by
Paul Muir
Saint Mary's University
Coauthors: Christina Christara, University of Toronto,
Mathematical
models in a great majority of applications involve the numerical
solution of partial differential equations (PDEs). Essentially all
such models are sufficiently complicated that the associated PDEs
typically must be solved using sophisticated numerical algorithms.
This minisymposium will bring together researchers who will discuss
recent developments in new computational algorithms for the numerical
solution of PDEs including highorder collocation methods in one
and two spatial dimensions, domain decomposition algorithms, and
symplectic finitedifference timedomain schemes, applied to problems
including modelling of intracellular signalling pathways (mathematical
biology), pricing of American options (computational finance), and
the numerical solution of Maxwell's equations (computational electromagnetism).
Confirmed
speakers:
Zhi
Li, Saint Mary's University
Bspline collocation software for 2D Parabolic PDEs, with
application to Cell Biology
In
this talk, we describe new software, BACOL2D, for solving two
dimensional timedependent parabolic partial differential equations
(PDEs), defined over a two dimensional rectangular region. The
numerical solution is represented as a bivariate piecewise polynomial
(using tensor product Bspline bases) with unknown time dependent
coefficients. These coefficients are determined by imposing collocation
conditions, i.e., by requiring the numerical solution to satisfy
the PDE at a number of points within the spatial domain. This
leads to a large system of timedependent
differential algebraic equations (DAEs), which we solve using
the high quality DAE solver, DASPK 2.0. BACOL2D is natural extension
of the onedimensional PDE solver BACOL and many of the algorithms
employed in BACOL are applicable to BACOL2D. We also describe
an algorithm for a fast block LU solver with modified alternate
row and column elimination with partial pivoting for the treatment
of the almost block diagonal linear systems that arise during
the numerical solution of the DAEs. We will also briefly consider
numerical results to demonstrate convergence rates for the collocation
solution and, in particular, the existence of superconvergent
points that may be useful for error estimation. We also demonstrate
the use of BACOL2D to solve a 2D cell biology model.
Duy
Minh Dang, University of Waterloo
Adaptive and highorder methods for pricing American options
We
develop spacetime adaptive and highorder methods for valuing
American options using a partial differential equation (PDE) approach.
The linear complementarity
problem arising due to the free boundary is handled by a penalty
method. Both finite difference and finite element methods are
considered for the space discretization of the PDE, while classical
finite differences, such as CrankNicolson, are used for the time
discretization. The highorder discretization in space is based
on an optimal finite element collocation method, the main computational
requirements of which are the solution of one tridiagonal linear
system at each time step, while the resulting errors at the gridpoints
and midpoints of the space partition are fourthorder. To control
the space error, we use adaptive gridpoint distribution based
on an error equidistribution principle. A time stepsize selector
is used to further increase the efficiency of the methods. Numerical
examples show that our methods converge fast and provide highly
accurate options prices, Greeks, and early exercise boundaries.
Dong Liang, York University,
New Energy Conservation Properties and EnergyConserved SFDTD
Schemes for Maxwell's Equations
Computational
electromaganetics has been playing a more and more important role
in many areas of electromagnetic industry, such as radio frequency,
microwave, integrated optical circuits, antennas, and wireless
engineering, etc. It is of special importance to develop efficient
highorder methods for effective and accurately simulating propagation
of electric and magnetic waves in large scale field and long time
duration. However, most previous ADI or splitting schemes break
the energy conservation of elctromegnetic waves. In this talk,
I will first give new energy conservation properties of electromagnetic
waves in lossless medium, and then provide our newly developed
energyconserved SFDTD schemes for Maxwell's equations. Both
theoretical analysis and numerical experiment will be presented
to show the efficiency of the new schemes.
Hans
De Sterck, University of Waterloo
Global approximation of singular capillary surfaces: asymptotic
analysis meets numerical analysis
Singular
capillary surfaces in domains with sharp corners or cusps are
well studied and the asymptotic series approximation of the solution
is known. However, the asymptotic series approximation is only
valid in a sufficiently small neighbourhood of the singularity,
and we wish to obtain a global approximation of the solution through
finite element approximation. Yet it is also known that the singularity
of the solution spoils the accuracy of standard finite element
approximations, which cannot reproduce the singularity accurately.
We show that an accurate numerical approximation can be obtained
through an appropriate change of variable combined with a change
of coordinates, motivated by the known
asymptotic behaviour. Using this accurate numerical approximation
methodology, we can numerically confirm the validity of the known
asymptotic expansions in great detail, and we can make two conjectures
on asymptotic behaviour of singular capillary surfaces at a cusp
for two open cases.
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