Teodora Mitkova

Explicit local time-stepping methods for Maxwell's equations

Explicit local time-stepping methods are derived for time dependent Maxwell equations in conducting and non-conducting media. By using smaller time steps precisely where smaller elements in the mesh are located, these methods overcome the bottleneck caused by local mesh refinement in explicit time integrators. When combined with a finite element discretisation in space with an essentially diagonal mass matrix, the resulting discrete time-marching schemes are fully explicit and thus inherently parallel. In a non-conducting source-free medium they also conserve a discrete energy, which provides a rigorous criterion for stability. Starting from the standard leap-frog scheme, local time-stepping methods of arbitrarily high accuracy are derived for non-conducting media. Numerical experiments with a discontinuous Galerkin discretisation in space validate the theory and illustrate the usefulness of the proposed time integration schemes.

Numerical treatment of free surface problems in ferrohydrodynamics

The numerical treatment of free surface problems in ferrohydrodynamics is considered. Starting from the general model, special attention is paid to field–surface and flow–surface interactions. Since in some situations these feedback interactions can be partly or even fully neglected, simpler models can be derived. The application of such models to the numerical simulation of dissipative systems, rotary shaft seals, equilibrium shapes of ferrofluid drops, and pattern formation in the normal-field instability of ferrofluid layers is given. Our numerical strategy is able to recover solitary surface patterns which were discovered recently in experiments. (Some figures in this article are in colour only in the electronic version)

High-order explicit local time-stepping methods for damped wave equations

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. Local time-stepping methods overcome that bottleneck by using smaller time-steps precisely where the smallest elements in the mesh are located. Starting from classical Adams-Bashforth multi-step methods, local time-stepping methods of arbitrarily high order of accuracy are derived for damped wave equations. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these local time-stepping methods.

Runge-Kutta Based Explicit Local Time-Stepping Methods for Wave Propagation

Locally refined meshes severely impede the efficiency of explicit Runge--Kutta (RK) methods for the simulation of time-dependent wave phenomena. By taking smaller time-steps precisely where the smallest elements are located, local time-stepping (LTS) methods overcome the bottleneck caused by the stringent stability constraint of but a few small elements in the mesh. Starting from classical or low-storage explicit RK methods, explicit LTS methods of arbitrarily high accuracy are derived. When combined with an essentially diagonal finite element mass matrix, the resulting time-marching schemes retain the high accuracy, stability, and efficiency of the original RK methods while circumventing the geometry-induced stiffness. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these LTS-RK methods.

Finite Element Methods for Coupled Problems in Ferrohydrodynamics

The interaction between a magnetisable fluid and an external magnetic field gives rise to several interesting phenomena in ferrohydrodynamics. Our mathematical models and solution strategies are mainly focused on cases in which the magnetic liquid exhibits a free surface which is not known a-priori. In particular, two special problems are taken into consideration: the behaviour of a ferrofluid drop in a rotary shaft seal and the generation of an ordered pattern of surface protuberances when the applied field exceeds a critical value.

Numerical Modelling of the Flow in Magnetic Liquid Seals

The dynamic behaviour of magnetic liquid seals can be described by a two-dimensional model, which consists of a convection diffusion- type equation for the azimuthal velocity and an incompressible Navier-Stokes equation for the velocity and pressure fields in the plane cross-section. A decoupling numerical solution strategy is proposed and moreover, a-priori error estimates for the discrete solutions are given.

Discontinuous Galerkin Methods and Local Time Stepping for Wave Propagation

Locally refined meshes impose severe stability constraints on explicit time‐stepping methods for the numerical simulation of time dependent wave phenomena. To overcome that stability restriction, local time‐stepping methods are developed, which allow arbitrarily small time steps precisely where small elements in the mesh are located. When combined with a discontinuous Galerkin finite element discretization in space, which inherently leads to a diagonal mass matrix, the resulting numerical schemes are fully explicit. Starting from the classical Adams‐Bashforth multi‐step methods, local time stepping schemes of arbitrarily high accuracy are derived. Numerical experiments validate the theory and illustrate the usefulness of the proposed time integration schemes.

Numerical Simulation of the Flow in Magnetic Fluid Rotary Shaft Seals

The hydrodynamical properties of the magnetic fluid in magnetic fluid seals are described by a coupled system of nonlinear partial differential equations in a three-dimensional domain with free boundaries. We propose a reduction of the three-dimensional model and describe the resulting two subproblems, the calculation of new boundaries for given flow and magnetic data, and the computation of the flow in a fixed domain, which have to be solved in an iterative manner. We consider in detail the finite element solving strategy for the flow part, which builds the main effort within the overall algorithm and show the results of a numerical test example.