- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 01:09:40
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=8&iss=2&update_since=2024-03-29
Interfaces and Free Boundaries
Interfaces Free Bound.
IFB
1463-9963
1463-9971
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
10.4171/IFB
http://www.ems-ph.org/doi/10.4171/IFB
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
8
2006
2
Second order phase field asymptotics for multi-component systems
Harald
Garcke
Universität Regensburg, REGENSBURG, GERMANY
Björn
Stinner
University of Warwick, COVENTRY, UNITED KINGDOM
Phase field models, sharp interface models, numerical simulations, matched asymptotic expansions
We derive a phase field model which approximates a sharp interface model for solidification of a multicomponent alloy to second order in the interfacial thickness $\varepsilon$. Since in numerical computations for phase field models the spatial grid size has to be smaller than $\varepsilon$ the new approach allows for considerably more accurate phase field computations than have been possible so far. In the classical approach of matched asymptotic expansions the equations to lowest order in $\varepsilon$ lead to the sharp interface problem. Considering the equations to the next order, a correction problem is derived. It turns out that, when taking a possibly non-constant correction term to a kinetic coefficient in the phase field model into account, the correction problem becomes trivial and the approximation of the sharp interface problem is of second order in $\varepsilon$. By numerical experiments, the better approximation property is well supported. The computational effort to obtain an error smaller than a given value is investigated revealing an enormous efficiency gain.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
131
157
10.4171/IFB/138
http://www.ems-ph.org/doi/10.4171/IFB/138
Analysis of the heteroclinic connection in a singularly perturbed system arising from the study of crystalline grain boundaries
Nicholas
Alikakos
University of Athens, ATHENS, GREECE
Paul
Fife
University of Utah, SALT LAKE CITY, UNITED STATES
Giorgio
Fusco
Università degli Studi dell'Aquila, L'AQUILA, ITALY
C.
Sourdis
University of Athens, ATHENS, GREECE
Mathematically, the problem considered here is that of heteroclinic connections for a system of two second order differential equations of Hamiltonian type, in which a small parameter $\e$ conveys a singular perturbation. The motivation comes from a multi-order-parameter phase field model developed by Braun et al \cite{BCMcFW} and \cite{T} for the description of crystalline interphase boundaries. In this, the smallness of $\e$ is related to large anisotropy. The existence of such a heteroclinic, and its dependence on $\e$, is proved. In addition, its robustness is investigated by establishing its spectral stability.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
159
183
10.4171/IFB/139
http://www.ems-ph.org/doi/10.4171/IFB/139
Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model
Theodore
Kolokolnikov
University of British Columbia, VANCOUVER, CANADA
Michael
Ward
University of British Columbia, VANCOUVER, CANADA
Juncheng
Wei
University of British Columbia, VANCOUVER, CANADA
Slow translational instabilities of symmetric $k$-spike equilibria for the one-dimensional singularly perturbed two-component Gray-Scott (GS) model are analyzed. These symmetric spike patterns are characterized by a common value of the spike amplitude. The GS model is studied on a finite interval in the semi-strong spike-interaction regime, where the diffusion coefficient of only one of the two chemical species is asymptotically small. Two distinguished limits for the GS model are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime it is shown analytically that $k-1$ small eigenvalues, governing the translational stability of a symmetric $k$-spike pattern, simultaneously cross through zero at precisely the same parameter value at which $k-1$ different asymmetric $k$-spike equilibria bifurcate off of the symmetric $k$-spike equilibrium branch. These asymmetric equilibria have the general form $SBB...BS$ (neglecting the positioning of the $B$ and $S$ spikes in the overall spike sequence). For a one-spike equilibrium solution in the intermediate regime it is shown that a translational, or drift, instability can occur from a Hopf bifurcation in the spike-layer location when a reaction-time parameter $\tau$ is asymptotically large as $\eps\to 0$. Locally, this instability leads to small-scale oscillations in the spike-layer location. For a certain parameter range within the intermediate regime such a drift instability for the GS model is shown to be the dominant instability mechanism. Numerical experiments are performed to validate the asymptotic theory.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
185
222
10.4171/IFB/140
http://www.ems-ph.org/doi/10.4171/IFB/140
Travelling front solutions arising in the chemotaxis-growth model
Mitsuo
Funaki
Hiroshima National College of Maritime Technology, HIROSHIMA, JAPAN
Masayasu
Mimura
Meiji University, KAWASAKI, JAPAN
Tohru
Tsujikawa
Miyazaki University, MIYAZAKI, JAPAN
Chemotaxis, travelling front, singular perturbation, interfacial stability
We consider a bistable reaction-diffusion-advection system describing the growth of biological individuals which move by diffusion and chemotaxis. We use the singular limit procedure to study the dynamics of growth patterns arising in this system. It is shown that travelling front solutions are transversally stable when the chemotactic effect is weak and, when it becomes stronger, they are destabilized. Numerical simulations reveal that the destabilized solution evolves into complex patterns with dynamic network--like structures.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
223
245
10.4171/IFB/141
http://www.ems-ph.org/doi/10.4171/IFB/141
A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth
Avner
Friedman
Ohio State University, COLUMBUS, UNITED STATES
Tumor growth, free boundary problems, hyperbolic equations, Stokes equation
We consider a tumor model with three populations of cells: proliferating, quiescent, and necrotic. Cells may change from one type to another at a rate which depends on the nutrient concentration. We assume that the tumor tissue is a fluid subject to the Stokes equation with sources determined by the proliferation rate of the proliferating cells. The boundary of the tumor is a free boundary held together by cell-to-cell adhesiveness of intensity $\gamma$. Thus, on the free boundary the stress tensor $T$ and the mean curvature $\kappa$ are related by $T\vec n=-\gamma\kappa\vec n$ where $\vec n$ is the outward normal. We prove that the coupled system of PDEs for the densities of the three types of cells, the nutrient concentration, and the fluid velocity and pressure have a unique smooth solution, with a smooth free boundary, for a small time interval.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
247
261
10.4171/IFB/142
http://www.ems-ph.org/doi/10.4171/IFB/142