Timothy Barth, NASA Ames Research Center
Error Representation for Time Dependent Compressible Flow Calculations
For better or worse, our physical world is constantly evolving
in time. Many important physical phenomena depend fundamentally
on time either deterministically or through dynamical system behavior.
These leads to a multitude of challenges associated with identifying,
quantifying, and controlling numerical errors in complex time dependent
numerical simulations. For example, in turbulent flow simulations
it is well known that the control of pointwise solution errors quickly
becomes intractable as the flow Reynolds number increases but the
control of errors occurring in statistics and space-time averaged
quantities may still be tractable [1].
In this presentation, we consider the representation and control
of numerical solution errors in space-time FE methods using standard
duality techniques as succinctly described in [2,3]. For computed
output quantities of interest that are mathematically described
as functionals, this technique exploits the precise relationship
between functional errors and weighted combinations of computable
element residuals. The associated non-perturbative theory provides
the exact form of these residual weights and elucidates via Galerkin
orthogonality why these weights depend not on the dual solution
but rather on the difference between the dual solution and any projection
of the dual solution into the primal approximation space. This latter
property often leads to a non-intuitive space-time dependence of
the local solution error on the accuracy of computed output functionals.
Finally, we briefly mention recent work in adapting established
error control techniques for ODEs to a space-time FEM setting. In
this approach, the normed-error in each time slab is adaptively
made smaller than some prescribed tolerance before the next time
slab interval is solved. If the time slab error tolerances are suitably
chosen, then error control after N time steps is achieved. The presentation
of numerical results will hopefully accentuate the challenges and
difficulties in space-time error representation as well as stimulate
fruitful discussions addressing the feasibility of genuine error
control for time dependent fluid flow problems.
[1] J. Hoffman and C. Johnson,"Adaptive Finite Element Methods
in
Incompressible Fluid Flow", LNCSE, Vol. 25, Springer-Verlag
Pub, 2002.
[2] K. Ericksson, D. Estep, P. Hansbo and C. Johnson,
Computational Differential Equations, Cambridge Press., 1996.
[3] R. Becker and R. Rannacher,"An Optimal Control Approach
to
A-Posteriori Error Estimation in Finite Element Methods, Acta
Numerica, 2001.
Marsha Berger, New York University
Status Report on Embedded Boundary Grids
The Cartesian grid embedded boundary approach has attracted much
interest in the last decade due to the ease of grid generation for
complicated geometries. This approach uses rectangular Cartesian
meshes over most of the domain, with irregular (cut) cells only
at the
intersection of the mesh with the boundary of a solid object. In
this talk we first briefly survey our approach to embedded boundary
computations, and describe what distinguishes it from level set
or other immersed boundary approaches. We then discuss some of the
algorithmic issues that arise in this approach. In particular we
concentrate on numerical discretizations that avoid loss of accuracy
and
stability at irregular boundary cells. We end with some computations
of 3D flows with collaborators at NASA Ames Research Center.
Martin Berzins, University of Utah
Nonlinear High-Order Space and Time Discretization Methods
The analysis of high order schemes based on ENO and WENO methods
will be considered. It will be shown that the use of high-order
provably data bounded polynomials, based on extensions of the work
of [Berzins SIAM Review December 07], provides a way to develop
positivity preserving (W)ENO methods of potentially very high orders.
A series of numerical experiments are used to determine optimal
orders for typical solution profiles. A similar idea is then used
to consider variable-order positivity preserving methods for time
integration, based on the variable-step variable-order backward
differentiation code DASSL.
Hans De Sterck, University of Waterloo
Efficient solution of stationary Euler flows with critical points
and shocks
It is well-known that stationary transonic solutions of the compressible
Euler equations are hard to compute using the stationary form of
the equations. Therefore, time marching methods with explicit or
implicit time integration are normally employed. However, the computational
complexity of the time marching approach is far from optimal, because
convergence tends to be slow and tends to slow down even more as
resolution increases. In this talk we explore the alternative of
solving the stationary equations directly, which is a viable approach
when the solution topology is known in advance. We first present
a solution method for one-dimensional flows with critical points.
The method is based on a dynamical systems formulation of the problem
and uses adaptive integration combined with a two-by-two Newton
shooting method. Example calculations show that the resulting method
is fast and accurate. A sample application area for this method
is the calculation of transonic hydrodynamic escape flows from extrasolar
planets and the early Earth, and the method is also illustrated
for quasi-one-dimensional nozzle flow and black hole accretion.
The method can be extended easily to handle flows with shocks, using
a Newton method applied to the Rankine-Hugoniot relations. Extension
to flows with heat conduction is also discussed. The presentation
will conclude with some thoughts on how the approach presented can
be generalized to problems in higher dimensions.
Clinton P. T. Groth, University of Toronto
Numerical Solution of Continuum and Non-Equilibrium Flows Using
Hyperbolic and Physically-Realizable Descriptions Based on Moment
Closures
The numerical solution of continuum and non-equilibrium flows by
using fully hyperbolic and realizable mathematical descriptions
which follow from a hierarchy of moment closures is described. A
somewhat novel hierarchy of both physically realizable and strictly
hyperbolic moment closures is considered which follows from a slight
modification to the more usual maximum-entropy closure hierarchies.
An
arbitrary-Lagrangian-Eulerian (ALE) parallel finite-volume scheme
with adaptive mesh refinement (AMR) and Riemann-solver based numerical
flux function is also described for solving the system of partial
differential equations arising from the closures on multi-block
meshes with embedded and possibly moving boundaries. The purely
hyperbolic nature of the moment equations is shown to be particularly
insensitive and therefore well suited to discretizations involving
grids with irregularities arising from embedded boundary and moving
interface treatments. The capabilities of the proposed hyperbolic
mathematical treatment and finite-volume scheme for predicting both
continuum and non-equilibrium flows are explored by considering
the a number of canonical problems, including one-dimensional shock
structure flows, Couette flow, flat plate flows, flow past a circular
cylinder, as well as transonic flow past a micro airfoil. The talk
will conclude with a brief summary of findings and a discussion
of the potential of the proposed hyperbolic descriptions for application
to a wider class of flow problems.
Jae-Hun Jung, SUNY Buffalo
Spectral collocation methods for hyperbolic equations with singular
sources
Hyperbolic equations with singular sources are found in many physical
and engineering applications. The solution of such
equations contains the local jump discontinuity which is moving
in time in general. Consequently its spectral approximation yields
the Gibbs phenomenon, that is, the solution is highly oscillatory
and the error does not converge in the maximum error sense.
Regularizing singular sources reduces the Gibbs oscillations but
the maximum error does not converge yet.
To minimize the Gibbs oscillations and improve convergence, we consider
a consistent approach to the singular source terms in
the spectral approximation. Singular sources represented as the
sum of a Dirac delta function and its derivative(s) are obtained
by the direct projection of the Heaviside functions on the collocation
points. The direct projection is obtained by the
collocative derivative of the Heaviside functions and the singular
source terms on the collocation points are highly oscillatory.
This approach, however, yields a convergent result for some hyperbolic
equations with spectral accuracy due to some cancellations
of the Gibbs oscillations on the collocation points.
Several hyperbolic equations with singular sources are considered
including the wave equations from the collision of two
black holes and a nonlinear Schr\"{o}dinger equation in homogeneous
medium with defects. These equations contain the delta
function type sources which are either moving or static. For the
moving sources, the given PDEs are also reduced into ODEs.
Numerical examples will be presented to verify the efficiency and
accuracy of the proposed method.
Smadar Karni, University of Michigan
A Hybrid Scheme for the Baer-Nunziato Two-Phase Flow Model
The Baer-Nunziato two phase flow model describes flame propagation
in gas-permeable reactive granular materials. We focus on the hydrodynamic
part of the system and neglect the terms due to combustion processes,
drag and heat transfer. The system is an averaged flow model, expressing
conservation of mass, and momentum and energy balance of the gas
and solid phases, plus an additional 'compaction' equation describing
the evolution of porosity. The system is only conditionally hyperbolic,
and may fail to have a complete set of eigenvectors. It is also
in non-conservation form due to momentum and energy exchange between
the phases. The presence of non-conservative terms has major consequences
both theoretically and computationally. If the porosities are piecewise
constant, the phases
'talk' to each other only across the porosity jump. Computing solutions
of the Riemann problem rest on capturing the jump conditions across
the porosity jump, but even for state-of-the-art numerical methods
this may prove a difficult task.
We have formulated the Baer-Nunziato model in terms of the Riemann
Invariants across the compaction wave, and propose a hybrid algorithm
that uses the Riemann Invariants formulation across the compaction
wave, and the conservative formulation away from the compaction
wave. The talk will describe the hybrid scheme and present numerical
results.
This is joint work with Gerardo Hernandez of the University of
Michigan.
Barbara Keyfitz, Ohio State University
Linear and Nonlinear Stability of Shocks
Abstract: Shocks (nonlinear discontinuities) separating a region
of hyperbolic states from a non-hyperbolic region can occur in one
of two ways in conservation laws. Steady transonic flow is a well-known
phenomenon, and transonic shocks appear to be both physically and
mathematically stable. Unsteady non-hyperbolic systems, on the other
hand, would seem to form ill-posed, even catastrophically ill-posed
problems. In this talk, I will review some work that Karen Ames
and I did that suggested a sense in which shocks in these ill-posed
systems may enjoy a type of stability. Later work with Milton Lopes
has confirmed our initial, linear results. These results can be
compared to more recent work with a number of co-authors on construction
of transonic shocks.
Lilia Krivodonova, University
of Waterloo
A space and time adaptive discontinuous Galerkin method
Adaptive schemes provide computational advantages over uniform
mesh computations by allocating resources in regions where they
are most needed. The computational mesh can be refined ($h$-refinement)
or the order of the scheme can be modified ($p$-refinement) to better
resolve the solution. As a result, mesh sizes and local CFL numbers
can vary greatly throughout the computational domain. This can be
inefficient as a few small cells impose a restrictive time step
on the whole mesh. We will present an adaptive high-order time integration
scheme for solving partial differential equations with the method
of lines and explicit Runge-Kutta integrators. A time step on each
cell will be defined by the local CFL condition. That is, adaptivity
in time will consist of taking a time step $dt$ on cells of size
$h$, two $dt/2$ steps on cells of size $h/2$,and so on. The total
number of function evaluations will be, therefore, reduced. The
interface conditions will be imposed on interelemental boundaries
so that the order of the underlying Runge-Kutta scheme is preserved.
The novelty of this algorithm is its small stencil. This makes it
suitable for unstructured grids, where multi-layer reconstructions
are difficult. We will show how this
algorithm can be combined with adaptive mesh-refinement to achieve
a significant speed-up for solution of transient flow problems.
Computational issues such as an efficient implementation of the
scheme, integration of spatial and temporal refinement strategies,
and limiting across the interfaces will be discussed.
Marc Laforest, Montreal Polytechnic
An adaptive version of the Illner-Rjasanow stochastic scheme
for the Boltzmann equation
To model the dynamics of materials far from thermodynamic equilibrium,
a kinetic model is often required, like the Boltzmann equation for
dilute gases. Deterministic schemes for the solution of the Boltzmann
equation are typically efficient and accurate but quite limited
with respect to the collision operator that can be modeled. On the
other hand, stochastic schemes, despite their slow convergence,
are much simpler to implement and adapt for even the most complex
collision kernels involving multiple complex species with long-range
interactions. The key issue for stochastic schemes is to reduce
the variance of the observed statistics while keeping the number
of degrees of freedom under control.
The original Random Discrete Velocity Model (RDVM) of Illner and
Rjasanow became the forerunner of what is now the most effective
stochastic scheme for the Boltzmann equation, namely the Stochastic
Weighted Particle Method of Rjasanow and Wagner. The RDVM scheme
works by using a discrete set of velocities in each cell of a structured
mesh. During the transport step, particles can be carried using
any high-order transport scheme. During the collision step, the
velocities are randomly subdivided into several small Discrete Velocity
Models (DVM) which determine how these subsets interact. In this
talk, we describe a new a posteriori error estimator for hierarchies
of DVMs and describe how it can be used it to determine the size
of the local discretization of velocity space in each cell. Using
numerical experiments, we demonstrate the accuracy of the error
estimator and it's efficiency at adapting the local discretization
of velocity space.
Joint work with Kondo Assi
Emmanuel Lorin, UOIT
About the reservoir technique for hyperbolic conservation laws
The talk is devoted to the reservoir technique coupled with finite
volume flux schemes for solving with low numerical diffusion nonlinear
hyperbolic conservation laws. I will present some convergence results,
simulations in 1d and 2d, and some results regarding the overall
algorithmic complexity of the method"
Carl Ollivier-Gooch, University of British
Columbia
High-Order Accurate Unstructured Mesh Finite-Volume Schemes
for Computational Aerodynamics
High-order finite-volume methods for unstructured mesh CFD are
reasonably mature as research tools, with demonstrable accuracy
and efficiency benefits compared with second-order methods. This
maturity comes as a result of recent advances in limiting, convergence
acceleration, and adaptive techniques as applied to high-order methods,
which will be the focus of this talk.
Steve Ruuth, Simon Frazer University
A Simple Technique for Solving Partial Differential Equations
on Surfaces
Many applications require the solution of time-dependent partial
differential equations (PDEs) on surfaces or more general manifolds.
Methods for treating such problems include surface parameterization,
methods on triangulated surfaces and embedding techniques. This
talk describes an embedding approach based on the closest point
representation of the surface and describes some of its advantages
over other embedding methods. Noteworthy features of the method
are its generality with respect to the underlying surface and its
simplicity. In particular, the method requires only minimal changes
to the corresponding three-dimensional codes to treat the evolution
of PDEs on surfaces.
Chi-Wang Shu, Brown University
Superconvergence and Time Evolution of Discontinuous Galerkin Finite
Element Solutions
In this talk, we study the convergence and time evolution of the
error between the discontinuous Galerkin (DG) finite element solution
and the exact solution for conservation laws when upwind fluxes
are used. We prove that if we apply piecewise polynomials to a one
dimensional scalar linear equation or a hyperbolic linear system,
the DG solution will be superconvergent towards a particular projection
of the exact solution. Thus, the error of the DG scheme will not
grow for fine grids over a long time period (for $t$ up to $O( \frac{1}{\sqrt{h}}
)$ where $h$ is the mesh size). We prove this result for general
meshes and arbitrary polynomial degree $k$, and give numerical examples
to demonstrate the superconvergence property, as well as the long
time behavior of the error for more general cases including nonlinear
equations and two-dimensional problems. Generalizations to local
discontinuous Galerkin (LDG) method for convection diffusion equations
will also be given. This is a joint work with Yingda Cheng.
Marek Stastna, University of Waterloo
Application of hyperbolic system methods to dispersive, nonlinear
internal waves and porous media acoustics
I will begin by describing the manner in which numerical methods
designed for hyperbolic systems improve the simulations of internal
waves in an incompressible, density stratified fluid, especially
in the high Reynolds number limit. I will pay particular attention
to situations in which density overturns occur for which limiting
has been termed "implicit Large Eddy Simulation (LES)".
The relationship between methods can be turned around, and results
for fully nonlinear internal waves can be used to improve the numerical
simulation of classical, weakly nonlinear corrections to hyperbolic
waves, such as the Korteweg de Vries equation. Finally I will consider
porous media acoustics for which the corrections to hyperbolic systems
are non-conservative and consist of absorbing terms. I will survey
some of the techniques available for the numerical solution of such
systems and demonstrate that classical, or Biot, frequency dependent
absorption is non-causal.
Allen M. Tesdall, CUNY
A two-phase Stefan problem for the unsteady transonic small disturbance
equations
Numerical solutions of weak shock reflection problems for the unsteady
transonic small disturbance equations, the nonlinear wave system,
and the compressible Euler equations at a set of parameter values
for which regular reflection is impossible contain a complex structure.
Instead of a mathematically inadmissible Mach reflection, as is
apparently observed in experiments, the solutions contain a cascade
of triple points and tiny supersonic patches behind a leading triple
point. A centered expansion wave originates at each triple point.
We call this structure of repeating supersonic patches and triple
points ``Guderley Mach reflection,'' or GMR.
At the upstream side of each patch in GMR, a sonic line separates
the patch from a region of subsonic flow. This sonic line can be
considered a free boundary in the formulation of a free boundary
problem, with the states on either side coupled through the boundary.
As a step towards the goal of formulating this free boundary problem,
we present a problem which retains its main features, but which
is simpler. We choose the simplest model of weak shock reflection,
the unsteady transonic small disturbance equations (UTSDE). We choose
initial data which results in an expansion wave that reflects off
a sonic line, similar to the reflection of a centered rarefaction
off the sonic line in a single supersonic patch in GMR. In both
GMR and our simpler problem for the UTSDE, the states on either
side of the free boundary are coupled through the boundary, and
the solutions in both the supersonic and subsonic regions are a
priori unknown. These features make these problems analogous to
two-phase Stefan problems for the heat equation. At the moment,
we have linearized the simplified problem and solved it exactly.
We have not yet formulated the free boundary problem, but we have
solved the full nonlinear simplified problem numerically. We will
present and describe our solutions and our solution method.
Tim Warburton, Rice University
Advances in Wave Propagation with the Discontinuous Galerkin
Method
A range of important features relating to the practical application
of discontinuous Galerkin (DG) method for wave propagation will
be
discussed.
Given the suitability of DG for solving Maxwell's equations and
their ability to propagate waves over long distance, it is natural
to seek
effective boundary treatments for artificial radiation boundary
conditions. A new family of far field boundary conditions will be
introduced which gracefully transmit propagating and evanescent
components out of the domain. These conditions are specifically
formulated with DG discretizations in mind, however they are also
relevant for a range of numerical methods.
There is an Achilles heel to high order discontinuous Galerkin
methods when applied to conservation laws. The methods are typically
constructed with polynomial field representations and unfortunatelythese
suffer from excess maximum gradients near the edges of elements.
I will describe a simple filtering process that allows us to reduce
these anomalous gradients and provably yield a dramatic increase
in the maximum allowable time step. Additional experiments with
local time stepping methods will also be presented.
Finally, I will discuss the use of GPU hardware to accelerate computation
for time-domain electromagnetics simulations.
