International Conference in Honor of
Saul Abarbanel's 80th Birthday
June 28 - 29, 2011, Tel Aviv University, Israel
Acknowledgement
We wish to thank the following for their contribution to the success of
this conference: European Office of Aerospace Research and Development,
Air Force Office of Scientific Research, United States Air Force
Research Laboratory (www.london.af.mil), and Tel Aviv University, Tel Aviv, Israel.
List of Abstracts
Speaker: Chi-Wang Shu, Division of Applied Mathematics, Brown University
Title: Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin and finite volume schemes for conservation laws
Abstract:
We construct uniformly high order accurate discontinuous Galerkin
(DG) and weighted essentially non-oscillatory (WENO) finite volume
(FV) schemes satisfying a strict maximum principle for scalar
conservation laws and passive convection in incompressible flows, and
positivity preserving for density and pressure for compressible Euler
equations. A general framework (for arbitrary order of accuracy) is
established to construct a limiter for the DG or FV method with first
order Euler forward time discretization solving one dimensional scalar
conservation laws. Strong stability preserving (SSP) high order time
discretizations will keep the maximum principle and make the scheme
uniformly high order in space and time. One remarkable property of
this approach is that it is straightforward to extend the method to
two and higher dimensions. The same limiter can be shown to preserve
the maximum principle for the DG or FV scheme solving two-dimensional
incompressible Euler equations in the vorticity stream-function
formulation, or any passive convection equation with an incompressible
velocity field. A suitable generalization results in a high order DG
or FV scheme satisfying positivity preserving property for density and
pressure for compressible Euler equations. Numerical tests
demonstrating the good performance of the scheme will be reported.
This is a joint work with Xiangxiong Zhang.
Speaker: Jennifer K. Ryan,
Delft Institute for Applied Mathematics,
Delft University of Technology
Title: Position-Dependent Smoothness-Increasing Accuracy-Conserving Filtering for
Discontinuous Galerkin Solutions
abstract
Speaker: Roger Temam,
Indiana University, Bloomington
Title: Implementation of non local boundary condtions in the equations
of the atmosphere and the oceans.
Abstract:
It has been known for a long time that the inviscid equations
of the atmosphere and the oceans, the so-called primitive equations,
necessitate nonlocal boundary conditions, unlike the Euler equations to
which they relate.
Description and implementation of appropriate boundary conditions raise
a number of theoretical and numerical difficulties which we will
describe and review in this lecture.
Speaker: Semyon Tsynkov,
NC State University (Raleigh, NC, USA)
Title: Quasi-Lacunae of Maxwell’s Equations.
Abstract:
Classical lacunae (in the sense of Petrowsky) in the solutions of
hyperbolic differential equations and systems are a manifestation of the
Huygens' principle. If the source terms are compactly supported in space
and time, then, at any finite location in space, the solution becomes
identically zero after a finite interval of time. In other words, the
propagating waves have sharp aft fronts. For Maxwell's equations though,
even if the currents that drive the field are compactly supported in
time, they may still lead to the accumulation of charges. In that case,
the solution won't have the lacunae per se. We show, however, that the
notion of classical lacunae can be generalized, and that even when the
steady-state charges are present, the waves still have sharp aft fronts.
Yet behind those aft fronts, there is a non-zero electrostatic solution
rather than identical zero. In this case, we refer to quasi-lacunae as
opposed to conventional lacunae.
Quasi-lacunae of Maxwell’s equations can be efficiently exploited in the
numerical context. Indeed, the performance of many well-known methods
used for the treatment of outer boundaries in computational
electromagnetism may deteriorate over long time intervals. The methods
found susceptible to this undesirable phenomenon include various
artificial boundary conditions and perfectly matched layers. We propose
a universal algorithm for correcting this problem. It relies on
quasi-lacunae and the Huygens’ principle, and guarantees a temporally
uniform error bound regardless of either why the deterioration has
originally occurred, or how it actually manifested itself.
In collaboration with S. Petropavlovsky (Moscow, Russia)
Speaker: Daniel Michelson, Department of Computer Science & Applied Mathematics, The Weizmann Institute of Science
Title: Non-linear Stability of Shock Waves for Multi-dimensional Viscous Conservation Laws
abstract
Speaker: Alina Chertock, North Carolina State University
Title: Interaction Dynamics of Singular Wave Fronts Computed by Particle Methods
Abstract:
Some of the most impressive singular wave fronts seen in Nature are the
transbasin oceanic internal waves, which may be observed from a space
shuttle, as they propagate and interact with each other. The
characteristic feature of these strongly nonlinear waves is that they
reconnect whenever any two of them collide transversely. The dynamics of
these internal wave fronts is governed by the so-called EPDiff equation,
which, in particular, coincides with the dispersionless case of the
Camassa-Holm (CH) equation for shallow water in one- and two-dimensions.
Typical weak solutions of the EPDiff equation are contact
discontinuities that carry momentum so that wavefront interactions
represent collisions in which momentum is exchanged. The equation admits
solutions that are nonlinear superpositions of traveling waves and
troughs that have a discontinuity in the first derivative at their peaks
and therefore are called peakons. Capturing these solutions numerically
is a challenging task especially when a peakon-antipeakon interaction
needs to be resolved.
In this talk, I will present a particle method for the numerical
simulation and investigation of solitary wave structures of the EPDiff
equation in one and two dimensions. I will show that the discretization
of the EPDiff by means of the particle method preserves the basic
Hamiltonian, the weak and variational structure of the original problem,
and respects the conservation laws associated with symmetry under the
Euclidean group. I will also present a convergence analysis of the
proposed particle method in 1-D.
Finally, I will demonstrate the performance of the particle methods on a
number of numerical examples in both one and two dimensions. The
numerical results illustrate that the particle method has superior
features and represent huge computational savings when the initial data
of interest lies on a submanifold. The method can also be effectively
implemented in straightforward fashion in a parallel computing
environment for arbitrary initial data.
Speaker: Gadi Fibich, Tel Aviv University
Title: Continuations of the nonlinear Schrodinger equation beyond the singularity
abstract
Speaker: Eli Turkel, Tel Aviv University
Title: High Accuracy Solution of the Helmholtz Equation in General Shaped Domains and Interfaces
abstract
Speaker: Moshe Goldberg, Technion
Title: The $Theta$-Method for Parabolic Initial-Value Problems
Abstract:
In this talk we present an updated review of a well known family
of finite-difference schemes for parabolic initial-value problems, known
as the $theta$-method.
Speaker: Wai Sun Don, Hong Kong Baptist University
Title: High Order Weighted Essentially Non-Oscillatory WENO-Z schemes for Hyperbolic Conservation Laws
abstract
Speaker: Zeev Schuss, Department of Mathematics, Tel-Aviv University
Title: THE NARROW ESCAPE PROBLEM
Abstract:
The narrow escape problem in diffusion theory, which goes back to Lord Rayleigh, is to calculate the mean first passage time, also called the narrow escape time (NET), of a Brownian particle to a small absorbing window on the otherwise reflecting boundary of a bounded domain. The renewed interest in the NET problem is due to its relevance in molecular biology and biophysics. The small window often represents a small target on a cellular membrane, such as a protein channel, which is a target for ions, a receptor for neurotransmitter molecules in a neuronal synapse, a narrow neck in the neuronal spine, which is a target for calcium ions, and so on. The leading order singularity of the Neumann function for a regular domain strongly depends on the geometric properties of the boundary. It can give a smaller contribution than the regular part to the absorption flux through the small window when it is located near a boundary cusp. We find the dependence of the absorption flux on the geometric properties of the domain and thus reveal geometrical features that can modulate the flux. This indicates a possible way to physiologically code information.
Joint work with Amit Singer (Department of Mathematics and PACM, Princeton) and D. Holcman and N. Hoze
(Département de Mathématiques et de Biologie, Ecole Normale Supérieure, France)
Speaker: Doron Levy,
Department of Mathematics and Center for Scientific Computation and Mathematical Modeling
University of Maryland, College Park
Title: Mathematical Models of Leukemia, Cancer Stem Cells, and Drug Resistance
Abstract:
Leukemia is a cancer of the blood that is characterized by an abnormal
production of white blood cells. Traditional approaches for treating
leukemia combine chemotherapy, radiotherapy, and bone marrow (or stem
cell) transplants. The treatment of Chronic Myelogenous Leukemia (CML)
was revolutionized over the past decade with the introduction of new
molecular-targeted drugs. Unfortunately, these drugs keep many patients
in remission but do not cure the disease.
In this talk we will also discuss our recent work on mathematical models
of cancer stem cells and their role in developing drug resistance. When
combined with clinical and experimental data, our mathematical analysis
of drug resistance provides new insights on how to approach treating
CML. This is a joint work with Peter Kim, Cristian Tomasetti, and Peter
Lee.
Speaker: Michael Sever,
Hebrew University, Jerusalem
Title: Higher order differencing of some second order hyperbolic systems
Abstract:
Leapfrog time differencing results in high order accuracy for a class
of symmetric hyperbolic systems. We show how coefficients depending
smoothly on the space variables is compatible with this approach and with
a minimal stencil for the space discretization.
The order of accuracy survives low regularity of the solution caused
by low regularity initial data. For some examples of nonsmooth
coefficients, some improvement in acuracy can be recovered by careful
space discretization.
Speaker: Dalia Fishelov,
Afeka- Tel-Aviv Academic College of Engineering
Title: The discrete biharmonic equation- optimal convergence in one dimension.
Abstract:
A fourth-order compact scheme for the one-dimensional biharmonic
problem has recently been used as a primary building block in discrete
approximations to the
two-dimensional biharmonic operator. In this talk it
is analyzed in detail, taking into account the effect of the boundary
points.
The elliptic property of (sharp) coercivity is verified for the
discrete operator; its interpretation in terms of stability
of the scheme leads (by an energy method) to a proof of convergence
of the discrete solution to the continuous one.
However, due to the low order of the truncation error near the
boundary, this proof yields a non-optimal rate of
convergence.
Using a matrix representation of the discrete
biharmonic operator and studying carefully its
spectral properties, optimal (fourth-order) convergence is established.
Speaker: Michael Bialy,
Tel-Aviv University
Title: Rich quasi-linear systems which are not in the evolution form.
Abstract:
The aim of this talk is to consider quasi-linear systems of conservation
laws which are not in the form of evolution equations. We propose a new,
more general, condition of Richness for such a
system and prove that the blow up analysis along characteristic curves can
be performed for it in an analogous manner. This opens a possibility to use
this ansatz not only for Hydro-dynamics but also for geometric problems.
Speaker: Jan Hesthaven,
Brown University
Title: Absorbing layer methods for the low-speed compressible Navier-Stokes equations.
Abstract:
While the development of perfectly matched layer (PML) methods have
had a profound impact on the ability to accurately model complex
linear wave problems in open domains, the general extension of such
techniques to nonlinear wave problems remains elusive.
In this talk we show how taking a step back and considering the BGK
approximation for Boltzmann equation suggests a path that allows the
development of PML style layers for the compressible Navier-Stokes
equations. While we shall focus on the low-speed case, it appears that
similar ideas are applicable to the transonic case with minor changes.
We shall discuss the different elements of these developments and
illustrate the performance of the absorbing layers for a number of
preliminary benchmarks.
This work is done in collaboration with Z. Gao (Ocean University of
China) and T. Warburton (Rice University).
Speaker: Hillel Tal-Ezer
Title: Highly Accurate Algorithm for Time-Dependent PDEs
abstract
Speaker: Sigal Gottlieb, UMass Dartmouth
Title: The development and state of the art of strong stability preserving methods.
Abstract:
Strong stability preserving time discretizations were developed for use with total variation
diminishing spatial discretizations, and proven useful for the time evolution of ordinary differential
equations resulting from a method-of-lines discretization of hyperbolic PDEs. In this talk,
I describe the historical development of strong stability preserving methods, and the current state
of the art in this field.
Speaker: Steve Schochet, Tel-Aviv University
Title: Remarks on CFL conditions and on Nonlinear Stability
Abstract:
In the first part, some precise CFL conditions will be explicitly
computed.
In the second part, an improved convergence theory for finite
difference schemes for quasilinear symmetric hyperbolic systems will
be described. The theory requires only the same smoothness needed
for the existence theory of the PDEs themselves. Most of the results in this part
will appear in the forthcoming PhD thesis of Liat Even-Dar Mandel.
Speaker: Adi Ditkowski
Title: Analysis of the Du Fort-Frankel methods
abstract
Speaker: David Levin, Tel-Aviv University
Title: Extrapolation Models for Acceleration and Extension
Abstract:
We discuss the role of linear models for two extrapolation problems.
The first is the extrapolation to the limit of infinite series, i.e.
convergence acceleration. The second is an extension problem: Given
function values on a domain D, possibly with noise, we would
like to extend the function to a larger domain.
In addition to smoothness at the boundary of D, the extension on
should also resemble behavioral trends of the
function on D, such as growth and decay or even oscillations. In
both problems we discuss the univariate and the bivariate cases, and
emphasize the role of linear models with varying coefficients.
