


Computational and Analytical Techniques in Modern Applications / Techniques numériques et analytiques dans les
applications modernes (Org: Peter Minev)
 JOHN BOWMAN, University of Alberta
The dual cascade in bounded and unbounded twodimensional fluids

Around 1967, Kraichnan, Leith, and Batchelor (KLB) independently proposed
the dual cascade theory, which is thought to describe turbulence in
unbounded twodimensional fluids. In a bounded domain, however, the upscale
energy cascade they discussed will be halted at the lowest wavenumber
(corresponding to the domain size). An upper bound on the ratio of the
total enstrophy to total energy derived by Tran and Shepherd [Physica
D, 2002] establishes that the energy must be dissipated at scales larger
than the forcing scale. This result is based on the assumption that the
square root of the ratio of mean enstrophy to mean energy injection is
spectrally localized to the forcing region. We investigate the conjecture
that turbulence driven by a spectrally localized temporally whitenoise
random forcing satisfies this assumption. We also provide numerical
evidence that energetic reflections at the lower spectral boundary may
eventually lead to a largescale k^{3} energy spectrum, in agreement
with the largescale k^{3} spectra observed in the atmosphere by Lilly
and Peterson [Tellus 35A, 379 (1983)]. A spectral constraint
derived by Tran and Bowman [Physica D, 2003] establishes that the two
inertialrange exponents must sum to 8. A largescale k^{3} spectrum
resulting from reflections at the lower spectral boundary would then
explain the smallscale k^{5} spectrum frequently observed in numerical
simulations of the enstrophy range. We propose that combined supergrid and
subgrid models based on Kolmogorov's hypothesis of selfsimilar energy (or
enstrophy) transfer could be used to mimic the behaviour of an unbounded
fluid in a doubly periodic domain, thereby allowing one to address the
validity of the classical KLB theory for unbounded fluids.
 LIQUN CAO, Institute of Computational Mathematics and ScienceEngineering
Computing, Academy of Mathematics and System Sciences,
Chinese Academy of Sciences, Beijing 100080, China
Multiscale mathematical methods and numerical simulations
in composite materials and porous media

The lecture begins with a short review of physical background of
composite materials and porous media like periodicity or randomness
(from static and kinetic mechanical problems of composite materials,
heat and mass transfer in porous media, wave propagation in heterogeous
media, etc.), and then introduces several recent mathematical
results on the asymptotic homogenization methods. In particular, I
would like to advance our methods and numerical results for the above
physical problems. Finally, some open problems are also presented in
corresponding sections.
 GRAEME FAIRWEATHER, Colorado School of Mines
Orthogonal spline collocation methods for partial
integrodifferential equations in two space variables

New efficient algorithms are formulated and analyzed for the solution
of a class of linear partial integrodifferential equations of
parabolic type in the unit square. In these methods, orthogonal
spline collocation (OSC) with C^{1} piecewise polynomials of degree
³ 3 is used for the spatial discretization. For the time stepping,
alternating direction implicit (ADI) methods based on the backward
Euler method, the Crank Nicolson method and the second order BDF scheme
are considered. Such methods reduce the multidimensional problem to
sets of independent onedimensional problems in which the OSC matrices
are easily determined since, unlike the finite element Galerkin case,
no integrals must be evaluated or approximated. The methods are shown
to be of first or second order accuracy in time and of optimal order
accuracy in the L^{2}, H^{1} and H^{2} norms in space. From the
analysis, a new optimal order H^{2} estimate is obtained for an ADI OSC
Crank Nicolson method for the heat equation in two space variables.
ADI OSC methods are also examined for the solution of a class of
evolution equations with a positive type memory term, and an optimal
order L^{2} estimate is derived for each method.
This is joint work with Amiya Pani and Bernard Bialecki.
 MARINA GAVRILOVA, University of Calgary, Calgary, Alberta
Exact computation library development: challenges
in manipulating floatingpoint numbers

There are numerous challenges faced by scientists when developing
algorithm libraries for scientific applications. Issues of
functionality, stability, performance, and flexibility depending on the
specific application areas are at the focus of researchers working in
both theoretical and applied areas. This talk discusses some of the
approaches to development and implementation of two and
threedimensional data structures for applications in molecular
biology, mechanical engineering and GIS. The talk concentrates on
issues of algorithm efficiency, precision of the result and numerical
stability. Challenges in the development of the ECL (Exact
Computational Library) for performing exact computations in the fixed
precision floatingpoint arithmetic are discussed. The ECLibrary is
based on the interval point arithmetic, iterative approximation
methods, reduction technique and algorithms for performing complex
transformations on floatingpoint numbers. The ESAE algorithm (Exact
Sign of Algebraic Expression) will be described. Some implementation
issues will be also discussed.
 ABBA GUMEL, Department of Mathematics, University of Manitoba, Winnipeg,
Manitoba R3T 2N2
Nonstandard finitedifference methods for some reallife problems

The use of standard numerical discretization techniques, such as
explicit RK methods, to integrate nonlinear differential equations
often leads to schemedependent instabilities and/or convergence to
spurious solutions when certain stepsizes or parameter values are used
in their simulations. This paper presents some nonstandard
finitedifference methods that are, in general, free of the
aforementioned drawbacks. These schemes are designed in such a way
that they preserve the important features/properties of the continous
model they approximate.
 DONG LIANG, Department of Mathematics and Statistics, York University, Toronto,
Ontario M3J 1P3
Nonstandard upwinding finite covolume methods for the convection
diffusion problems

The convection diffusion equations, which describe many realistic
procedures in many problems of science and technology; eg., fluid
mechanics, heat and mass transfer, groundwater modelling, petroleum
reservoir simulation and environmental protection, are very important
and difficult in numerical simulation. The standard finite difference
methods or finite element methods will introduce severe nonphysical
oscillations into the numerical solutions since the corresponding
discrete schemes are unstable for the problems. Because of satisfying
both the stability and the conservation of mass, the methods of
finitevolumetype with upwinding techniques have obtained high
successes in the numerical simulation of the convection diffusion
problems. However, the standard upwinding technique treating convection
terms usually derives lowerorder accuracy schemes for the problems. In
this talk, we will present the nonstandard highorder upwinding finite
covolume methods for the convection diffusion problems. The
conservation law of mass and the unconditional stability are analyzed,
the highorder error estimates are obtained for the methods. Numerical
experiments are given to demonstrate the performance of the schemes.
 TAO LIN, Virginia Tech, Blacksburg, Virginia 24060
An immersed finite element method for axial symmetric
3D nonlinear interface problems

We will discuss an immersed finite element method for axial symmetric
three dimensional nonlinear interface problems. The basis functions in
this method are piecewise linear polynomials satisfying the jump
conditions approximately (or even exactly in many situations). In
addition, the mesh in this method does not have to be aligned with the
interface because the interface is allowed to pass through the
elements. Therefore, structured Cartesian meshes can be used in this
method to facilitate efficient numerical solutions. We will show that
this method has the usual second and first order convergence rates in
L^{2} and H^{1} norms, respectively. Numerical examples for a nonlinear
interface problem arising from ion optics modelling in composite
structures will be provided to illustrate features of this method.
 DONGWOO SHEEN, Seoul National University
Nonconforming elements on quadrilaterals

In this talk we will survey on recent results on nonconforming finite
elements on quadrilaterals (http://www.nasc.snu.ac.kr/). These
elements, originally developed for elliptic problems, are applied to
solving Stokes, elasticity, Helmholtz, and Maxwell's equations. Error
estimates and selected numerical results will be shown.
 S. SIVALOGANATHAN, University of Waterloo, Waterloo, Ontario N2L 3G1
Tracking the motion of the vevtricular wall in shunted
hydrocephalus

Although interest in the biomechanics of the brain goes back over
centuries, mathematical models of hydrocephalus and other brain
abnormalities are still in their infancy and a much more recent
phenomenon. This is rather surprising, since hydrocephalus is still an
endemic condition in the pediatric population. Treatment has
dramatically improved over the last 30 years thanks to the introduction
of CSFshunts. This too, however, is not without problems and the shunt
failure rate at 2 years post shunt insertion is 50common causes of shunt failure is due to shunt obstruction. Common
sense suggests that the optimal shunt location would be in the
ventricular region that remains largest after ventricular decompression
drainage. In this talk, we will report on some recent progress towards
the solution of this problem.
 MANFRED TRUMMER, Department of Mathematics, Simon Fraser University, Burnaby,
British Columbia V5A 1S6
Adaptive multiquadric collocation for boundary layer problems

An adaptive collocation method based upon radial basis functions is
presented for the solution of singularly perturbed twopoint boundary
value problems. We combine a multiquadric integral formulation with a
previously employed coordinate stretching technique. A new error
indicator function accurately captures the regions of the interval with
insufficient resolution. This indicator is used to adaptively add data
centres and collocation points. Our method resolves extremely thin
layers accurately with fairly few basis functions. We demonstrate the
effectiveness and the robustness of our new method on a number of
examples.
Joint work with Leevan Ling, SFU.
 H. VAN ROESSEL, Alberta

 ERIK VAN VLECK, University of Kansas
Spatially discrete models of nerve impulses

We consider spatially discrete FitzHughNagumo equations as a model for
ionic conductances that generate the action potential of nerve fibers
in motor nerves of vertebrates. Existence results for front and pulse
solutions are discussed. Numerical techniques are considered to
approximate solutions to mixed type functional differential equations
obtained when considering traveling fronts and pulses of these
equations. Extensions to coupled systems of spatially discrete
FitzHughNagumo equations corresponding to bundles of nerve fibers will
be discussed.
 ZHIMIN ZHANG, Wayne State University
A posteriori Error estimator based on polynomial preserving
recovery

A new gradient recovery technique (PPR) is introduced to construct a
posteriori error estimator. The recovery operator is polynomial
preserving and insensitive to mesh distortion. Some theoretical results
are provided regarding the recovery operator. In addition, some
numerical results are provided in comparison with the ZienkiewiczZhu
error estimator based on the superconvergent patch recovery (SPR).

