# Abstracts – invited speakers *(in the order of the presentations)*

**Ian Sloan **(I.Sloan@unsw.edu.au)
University of New South Wales, Australia

*Title: *PDE with random coefficients – a
high-dimensional problem

*Abstract: *This talk describes recent
computational developments in partial differential equations with random
coefficients treated as a high-dimensional problem. The prototype of such
problems is the underground flow of water or oil through a porous medium, with
the permeability of the material treated as a random field. (The stochastic
dimension of the problem is high if the random field needs a large number of
random variables for its effective description.) There are many approaches to
the problem, ranging from the polynomial chaos method initiated by Norbert
Wiener to the Monte Carlo and (of particular interest to my group) Quasi-Monte
Carlo methods. In recent years there have been significant progress in the
development and analysis of algorithms in these areas.

**Rob Falgout **(rfalgout@llnl.gov)
Lawrence Livermore National Laboratory, USA

*Title: *Multigrid Methods: Scalable
algorithms for extreme-scale computing

*Abstract: *Multigrid methods are
important techniques for efficiently solving huge linear systems and they have
already been shown to scale effectively on parallel computers with millions of
cores. Future exascale architectures will require solvers to exhibit even higher
levels of concurrency (1B cores), minimize data movement, exploit machine
heterogeneity, and demonstrate resilience to faults. While considerable
research and development remains to be done, multigrid approaches are ideal for
addressing these challenges. In this talk, we will introduce the multigrid
method and discuss its essential features. We will begin with classical
geometric multigrid and then move on to algebraic multigrid (AMG). We will also
discuss the added complexity of developing parallel multigrid methods and
software, especially in the context of the exascale machines on the horizon,
and touch on some current research topics as well.

**Remi Abgrall **(remi.abgrall@math.uzh.ch) Universität Zürich, Switzerland

*Title: *Recent advances in numerical
approximation of hyperbolic systems: some history and future

*Abstract: *In this talk, I will explain
some problems arising from the numerical approximation of hyperbolic systems by
means of finite element-like techniques. In order to give the ideas, I will
mostly focus on simple scalar problems, and show the extension to systems in
the end. I will first start by some simple facts on these problems, recall some
classical techniques and explain what are the issues. In a second part, I will
provide some hints on recent advances: parameter free methods, unsteady
problems, shock stabilisation.

**Jacob Schroeder
**(schroder2@llnl.gov) Lawrence
Livermore National Laboratory, USA

*Title: *Multigrid Reduction in Time: A
flexible and scalable approach to parallel-in-time

*Abstract: *The need for parallel-in-time
algorithms is currently being driven by the rapidly changing nature of computer
architectures. Future speedups will come through ever increasing numbers of
cores, but not faster clock speeds, which are stagnant. Previously, increasing
clock-speeds could compensate for traditional sequential time stepping
algorithms as the problem size increased. However this is no longer the case,
leading to the sequential time integration bottleneck and the need to
parallelize in time. In this talk, we examine an optimal-scaling parallel time
integration method, multigrid reduction in time (MGRIT). MGRIT applies
multigrid to the time dimension by solving the (non)linear systems that arise
when solving for multiple time steps simultaneously. The result is a versatile
approach that is nonintrusive and wraps existing time evolution codes. MGRIT
allows for various time discretizations (e.g., Runge-Kutta and multistep) and
for adaptive refinement/coarsening in time and space. Nonlinear problems are
handled through full approximation storage (FAS) multigrid. Some recent
theoretical results, as well as practical results for a variety of problems
will be presented, e.g., explicit/implicit time integration, nonlinear
diffusion and compressible Navier-Stokes.

**Neela Nataraj **(neela@math.iitb.ac.in) Indian Institute
of Technology Bombay, India

*Title: *Finite element methods for Plate
Bending Problems

*Abstract: *In this talk, after giving a
brief introduction to some plate models,
we consider the von Karman equations that describe the bending of thin elastic
plates. Conforming and non-conforming finite element methods are employed to
approximate the displacement and Airy stress functions. Techniques for deriving
optimal order theoretical error estimates are explained. The results of
numerical experiments which justify the theoretical estimates are presented.

**Clinton Groth **(groth@utias.utoronto.ca)
University of Toronto, Canada

*Title: *High-Order Anisotropic Adaptive
Mesh Refinement Finite-Volume Schemes for Multi-Scale Physically-Complex Flows

*Abstract: *A family of high-order central
essentially non-oscillatory (CENO) finite-volume schemes with adaptive mesh
refinement (AMR) are described for the prediction of a range of multi-scale
physically-complex flows having both disparate and anisotropic spatial
scales. The CENO schemes are based on a
hybrid solution reconstruction procedure that combines an unlimited high-order
k-exact, least-squares reconstruction technique, following from a fixed central
stencil, with a monotonicity preserving limited piecewise linear least-squares
reconstruction. Switching in the hybrid
procedure is determined by a solution smoothness indicator that detects whether
or not the solution is accurately represented on the mesh. The solution
smoothness indicator can also used in the formulation of refinement criteria
for directing mesh adaptation. The
proposed approach avoids some of the complexities associated with the original
essentially non-oscillatory (ENO) and other weighted ENO (WENO) schemes and is
thereby well suited for solution reconstruction on irregular and unstructured
mesh. The development of the high-order
finite-volume approach for both multi-block body-fitted and more generally
unstructured meshes in three dimensions is considered. In the case of the former, the scheme has
been developed and applied in conjunction with an efficient and highly scalable
anisotropic AMR that uses an unstructured binary tree hierarchical data
structure to permit local anisotropic refinement of the grid in a preferred
coordinate direction. The anisotropic
AMR scheme and block connectivity permits coarsening of the grid blocks in a
manner that is independent of refinement history and allows the mesh to rapidly
re-adapt for unsteady applications.
Applications will be discussed for a range of problems including
high-speed compressible flows, viscous incompressible and compressible flows,
as well as reactive flows. The potential of the combined CENO and anisotropic
AMR schemes for the simulation of physically-complex flows in an efficient and
accurate manner will be demonstrated.

**Markus Hegland **(Markus.Hegland@anu.edu.au)**
**The Australian National University, Australia

*Title: *A review of the sparse grid
combination technique for the solution of partial differential equations

*Abstract: *The sparse grid combination
technique uses extrapolation to enhance the performance for given numerical
solvers. In particular the approximation takes the form

where u(γ) is the given numerical solution with numerical parameters γ and
where c_{γ} are scalar factors. This method is in particular well
suited for parallel computing and high-dimensional problems. In the talk this
approach will be illustrated for the solution of the gyrokinetic equations of
plasma physics based on the given solver GENE developed at the TU Munich. The
performance of the method depends on the choice of the parameters γ and the
coefficients. It will be seen how the choice of the γ and the c_{γ}
leads to quasi optimal approximations and may even be used to deal with
computer hardware faults. We will also consider the solution of elliptic PDEs
and eigenvalue problems and if time permits to PDEs originating in the
determination of density estimators and machine learning.

**Jörg Frauendiener **(joergf@maths.otago.ac.nz)** **University of Otago, New Zealand

*Title: *Computational Gravity

*Abstract: *Computational gravity is the
part of computational physics that is concerned with the solution of Einstein's
field equations of general relativity. This is a theory describing space, time
and matter on a very fundamental and geometrical level. The geometric aspect of
the theory entails several problems for the numerical treatment. In this talk I
will discuss these fundamental issues and show some of the applications that
have been developed over the recent years.

*Title: *The Mimetic Finite
Difference Method and its application to diffusion problem

*Abstract: *Mimetic discretizations
provide a mathematical framework that allows the construction of families of
schemes for the numerical resolution of partial differential equations. From an
historical viewpoint, the mimetic approach dates back to the work on Geometric
Integration of Whitney and the discretization differential operators by a
duality principle from the Russian school of Samarskii. In this talk we review the Mimetic Finite
Difference (MFD) method and its application to diffusion problems [1]. This MFD
method works on unstructured meshes with cells of very general geometric shape,
polygons in 2D and polyhedra in 3D. The construction of the method is based on
the two main concepts of polynomial consistency and local stability, which
respectively determine the order of accuracy of the approximation and the
well-posedness of the discrete problem. In particular, any order of accuracy
can be attained by changing the degree of the polynomials that are used to
formulate the method. The duality principle is used to establish a variational
form of the method, which allows an efficient implementation by a local
construction on each cell and a global assembly as in the finite element
method. A variational reformulation in a finite element setting is possible and
provides an equivalent family of methods called "virtual
elements". The MFD method also
presents very strong connections with other families of numerical schemes whose
design is based on similar principles. These connections are also in discussed
in this talk.

[1] L. Beirao da Veiga, K.
Lipnikov, G. Manzini, "The Mimetic Finite Difference Method for Elliptic
Problems", Modeling Simulations & Applications Series, Springer 2014

# Abstracts – contributed talks *(in the order of the presentations)*

**William McLean** (w.mclean@unsw.edu.au)
Univeristy of New South Wales, Australia

*Title: *Subdiffusion in a nonconvex polygon

*Abstract: *We consider the spatial
discretisation of a time-fractional diffusion equation in a polygonal domain
using continuous, piecewise-linear finite elements. If the domain is convex,
then the method is known to be second-order accurate in L^{2} , but if
the domain has a re-entrant corner then the error analysis breaks down because
the associated Poisson problem is no longer H^{2} -regular. For a
quasi-uniform family of triangulations with mesh parameter h, the error is of
order h^{1+β} if largest re-entrant corner has angle π/β with 1/2 <
β < 1, but a suitable local refinement strategy restores h^{2}
convergence. Analogous results for the classical heat equation were proved in
2006 by Chatzipantelidis, Lazarov, Thomée and Wahlbin.

This is joint work with Bishnu
Lamichhane (Newcastle) and Kim-Ngan Le (UNSW).

**Kyle Talbot **(kyle.talbot@monash.edu)
Monash University, Australia

*Title: *Uniform temporal convergence of
numerical schemes for miscible displacement through porous media

*Abstract: *The single-phase, miscible
displacement through a porous medium of one incompressible fluid by another is
described by a nonlinearly-coupled elliptic parabolic system. Convergence
analyses exist for a variety of methods for the numerical approximation of the
solution to this system, including finite elements, finite volumes and
discontinuous Galerkin. These analyses typically demonstrate that the
approximation to the concentration variable converges in a space-time averaged
sense, e.g. in . I will illustrate that for a family of numerical methods that
includes hybrid finite volumes, mixed finite volumes and mimetic finite
differences, the concentration can be approximated uniformly in time, i.e. in , thereby providing an admissible approximation to the concentration
at any given point in time. This convergence is possible without assuming
uniqueness or regularity of the solution to the continuous problem.

**Johannes Reiner** (j.reiner@uq.edu.au)
University of Queensland, Australia

*Title: *Progressive Failure Modelling in
Composite Laminates

*Abstract: *Reliable and efficient failure
simulation within finite elements (FE) is an ongoing and challenging task. The
Phantom Node Method (PNM) allows for arbitrary modelling of discontinuities
while preserving elemental locality and standard FE techniques. It accounts for
geometrical and material nonlinearities by incorporating the total Lagrangian
formulation for large deformation and the cohesive concept at discontinuity
surfaces respectively. PNM is further extended to simulate different failure
modes and their interaction in composite laminates. It is shown that the
advanced PNM is able to predict typical fracture measures such as crack density
or stiffness reduction with good accuracy. Furthermore, progressive failure
interaction is quantitatively and qualitatively evaluated. Results agree well
with experimental findings.

**Krishna Saxena** (ksax995@aucklanduni.ac.nz)
University of Auckland, New Zealand

*Title: *Finite Element Modelling of
Auxetic Metamaterials

*Abstract: *Auxetic materials exhibit
negative Poisson’s ratio i.e. they display a lateral expansion when stretched
longitudinally and vice versa. The classical theory of elasticity restricts the
negative Poisson’s ratio to be -1 for isotropic solids. On the other hand,
metamaterials can display extreme negative Poisson’s ratio by manipulating the
geometry of unit cell.

A family of 2D and 3D auxetic
structures were created using computer aided design. They were simulated using
Finite Element analysis package Abaqus to determine the presence of negative
Poisson’s ratios and to test if they are a possible solution for potential
applications. The effect of element type on the negative Poisson’s ratio of
these structures was studied. The effect of geometry of unit cell on the
negative Poisson’s ratio of these structures was also investigated using FEM.
These structures were then tested for syncelasticity (out of plane bending)
using FEM. With the use of finite element approach, we will demonstrate how
material properties can be tuned by changing the geometry irrespective of their
composition. The presentation will discuss an overview of finite element
modelling of auxetic materials and structures.

**Duy Minh Dang** (duyminh.dang@uq.edu.au)
University of Queensland, Australia

*Title: *Optimal mean-variance portfolio
allocation: a Hamilton-Jacobi-Bellman PDE approach

*Abstract: *In this talk, we discuss a
numerical Hamilton-Jacobi-Bellman partial differential equation approach for
the mean-variance portfolio allocation problem under jump diffusion models. The
focus of this talk is on how to handle realistic portfolio constraints, jumps
in the risky asset, and a semi-self-financing strategy which involves positive
cash withdrawals but gives superior results in terms of mean-variance criteria.
Tests based on estimation of parameters from historical time series show that
the strategy is robust to estimation ambiguities.

**Quoc Thong Le Gia **(qlegia@unsw.edu.au)
University of New South Wales, Australia

*Title:* Higher order Quasi-Monte Carlo
integration for Bayesian Estimation

*Abstract: *We analyze Quasi-Monte Carlo
numerical integration methods in Bayesian estimation of solutions to parametric
operator equations with holomorphic dependence on the parameters. Such problems
arise in numerical uncertainty quantification and in Bayesian inversion of
operator equations with distributed uncertain inputs, such as uncertain
coefficients, uncertain domains or uncertain source terms and boundary data.

We establish error bounds for
higher order, Quasi-Monte Carlo quadrature for the Bayesian estimation. It
implies, in particular, regularity of the parametric solution and of the
parametric Bayesian posterior density in SPOD weighted spaces. This, in turn,
implies that the Quasi-Monte Carlo quadrature methods are applicable to these
problem classes, with dimension-independent convergence rates O(N^{-1/p})
of N-point HoQMC approximated Bayesian estimates where 0<p<1 depends only
on the sparsity class of the uncertain input in the Bayesian estimation.

This is a joint work with Josef Dick
(UNSW) and Robert Gantner and Christoph Schwab (ETH)

**Michael Feischl** (m.feischl@unsw.edu.au)
University of New South Wales, Australia

*Title: *A posteriori error estimates for
the Eddy-Current-LLG equations

*Abstract: *We analyze a numerical method
for the coupled system of the eddy current equation in three space dimensions
with the Landau-Lifshitz-Gilbert equation in a bounded domain. The unbounded
domain is discretized by means of finite-element/boundary-element coupling.
Even though the considered problem is strongly nonlinear, the numerical
approach is constructed such that only two linear systems per time step have to
be solved. We prove unconditional weak convergence (of a subsequence) towards a
weak solution as well as strong convergence with a priori error estimates if a
sufficiently smooth strong solution exists. In this case, the strong solution
is unique and coincides with each weak solution.

**Alexander Howse** (ajmhowse@gmail.com)
University of Waterloo, Canada

*Title: *Nonlinearly Preconditioned
Optimization on Grassmann Manifolds for Tucker Tensor Approximations

*Abstract: *Two new accelerated
optimization algorithms are presented for computing approximate Tucker tensor
decompositions by minimizing error measured in the Frobenius norm, subject to
orthonormality constraints on factor matrices.

The first is a nonlinearly
preconditioned conjugate gradient (NPCG) algorithm, wherein a nonlinear
preconditioner is used to generate a direction which replaces the gradient in
the standard nonlinear conjugate gradient (NCG) iteration. The second is a
nonlinear generalized minimal residual (N-GMRES) algorithm, in which a linear
combination of past iterates and a tentative new iterate, generated by a
nonlinear preconditioner, is minimized to produce an improved search direction.
The higher order orthogonal iteration (HOOI), the standard workhorse algorithm
for computing approximate Tucker decompositions, is used as the nonlinear
preconditioner in NPCG and N-GMRES.

The Euclidean versions of these
methods are extended to the manifold setting, where optimization over a
Cartesian product of Grassmann manifolds is used to handle orthonormality
constraints and to allow isolated minimizers. A Grassmann manifold, Gr(n,p), is
the set of all p-dimensional subspaces of R^n, and a given element may be
represented by an orthonormal matrix. Several modifications are required for
use on manifolds: logarithmic maps are used to determine required tangent
vectors, retraction mappings are used in the line search update step, vector
transport operators are used to compute linear combinations of tangent vectors,
and the Euclidean gradient is replaced by the manifold equivalent.

Several variants are provided for
the update parameter in NPCG, two for each of the Polak-Ribiere,
Hestenes-Stiefel, and Hager-Zhang formulae. NPCG and N-GMRES are compared to
HOOI, NCG, a limited memory BFGS quasi-Newton algorithm, and a manifold trust
region algorithm using randomly generated and real life tensor data with and
without noise, arising from applications in computer vision and handwritten
digit recognition. Numerical results show that N-GMRES and NPCG with update
parameters determined by modified Polak-Ribiere and Hestenes-Stiefel rules
accelerate HOOI significantly for large tensors, in cases where there are
significant amounts of noise in the data, and when high accuracy results are
required, and are a clear improvement in state-of-the-art methods.

**Linda Stals** (linda.stals@anu.edu.au)
Australian National University, Australia

*Title: *Adaptive refinement recovery
after fault simulation

*Abstract: *The use of adaptive refinement
techniques in combination with finite element methods is well established.
Furthermore, iterative techniques that incorporate information about the grid
structure, such as the multigrid method, have been shown to be a very efficient
approach to solving various types of partial different equations. These
techniques now form an integral part of many sophisticated parallel software
packages. However, the advent of larger and larger parallel machines leads to a
very modern twist of this tale, and that is how to recover if a fault occurs in
one of the processors.

In this talk we present a parallel
adaptive multigrid method that uses dynamic data structures to store a nested
sequence of meshes and the evolving solution. After a fail-stop fault, the data
residing on the faulty processor will be lost. However, the neighboring
processors contain enough information such that a consistent mesh can be
reconstructed in the faulty domain with the goal of resuming the computation
without having to restart from scratch.

I will briefly introduce the
foundation, the proposed mesh refinement methods, accuracy and reliability
verifications, and computational complexity of the new computational methods in
my presentation. The comparisons for singular points and asymptotic lines
between exact and numerical results for analytical velocity fields will be
presented by illustrations. Some of the comparisons between the benchmarks and
numerical results for lid-driven flow will also be provided. A number of
examples and demonstrations are used for explanations. Possible applications in
practice and future research in computational science, computing science and
other relevant disciplines will be introduced at the end. If the time provided
is not enough, I will delete some of the parts listed above in my presentation.

**Jesse Chan** (jchan985@gmail.com) Virginia Tech, USA

*Title: *GPU-accelerated Bernstein-Bezier
DG methods for wave problems

*Abstract: *The computationally intensive
nature of time-explicit nodal discontinuous Galerkin methods is well-suited to
implementation on Graphics Processing Units (GPUs). We evaluate the use of
Bernstein-Bezier bases as an alternative to nodal polynomials for discontinuous
Galerkin discretizations and show how to exploit properties of derivative and
lift operators specific to Bernstein polynomials. Issues of efficiency and
numerical stability are discussed in the context of a model wave propagation
problem, and computational experiments comparing high-order nodal bases and
high-order Bernstein bases are presented.

**Ryan McClarren** (rgm@tamu.edu)
Texas A&M, USA

*Title: *High Fidelity, Moment-Based
Methods for Particle Transport: The confluence of PDEs, Optimization, and HPC

*Abstract: *The calculation of the
transport of particles is important in many applications including rarefied gas
dynamics, plasma physics, and the nuclear energy systems. In this talk I will
motivate the choice of moment-based methods for solving particle transport
problems, and discuss the difficulties such approaches have. To obtain
physically-meaningful solutions the discretization of the original PDEs can
depend on the solution to an optimization problem. I will show how these
optimization problems arise, what methods perform best in terms of cost and
accuracy, and how these problems can be well-suited for high performance
computing.

**Tony Roberts** (anthony.roberts@adelaide.edu.au) University of
Adelaide, Australia

*Title:* Modeling, analysis and scientific
computation of complex multiscale systems

*Abstract: *We are developing a systematic
approach, both analytic and computational, to extract compact, accurate, system
level models of complex physical and engineering systems. The wide ranging
methodology is to develop and support the patch scheme which empowers large
scale simulation and prediction through computations on only small
well-separated patches of microscale simulators. The continuing challenge is to
couple these microscale simulations on microscale patches across un-simulated
space and establish efficiency, accuracy, consistency and stability on the
macroscale. Comprehensively accounting for multiscale interactions between
subgrid processes among macroscale variations ensures stability and accuracy.
In particular, I will discuss meso-time coupling between patches designed for
exascale computing. Based on dynamical systems theory and analysis, our approach
empowers systematic analysis and understanding for optimal macroscopic
simulation for forthcoming exascale computing.

**Yahya Alnashri** (yahya.alnashri@monash.edu)
Monash University, Australia

*Title: *A generic framework for
variational inequalities

*Abstract: *Gradient schemes is a generic
framework, which offers the unified convergence analysis of many conforming and
non conforming numerical methods for 2nd order PDEs.

In this talk I will apply the gradient
schemes framework to different kinds of elliptic variational inequalities which
have various applications, such as fluid dynamics, elasticity, biomathematics.
With the theoretical results of this framework, we can recover the convergence
rate for some methods previously performed for the variational inequalities. I
will also focus on completely new results coming from establishing the Hybrid
Mixed Mimetic method (HMM method) for variational inequalities. Finally, beside
providing test-cases taken for the literature, I will present an entirely
different test-case, which is of an available exact solution.

**Alexander Gilbert** (alexander.gilbert@student.unsw.edu.au)
University of New South Wales, Australia

*Title: *Applying quasi-Monte Carlo
methods to an eigenproblem with a random coefficient

*Abstract: *In this talk the coefficient
depends on a finite, but possibly high, number of stochastic parameters and as
such the eigenvalues and corresponding eigenfunctions will also depend on this
stochasticity. The aim is to approximate the expected value of the principal
eigenvalue by formulating it as a high-dimensional integral, so that
quasi-Monte Carlo (QMC) quadrature may be used. First we discretise in space
using finite element (FE) methods and then apply QMC methods to the FE
approximations. We show that the principal eigenvalue belongs to the spaces
required for QMC theory and provide numerical results.

**Jordan Pitt** (jordan.pitt@anu.edu.au)
The Australian National University, Australia

*Title: *Numerically Solving the 1D Serre
Equations in the Presence of Discontinuities

*Abstract: *The Serre equations are a
shallow water approximation to the incompressible Euler equations that retain
the terms of the Shallow Water Wave Equations while introducing dispersive
terms that make the Serre equations more relevant when wave amplitude is
significant compared to water depth. Most of the literature numerically solves
these equations for smooth initial conditions, however, in real world
applications such as the Dam-Break problem it is important to handle
discontinuous initial conditions.

Thus the numerical scheme of O. Le
Metayer, et.al. (2010) has been extended to build second- and third-order
methods to investigate the capabilities of this scheme in the presence of
discontinuities, which we expect to be good since it utilises a Finite Volume
Method.

These methods were validated and
their order of convergence was confirmed for smooth initial conditions using
the analytic soliton solution. The methods also compared well with the
experimental results of Hammack and Segur (1978) which contains a
discontinuity.

To further investigate the
behaviour of discontinuities a smooth approximation of the Dam-Break problem
was used to observe how the numerical solutions of the smooth dam break problem
behaved as the smooth initial conditions approached a discontinuous change in
water depth. The results of these methods were compared to the results of two
second-order finite difference methods. One being the second-order centred
finite difference approximation to the Serre equations and the other from
Grimshaw, et.al (2006). These schemes showed the same behaviour in the presence
of steep gradients as those derived from the numerical scheme of interest and
included all the observed behaviour for discontinuous and smooth initial
conditions thus observed far in the literature.

References:

Hammack, J. L. and Segur, H.
(1978). “The Korteweg-de Vries equation and water waves.

Part 3. Oscillatory waves.”
Journal of Fluid Mechanics, 84(2), 337–358.

El, G., Grimshaw, R. H. J., and
Smyth, N. F. (2006). “Unsteady undular bores in fully206 nonlinear
shallow-water theory.” Physics of Fluids, 18(027104).

Le Metayer, O., Gavrilyuk, S., and
Hank, S. (2010). “A numerical scheme for the Green-Naghdi model.” Journal of
Computational Physics, 229(6), 2034–2045.

**Zhenquan Li** (jali@cdsu.edu.au)
Charles Sturt University, Australia

*Title:* A new computational technique for
fluid flows

*Abstract: *Mathematicians and physicists
believe that the explanation and prediction of flows can be found through an
understanding of solutions to the Navier-Stokes equations or their extensions
such as k-ϵ model for turbulence. Currently the analytical solutions of the
Navier-Stokes equations or their extensions are not available. After extensive
accuracy analysis, it was found that meshing is one of the main issues in
finding accurate numerical solutions of differential equations. I have proposed
two mesh refinement methods (one for both 2D and 3D) and two streamline
tracking methods for computational (or CFD) velocity fields based on the
qualitative theory of differential equations. I have obtained positive results
when verifying the computational accuracy of these proposed methods with
analytical velocity fields. I have also conducted a sensitivity analysis which
examines if the same results for analytical velocity fields are kept by
numerical solutions of the Navier-Stokes equations. The comparisons of the
outputs from proposed methods with analytical velocity fields and numerical
benchmarks show that proposed methods can identify singular points, asymptotic
lines (planes) and separation curves. In Summary, we have achieved positive
outcomes on accuracy and reliability of the proposed methods. After the
development of the computer programs, the proposed methods can be widely applied
to many problems related to fluid flows in our daily life.

I will briefly introduce the
foundation, the proposed mesh refinement methods, accuracy and reliability
verifications, and computational complexity of the new computational methods in
my presentation. The comparisons for singular points and asymptotic lines
between exact and numerical results for analytical velocity fields will be
presented by illustrations. Some of the comparisons between the benchmarks and
numerical results for lid-driven flow will also be provided. A number of
examples and demonstrations are used for explanations. Possible applications in
practice and future research in computational science, computing science and
other relevant disciplines will be introduced at the end. If the time provided
is not enough, I will delete some of the parts listed above in my presentation.

**Santosh Kumar** (santosh2365@gmail.com)
Thapar University, India

*Title:* Finite volume approximation and
analysis of conservation laws arising in neuronal variability

*Abstract: *The objective of this paper is
to present and analyze numerical approximation of a single neuronal model.
Firstly we derive a hyperbolic conservation law for the distribution of
neuronal firing interval containing pointwise delay as well as advance. We have
modified the classical neuronal model and included the intensity of the
incoming current. Thereafter we propose a numerical approximation based on the
finite volume scheme for conservation laws with source term. In this scheme
homogeneous part is solved by finite volume approximation and the source term
is approximated by a linear interpolation. The developed numerical method is
analyzed for stability and convergence. We have proved the bounded variation
stability and also find the convergence estimates. In the last section, we
perform some numerical experiments to verify the predicted theory of the
numerical approximation constructed in this paper.

**Hans De Sterck **(Hans.DeSterck@monash.edu)
Monash University, Australia

*Title: *High-Order Finite Volume Methods
for Magnetohydrodynamics on Adaptive Cubed-Sphere Grids

*Abstract:* Simulations in spherical
geometries have important applications in space physics and geoscience. We
describe highly accurate finite-volume methods for hyperbolic conservation laws
on parallel dynamically adaptive cubed-sphere grids. In their most simple form,
cubed-sphere grids are obtained starting from a Cartesian grid on a cube that
is deformed into a sphere, resulting in a grid with quasi-uniform spacing and
without polar singularities. We develop a fourth-order central essentially
non-oscillatory (CENO) finite-volume scheme for hyperbolic conservation laws on
these adaptive cubed-sphere grids. Specific challenges include formulating
high-order accurate discretizations on computational cells with non-planar
surfaces, maintaining high-order accuracy at the sector boundaries and corners
of the adaptive cubed-sphere grid, and maintaining the divergence-free property
of the magnetic fields in the magnetohydrodynamics equations that are of
special interest in our applications. The 3D CENO scheme is implemented in a
parallel dynamically adaptive simulation framework. Numerical tests demonstrate
accuracy and efficiency of the approach and show excellent parallel scalability
on thousands of computing cores. This is joint work with Lucian Ivan and
Clinton Groth.

**Marian Moldenhauer **(moldenhauer@zib.de) Konrad-Zuse-Zentrum für Informationstechnik (ZIB), Germany

*Title*: Optimal Hip Implant Positioning

*Abstract: *In an aging society where the
number of joint replacements rise, it is important to also increase the
longevity of implants. In particular hip implants have a life-time of at most
15 years. This derives primarily from pain due to migration, wear,
inflammation, and dislocation, which is affected by the positioning of the
implant during the surgery. Current joint replacement practice uses 2D software
tools and the experience of surgeons. Especially the 2D tools fail to take the
patients natural range of motion as well as stress distribution in the joint
induced by different daily motions into account. Optimizing the hip joint implant
position for all possible parametrized motions under the constraint of a
dynamic contact problem is prohibitively expensive as there are too many
motions and every position change demands a recalculation of the contact
problem. For the reduction of the computational effort, we use adaptive
refinement on the parameter domain. A coarse initial grid is to be locally
refined using goal-oriented error estimation. This approach will be combined
with multi-grid optimization such that numerical errors are reduced.

**Jerome Droniou** (jerome.droniou@monash.edu)
Monash University, Australia

*Title: *An arbitrary-order scheme for
convection-diffusion equations

*Abstract: *Convection-diffusion equations
permeate a variety of fluid flows models, including in particular flows in
porous media. In such models, the natural diffusion can be in some places much
smaller than the convection driven by the Darcy velocity, and it is therefore
essential to dispose of numerical methods that can automatically and locally
adapt to the flow regime (diffusion-dominated or convection-dominated). Some
practical constraints must also be taken into account, such as e.g. the
capacity for the method to be efficiently implemented in a parallel environment.

In this talk, we will present a
numerical scheme of arbitrary order to deal with convection-diffusion
equations. This scheme uses separate degrees of freedom on cells and faces, and
has a local connectivity (each cell is only connected to its faces) which makes
it a good candidate for parallel implementations. The discretisation of the
convective terms uses a stabilisation which automatically adjusts to all
regimes (including vanishing viscosity). The error estimates we obtain are
optimal in all regimes, thanks to the usage of local Péclet numbers.

This is a joint work with D. Di
Pietro and A. Ern.