(div)-conforming, locally momentum conserving and energy stable. Moreover, they produce pointwise divergence free velocity fields. Previous work in the area of numerical methods applied to free surface problems, has resulted in issues with stability due to the discontinuous approximation of the free surface, as a result of the discontinuous polynomial approximation of DG methods. For this reason we propose using a space-time HDG method to discretize the system of partial differential equations, but a space-time EDG to discretize the free surface boundary conditions, which allows to obtain a continuous approximation to the wave height. In this talk, we discuss the challenges associated to solving free surface problems as well as our proposal for solving said problems. We present a HDG discretization of Laplace’s equation with linear free surface boundary conditions. This HDG method produces approximations to the velocity potential and its gradient that converge with optimal rate. Additionally, an element-by-element postprocessing can be performed in order to obtain superconvergence of the scalar variable. We present a numerical test that shows the rates of convergence of the method, and also a simulation of water waves generated by a wave maker.