Hybrid schemes for Euler equations in Lagrangian coordinates

P. Bureš and R. Liska

Czech Technical University in Prague, Czech Republic

Abstract: Many fluid dynamics problems modeled by Euler equations involve large changes of volume or size of computational domain or moving boundaries and thus have to be treated in moving Lagrangian coordinates. Composite schemes are defined by global composition of Lax-Wendroff (LW) and Lax-Friedrichs (LF) schemes. The hybrid methods extends composite schemes in the way, that the combination of the LW and LF scheme is done locally in the form of an affine combination of its numerical fluxes with a local shock switch which guarantees second order accuracy on smooth solutions. Hybrid schemes are developed in 1D and 2D. Numerical results of several test problems are presented.

Applications of Cahn-Hilliard Equation in Image Processing

V. Chalupecký

Czech Technical University in Prague, Czech Republic

Abstract: In our talk we present a numerical scheme for solving the Cahn-Hilliard equation which models the phase separation in binary alloys. We consider the Cahn- Hilliard equation both with constant mobility which corresponds to Mullins- Sekerka problem, and with degenerate mobility which approximates the motion by surface diffusion.

The presented numerical scheme based on the method of lines consists of the finite-difference discretization in space leading to a system of ordinary differential equations. This system is solved by the Cash-Karp modification of the Runge-Kutta method which enables us to adaptively change the time step. Care has to be taken when treating the case with degenerate mobility. We present results concerning the experimental order of convergence of our numerical scheme. Finally, we show some results demonstrating possible applications of the Cahn- Hilliard equation in image processing.

Smooth bifurcation, bifurcation direction and exchange of stability for variational inequalities based on Lagrange multipliers

J. Eisner

Mathematical Institute, Academy of Sciences of the Czech Republic

Abstract: We prove a bifurcation theorem of Crandall-Rabinowitz type (local bifurcation of smooth families of nontrivial solutions) for general variational inequalities on possibly non-convex sets with infinite-dimensional bifurcation parameter.

Under the additional nondegeneracy conditions the direction of bifurcation branches is shown in a neighbourhood of bifurcation points. In the case of potential operators, also the stability and instability of bifurcating solutions and of the trivial solution is described in the sense of minima of the potential. In particular, an exchange of stability is observed.

The result is based on a local equivalence of the variational inequality to a smooth equation with Lagrange multipliers, on scaling techniques and on an application of the Implicit Function Theorem. As an example, we consider a semilinear elliptic PDE with inequality conditions on the boundary of the domain.

Decay estimate of Stokes equations in an exterior domain and some stability theory of constant state of the compressible fluid flow

Y. Enomoto

Waseda University Tokyo, Japan

Abstract: I will talk about so called L^p-L^q estimate of solutions to the linearized equation of the Navier-Stokes equations for the compressible fluid flow in the exterior domain. And, I will talk about some decay property of solutions to the compressible fluid flow around the constant state. The result was obtaine in the three dimansional exterior domain cases by Kobayashi-Shibata. I extend this result to n-dimensional case (n ≥ 2).

Fluid Flow in Arbitrary Unbounded Domains

R. Farwig

Darmstadt University of Technology, Germany

Abstract: It is known by counter-examples that the usual L^q-approach to the Stokes equations, well known e.g. for bounded and exterior domains, cannot be extended to general domains $\Om\subset R³$ without any modification for q ≠ 2. In a joint paper with H. Kozono (Tohoku University, Sendai) and H. Sohr (University of Paderborn) we show that important properties like Helmholtz decomposition, analyticity of the Stokes semigroup, and the maximal regularity estimate of the nonstationary Stokes equations remain valid for general domains even for q ≠ 2 if we replace the space L^q for 2 ≤ q < ∞ by the intersection L² ∩ L^q and for 1 < q < 2 by the sum space L² + L^q.

Benchmark Solutions for the Two-Phase Porous-Media Flow

R. Fučík

Czech Technical University in Prague, Czech Republic

Abstract: The contribution presents a discussion of one- and two-dimensional models of the two-phase flow through porous media used in verification of more complex numerical models. We give information how a particular solution of the simplified model based on the McWorther-Sunada formula can be generalized. To enlarge the class of admissible boundary and initial conditions, we offer a numerical algorithm solving the transport equation for saturation, which is based on the Finite-Difference Method in space and time and yields values of the solution at given time levels and on a spatial grid of positions. The use of the algorithm is demonstrated on a series of computations in one-dimensional spatial domain.

Singular limit of a reaction-diffusion system for the invasion of bacteria in burn wounds

D.Hilhorst, J. R. King, M. Röger

Université de Paris-Sud, Orsay, France

Abstract: A mathematical model for the penetration of healthy tissue by bacteria from a burn wound was proposed by John King et al. The model equations take the form

where u_k,v_k are time and space dependent functions and γ, k>0 are positive constants. Here u_k corresponds to the concentration of degradative enzymes produced by the bacteria, and v_k corresponds to the volume fraction of healthy tissue, the population density of bacteria being taken to be proportional to v_k. The key parameter k>0 is typically very large and governs the degradation ratio of the tissue. The domain of the space variable x is the upper half-space of R³ and an initial burn wound is modelled by an initial condition v_k(0,.)=\bar{v}₀, for example by taking \bar{w}₀ to be the characteristic function of a flat cylindrical piece {|(x₁,x₂)| < l, 0 < x₃ < δ} with l>0 and 0 < δ << 1. By an asymptotic expansion analysis, a free boundary problem was obtained by King et al in the limit k→∞; for unknowns u,w, the upper half space splits in a region where u=0, and a region where u>0 and w=1. The common boundary of these two regions moves according to a Stefan-like condition.

We present a rigorous analysis in arbitrary space dimensions and include the possibility of a diffusion term in both equations.

Flow Over a Given Profile in a Channel with Dynamical Effects

R. Honzátko, J. Horáček and K. Kozel

Czech Technical University in Prague, Czech Republic

Abstract: The work deals with a numerical solution of 2D inviscid and viscous incompressible flow over the profile NACA 0012 in a channel. The finite-volume method in a form of cell-centered scheme at quadrilateral C-mesh is used. Governing system of equations is the system of Euler equations and Navier-Stokes equations respectively. Some numerical results are compared with experimental data. Steady state solutions of the flow and also unsteady flows caused by the prescribed oscillation of the profile were computed. The small disturbance theory and the Arbitrary Lagrangian-Eulerian method are applied. The method of artificial compressibility and the time dependent method are used for computation of steady state solutions.

On the Eguchi-Oki-Matsumura model for phase separation

N. Ishimura

Hitotsubashi University Kunitachi, Tokyo, Japan

Abstract: Japanese physists Eguchi, Oki, and Matsumura introduced a model in order to describe the phase separation phenomena from the viewpoint of thermodynamics of irreversible processes. The model consists of a coupled system of the local concentration and the local degree of order. In this talk I will review our recent mathematical research on this model. The existence and the convergence of solutions are shown, and the singular perturbation problem is discussed.

A finite element analysis of macroscopic models for superconductivity in 3D

Y. Kashima

University of Sussex, Brighton

Abstract: A finite element analysis of macroscopic models for type II superconductor in 3D configuration is considered. We formulate the magnetic field induced by the supercurrent flowing though the bulk superconductor in an evolution variational inequality. Introducing a magnetic scalar potential and a penalized energy leads to an unconstrained minimization problem. The discretization is carried out by employing curl conforming edge element. We argue the convergence property to the analytical solution by passing mesh size, time step and penalty coefficient to zero. Some numerical results will be reported.

On a Reaction - Diffusion System Arising in Biology

J. Kodovský

Czech Technical University in Prague, Czech Republic

Abstract: We present quantitative results concerning the study of a reaction-diffusion system based on the Gray-Scott interaction scheme. We observe a rich variety of dynamics and pattern motion, interaction and splitting. The numerical scheme solving the nonlinear system of PDEs is based on the method of lines and on the FDM. We perform numerical analysis of the semi-discrete scheme and experimentally investigate convergence behaviour, namely in case of complex pattern evolution.

Arbitrary Lagrangian Eulerian Simulations of High-Velocity Impact Problem

M. Kuchařík, J. Limpouch, R. Liska and P. Váchal

Czech Technical University in Prague, Czech Republic

Abstract: Our newly developed ALE code on 2D quadrilaterals is employed here in the numerical simulations of the crater formation during impact of small accelerated object on bulk target. The simulation of the high-velocity impact problem by the Lagrangian hydrodynamical codes leads in later stages to severe distortion of the Lagrangian grid which prevents continuation of computation. Such situations can however be treated by the Arbitrary Lagrangian-Eulerian (ALE) method. In order to maintain the grid quality, the ALE method applies grid smoothing regularly after several time steps of Lagrangian computation. After changing the grid, the conservative quantities have to be conservatively remapped from the old grid to the new, better one. After remapping, Lagrangian computation can continue.

Our numerical simulations are conducted for parameters near to recent experiments performed by Polish group at PALS laser facility. In these experiments, intense laser beam was focused normally on a thin foil or a disc target. The unevaporated part of the material was ablatively accelerated to high velocities up to 100 km/s in a vacuum gap. The accelerated mass struck normally into a thick solid target where a crater was formed. Such a laboratory realization of high-velocity impact has important applications in the construction of cosmic aircrafts, and also in the inertial confinement fusion research.

Modelling Multiphase Flow in Heterogeneous Porous Media

J. Mikyška

Czech Technical University in Prague, Czech Republic

Abstract: Multi-phase models that simulate the behavior of non-aqueous phase liquids in porous media can be used to obtain fundamental understanding of the complex behavior and predict the fate of waste chemicals in the subsurface. Existing models have limitations in simulating highly heterogeneous systems to be able to represent realistic field conditions. The presentation reports development of a new multiphase flow code called VODA. It starts with a brief introduction of the mathematical model of the multiphase flow in porous media. Then, the Control Volume Finite Element (CVFE) discretization is described and finally, several numerical experiments are presented concerning validation of the developped code. Experimental convergence analysis is carried out for two well-known one-dimensional two-phase flow problems. Examples of several further two-phase flow computations in a heterogeneous medium will be also given.

Phase field approximation of precipitation pattern (Liesegang rings): Modeling and simulation

M. Mimura

Meiji University, Tamaku, Kawasaki, Japan

Abstract: In 1896, A colloidal chemist and photographer L. E. Liesegang surprisingly discovered a regularized discontinuous precipitation in a simple reaction process. Since then, many experiments as well as theories have been proposed to understand the mechanism behind the phenomenon. However, as far as I know, it has not yet been completely understood. For this reason, we propose a phase field model to this very old problem.

Numerical Simulation of Dislocation Dynamics

V. Minárik, J. Kratochvíl, K. Mikula and M. Beneš

Czech Technical University in Prague, Czech Republic,
Slovak Technical University, Bratislava

Abstract: The aim of this contribution is to present the current state of our research in the field of numerical simulation of dislocations moving in crystalline materials. The simulation is based on recent theory treating interactions of dislocation curves and dipolar loops, both occurring in the material and interacting by means of forces of elastic nature. The mathematical model describes the motion and interaction laws for one dislocation curve and finite number of dipolar loops placed in 3D space. The interactions occur not only between the dislocation curve and dipolar loops but also between dipolar loops themselves, which makes the model more complex. Equations of motion for a parametrically described dislocation curve are discretized by the flowing finite volume method in space. The interaction force is computed for each dipolar loop and along the discretized curve. The resulting system of ordinary differential equations is solved by a higher order time solver.

Relaxation oscillation in assemblies of point vortices

T. Nakaki

Kyushu University, Fukuoka, Japan

Abstract: The motion of assembly of point vortices in the two-dimensional Euler fluid is one of classical problems, and treated in some textbooks on fluid dynamics. Under some conditions on the circulations and initial configurations, the vortices exhibit a relaxation oscillation. In this talk, we consider such a oscillation from analytical and numerical points of view.

Some remarks on the steady fall of a rigid body in viscous fluids

Š. Nečasová

Mathematical Institute of the Academy of Sciences, Czech Republic

Abstract: The study of the motion of small particles in a viscous liquid has become one of the main focuses of the applied research over the last 40 years. This problem with applications e.g. in manufacturing of short-fiber composites, separation of macromolecules by electrophoresis requires study of the existence,stability and attainability of terminal states that are to be eventually achieved (as time goes to infinity) by rigid body of negative buoyancy that is dropped from rest in Navier-Stokes liquid. The terminal state means of the motion of the body with constant translational and angular velocities with respect to an inertial frame and the steady flow of the liquid as observed from a frame attached to the body.

To understand the problem, the linearized case was investigated. It leads to solving the Stokes problem or the Oseen problem with additional terms (ω × x) \cdot \nabla u and ω × u.

We consider the following type of the model

where Ω is the whole space R³ or an exterior domain in R³ and ω × v is the Coriolis force. We deal with the existence, uniqueness of the strong solution of the problem.

On a degenerate Allen-Cahn/Cahn-Hilliard system

Amy Novick-Cohen

Technion-IIT, Department of Mathematics, Haifa 32000, Israel

Abstract: A degenerate Allen-Cahn/Cahn-Hilliard system is considered which couples fourth order and second order dynamics. Formally it is possible to demonstrate that in a long time scaling limit, the predicted motion should be given by combined surface diffusion and motion by mean curvature which couple at triple junction. The angles governing the structure of the triple junction are determined by the underlying free energy. We explore the variational considerations which determine these angles, and in particular focus on possible degeneracies which may occur, and how this is related to wetting and prewetting.

Flux-free finite element method for immiscible two-fluid flows

Katsushi Ohmori

Toyama University, Toyama, Japan

Abstract: In our talk we present the flux-free finite element method based on the Eulerian framework for immiscible incompressible two-fluid flows, which ensures the mass conservation of the fluid. This method is derived from the variational formulation including the flux-free constraint for the Navier-Stokes equations by the Lagrange multiplier technique. We also give some numerical results to validate our method.

Numerical Recovery of the Signed Distance Function

T. Oberhuber

Czech Technical University in Prague, Czech Republic

Abstract: In our presentation, we discuss a new method for recovering the signed distance function used in the levelset methods. Out algorithm is efficient in case when the initial guess is close to the desired signed distance function as well as in case when the initial guess only defines the levelset of our interest. The approach is based on an evolution PDE of eikonal type. We derive new schemes to overcome the problems with discontinuities of derivatives. As the theoretical background for the equation is not zet developed, we emphasize features of the numerical algorithm and give several computational examples.

Asymptotic Behaviour of Solutions to Quasilinear Partial Differential Equations on Unbounded Spatial Intervals

L. Poul

Charles University, Prague, Czech Republic

Abstract: The talk concerns with the long-time behaviour of solutions to a quasilinear parabolic problem u_t=(F(u_x))_x+h(u) on a half-line, supplemented with the Dirichlet boundary conditions.

We will find, that there exists a solution with bounded energy, which does not converge. In particular, this solution converges to the travelling wave of solution to the corresponding stationary problem considered on the whole R.

The main tools used are the Zero Number Theory by Angenent and Concentrated Compactness by Lions.

Numerical Solution of Flows in Atmospheric Boundary Layer

K. Seinerová, K. Kozel, L. Beneš and T. Bodnár

Czech Technical University in Prague, Czech Republic

Abstract: The work deals with a numerical solution of viscous flows in the atmospheric boundary layer. The mathematical model is based on the system of Navier-Stokes equations. Numerical solution uses the finite difference method. Numerical results over 2D and 3D single hill are presented and discussed.

Some decay estimate and L_p maximal regularity of Stokes equations with non-slip, slip and Neumann boundary condition

Y. Shibata

Waseda University Tokyo, Japan

Abstract: I will talk about L_p-L_q estimate of the Stokes semigroup in an exterior domain and the L_p maximal regularity of the Stokes semigroup in a bounded domain in the case of non-slip, slip and Neumann boundary condition cases.

Mathematical model of NO production in a pulverized coal furnace

R. Straka

Czech Technical University in Prague, Czech Republic

Abstract: In this paper we deal with the NO formation and destruction during combustion processes in a pulverized coal furnace. The thermal (Zeldovich) route and the fuel-NO route are taken into account for numerical simulations. The fuel-NO route is divided to HCN and NH₃ intermediators for the volatile part of burning coal particles. For the char part two different mechanisms of the NO production is considered together with the NO destruction on the char surface.