Maria Garzon

Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model

Summary. A positivity-preserving numerical scheme for a strongly coupled cross-diffusion model for two competing species is presented, based on a semi-discretization in time. The variables are the population densities of the species. Existence of strictly positive weak solutions to the semidiscrete problem is proved. Moreover, it is shown that the semidiscrete solutions converge to a non-negative solution of the continuous system in one space dimension. The proofs are based on a symmetrization of the problem via an exponential transformation of variables and the use of an entropy functional.

A coupled level set-boundary integral method for moving boundary simulations

A numerical method for moving boundary problems based upon level set and boundary integral formulations is presented. The interface velocity is obtained from the boundary integral solution using a Galerkin technique for post-processing function gradients on the interface. We introduce a new level set technique for propagating free boundary values in time, and couple this to a narrow band level set method. Together, they allow us to both update the function values and the location of the interface. The methods are discussed in the context of the well-studied two-dimensional nonlinear potential flow model of breaking waves over a sloping beach. The numerical results show wave breaking and rollup, and the algorithm is verified by means of convergence studies and comparisons with previous techniques. 1. Introduction and overview

Numerical simulation of non-viscous liquid pinch-off using a coupled level set-boundary integral method

Simulations of the pinch-off of an inviscid fluid column are carried out based upon a potential flow model with capillary forces. The interface location and the time evolution of the free surface boundary condition are both approximated by means of level set techniques on a fixed domain. The interface velocity is obtained via a Galerkin boundary integral solution of the 3D axisymmetric Laplace equation. A short-time analytical solution of the Raleigh-Taylor instability in a liquid column is available, and this result is compared with our numerical experiments to validate the algorithm. The method is capable of handling pinch-off and after pinch-off events, and simulations showing the time evolution of the fluid tube are presented.

Wave Breaking over Sloping Beaches Using a Coupled Boundary Integral-Level Set Method

We present a numerical method for tracking breaking waves over sloping beaches. We use a fully non-linear potential model for incompressible, irrotational and inviscid flow, and consider the effects of beach topography on breaking waves. The algorithm uses a Boundary Element Method (BEM) to compute the velocity at the interface, coupled to a Narrow Band Level Set Method to track the evolving air/water interface, and an associated extension equation to update the velocity potential both on and off the interface. The formulation of the algorithm is applicable to two- and three-dimensional breaking waves; in this paper, we concentrate on two-dimensional results showing wave breaking and rollup, and perform numerical convergence studies and comparison with previous techniques.

A three-dimensional coupled Nitsche and level set method for electrohydrodynamic potential flows in moving domains

In this paper we present a new algorithm for computing three-dimensional electrohydrodynamic flow in moving domains which can undergo topological changes. We consider a non-viscous, irrotational, perfect conducting fluid and introduce a way to model the electrically charged flow with an embedded potential approach. To numerically solve the resulting system, we combine a level set method to track both the free boundary and the surface velocity potential with a Nitsche finite element method for solving the Laplace equations. This results in an algorithmic framework that does not require body-conforming meshes, works in three dimensions, and seamlessly tracks topological change. Assembling this coupled system requires care: while convergence and stability properties of Nitsches methods have been well studied for static problems, they have rarely been considered for moving domains or for obtaining the gradients of the solution on the embedded boundary. We therefore investigate the performance of the symmetric and non-symmetric Nitsche formulations, as well as two different stabilization techniques. The global algorithm and in particular the coupling between the Nitsche solver and the level set method are also analyzed in detail. Finally we present numerical results for several time-dependent problems, each one designed to achieve a specific objective: (a) The oscillation of a perturbed sphere, which is used for convergence studies and the examination of the Nitsche methods; (b) The break-up of a two lobe droplet with axial symmetry, which tests the capability of the algorithm to go past flow singularities such as topological changes and preservation of an axi-symmetric flow, and compares results to previous axi-symmetric calculations; (c) The electrohydrodynamical deformation of a thin film and subsequent jet ejection, which will account for the presence of electrical forces in a non-axi-symmetric geometry.

Numerical Simulation of Flows Involving Singularities

Many interesting fluid interface problems involve singular events, as breaking-up or merging of the physical domain. In particular, wave propagation and breaking, droplet and bubble break-up, electro-jetting, rain drops, etc. are good examples of such processes. All these mentioned problems can be modeled using the potential flow assumptions, in which an interface needs to be advanced by a velocity determined by the solution of a surface partial differential equation posed on this moving boundary. The standard approach, the Lagrangian-Eulerian formulation together with some sort of front tracking method, is prone to fail when break-up or merging processes appear. The embedded formulation using level sets seamlessly allows topological breakup or merging of the fluid domain. In this work we present the numerical approximation of the embedded model and some computational results regarding electrohydrodynamic applications.

Computing Through Singularities in Potential Flow with Applications to Electrohydrodynamic Problems

Many interesting fluid interface problems, such as wave propagation and breaking, droplet and bubble break-up, electro-jetting, rain drops, etc. can be modeled using the assumption of potential flow. The main challenge, both theoretically and computationally, is due to the presence of singularities in the mathematical models. In all the above mentioned problems, an interface needs to be advanced by a velocity determined by the solution of a surface partial differential equation posed on this moving boundary. By using a level set framework, the two surface equations of the Lagrangian formulation can be implicitly embedded in PDEs posed on one higher dimension fixed domain. The advantage of this approach is that it seamlessly allows breakup or merging of the fluid domain and therefore provide a robust algorithm to compute through these singular events. Numerical results of a solitary wave breaking, the Rayleigh-Taylor instability of a fluid column, droplets and bubbles breaking-up and the electrical droplet distortion and subsequent jet emission can be obtained using this levelset/extended potential model.