**Yasunori Aoki**, University of Waterloo

*High-order Finite Volume Element Method for Laplace Young Capillary
Equation*

Our aim is to construct a fast converging numerical method for
the Laplace-Young Capillary Equation (LYE), which is a scalar nonlinear
Partial Differential Equation (PDE) of elliptic type. We have extended
the Linear Finite Volume Element Method described by Scott, Sander
and Norbury to a high-order method. For bounded solutions, the high-order
FVEM leads to the expected improvement in numerical convergence
order. However, the solution of the LYE is not necessarily bounded,
and our standard method does not lead to improved order of convergence
for these singular solutions. Numerical convergence studies are
presented and avenues for further research are discussed.

**Dale Connor**, University of Waterloo

*Applying WENO Ideas to Interpolation of Two-dimensional Curves*

We propose a method for interpolating a two-dimensional curve using
Euler spirals. The algorithm uses an adaptive stencil to prevent
unwanted oscillations from appearing in the approximation. The interpolating
curve is C1 in well-behaved regions, and is C in regions poorly
approximated by a smooth curve. This allows the approximation to
accurately model shapes which are smooth with the exception of specific
points. The algorithm uses a lower order technique to approximate
the tangents at each point, these tangents are required to define
a unique Euler Spiral which is used as the interpolating segment
connecting consecutive points. Since the interpolation basis consists
of geometric shapes, it is well suited to interpolation of both
one- and two-dimensional problems.

**Georges Djoumna**, University of Waterloo

*The influence of numerical quadrature on the characteristic-based
advection schemes*

The advection equation remains difficult to solve numerically due
to the fact that the transport process takes place along the characteristic
lines and that the information comes from the past. A semi-Lagrangian
temporal discretization of the total derivative is used to obtain
the characteristic-Galerkin formulation. It is shown that tracking
the characteristics backward from the quadrature nodes provides
an accurate treatment of the advection terms. Numerical stability
and analytical results quantifying the amount of artificial viscosity
induced by the method are presented in the case of the one dimensional
linear advection equation, based on the modified equation approach.
Is the order of convergence in the absence of numerical integration
unaltered by the effect of numerical integration? This question
is under investigation, some preliminary results are analyzed. Key
Words: advection equation; characteristic backward tracking; modified
equation; finite elements; semi-Lagrangian methods.

**Lucian Ivan**, University of Toronto Institute for Aerospace
Studies

*Adaptive High-Order Central ENO Method for Hyperbolic Conservation
Laws*

High-order schemes are currently being actively pursued in an effort
to reduce the cost of large-scale scientific computing applications.
Moreover, for numerical simulations of physically complex flows
having a wide range of spatial and temporal scales both high-order
discretizations and adaptive mesh refinement (AMR) are often demanded.
For hyperbolic conservation laws the challenge has been to achieve
accurate discretizations while coping in a reliable and robust fashion
with discontinuities and shocks. A high-order central essentially
non-oscillatory (CENO) finite-volume procedure with adaptive mesh
refinement is presented for the solution of hyperbolic systems of
equations. The CENO spatial discretization is based on a hybrid
solution reconstruction procedure that combines the unlimited high-order
k-exact least-squares reconstruction technique of Barth based on
a fixed central stencil with a monotonicity preserving limited piecewise
linear least-squares reconstruction algorithm. Switching in the
hybrid procedure is determined by a solution smoothness indicator
that indicates whether or not the solution is resolved on the computational
mesh. The limited reconstruction procedure is applied to computational
cells with under-resolved and/or non-smooth solution content and
the unlimited k-exact reconstruction scheme is used for cells in
which the solution is fully resolved. An h-refinement criterion
based on the solution smoothness indicator is defined and used to
control refinement of the body-fitted multi-block AMR mesh.

**James McDonald**, University of Toronto

*Realizable Hyperbolic Moment Closures for Gaseous Flows *
First-Order Hyperbolic partial differential equations describing
viscous heat-conducting gas behaviour, such as those resulting from

moment methods of gaskinetic theory, have several computational
advantages over traditional fluid-dynamic equations, such as the

Navier-Stokes equations (which are partially elliptic in nature).
For example, their first-order nature leads to the narrowest possible

stencil requirements and make computational solutions less sensitive
to grid irregularities. One hierarchy of moment systems which seem
to

have many desirable mathematical properties are those which assume
the distribution function is always that which maximizes entropy

while describing a given set of moments. These maximum-entropy systems,
however, suffer from a major drawback: there exist physically

realizable moment values for which a maximum-entropy distribution
function does not exist. In these regions the moment equations

breakdown and become ill-posed. A remedy for this problem consisting
of a perturbation to the assumed form of the distribution function

which expands the region of realizability of the closure to include
all physically possible states is presented.

**Ruibin Qin**, University of Waterloo

*A Ghost Cell Approach to Imposing Solid Wall Boundary Conditions
on Cut Cells*

We propose a new technique for implementation of reflective solid
wall boundary conditions for solutions of one-dimensional Euler
equations using the discontinuous Galerkin method. We consider a
case where the cell adjacent to the solid wall is much smaller than
regular interior cells. This results in a very small global time
step when an explicit time integrator is used. Such situations frequently
arise with embedded boundary methods. A technique for reconstruction
of the solution on an extended ghost cell is proposed which allows
us to use a regular time step defined by inner cells throughout
the domain. Numerical examples demonstrate validity of the approach.

**Scott Rostrup**, University of Waterloo

*Simulating the Shallow Water Equations on Accelerator Architectures*

Though originally designed with video games and graphics, accelerator
architectures such as the Cell Processor (PS3) and GPUs (Graphics
Processing Unit) are beginning to be adopted by the scientific,
financial, and engineering communities. They are highly parallel
architectures suitable for a wide array of data-parallel computing
tasks including the numerical methods suitable for

simulating hyperbolic partial differential equations. These architectures
are explored as accelerators for an explicit wave based Godunov
scheme applied to the shallow water equations.

**Christopher Subich**, University of Waterloo

*High order moving mesh methods*

High order spectral or pseudospectral methods have excellent convergence
properties for differential equations with smooth solutions, but
their accuracy deteriorates in the presence of sharp, local features
-- most notably shocks. In contrast, moving finite difference methods
allow for dynamic resolution of sharp features, but finite difference
methods are inherently lower-order. This work unifies both approaches,
creating a uniformly high-order moving mesh method to solve the
viscous Burgers equation to high accuracy.

**L.T. Tran**, University of Utah

*An Improved ICE Method for Compressible Flow Problems in Uintah*

The Implicit Continuous-fluid Eulerian(ICE), a semi-implicit finite-volume
solver, is used for simulating problems in multiphase flow which
span a wide area of science and engineering. ICE is utilized by
the C-SAFE code Uintah at the University of Utah to simulate explosions,
fires and other fluid and fluid-structure interaction phenomena.
The implementation of ICE in the CSAFE code Uintah invokes operator
splitting in which the solution consists of a separate Lagrangian
phase and Eulerian phase. The choices of operators in these phases
effect the behavior of the ICE numerical solution. We discuss the
implementation of ICE method for the numerical solutions to systems
of conservation laws. The implementation of ICE used in Uintah is
given in many papers by Kashiwa at Los Alamos and extended to solve
multifield cases by Harman at Utah. In its original form the ICE
algorithm does not perform as well as the best current methods for
compressible flow problems. The poster to be presented shows how
the ICE algorithm may be modified so that it provides good solutions
for compressible high speed flow. This involves using limiters for
calculating fluxing velocities at faces, and applying a gradient
limiter

for approximating the advected quantities among cells. Computational
results for a number of test problems are used to demonstrate the
effectiveness of the approach.

Acknowledgement: This work was supported by the University of Utah's
Center for the Simulation of Accidental Fires and Explosions (C-SAFE)
and funded by the Department of Energy under subcontract No. B524196.

Joint work with M.Berzins