REVISITING THE CHANDRASEKHAR EQUATIONOF RADIATIVE TRANSFER.

Saul Abarbanel∗, Adi Ditkowski∗†, David Gottlieb‡

October 21, 2009

Abstract

We consider the Chandrasekhar equation of radiative transfer in the non-conservative casefor axially symmetric problems in semi-infinite atmosphere:

µ∂ I(τ ,µ)

∂τ= I(τ ,µ) − 1

2ω0

∫ 1

−1I(τ, µ)dµ ,

where µ is cos θ and θ is the angle from the normal to the plane τ = 0. µ > 0 andµ < 0 indicate, respectively, right and left direction of radiative propagation. I(τ ,µ) is theflux intensity and ω0 is the albedo due to atmospheric scattering. The above equation may beconsidered either as:

I. An initial value problem with initial condition given by I(0,µ) = I0(µ).

II. Initial/final value problem, where the initial and the final values are given by:

I+(L,µ) = I+L (µ) ; where I+(τ,µ) = I(τ,µ) ; µ > 0

I−(0,µ) = I−0 (µ) ; where I−(τ,µ) = I(τ,µ) ; µ < 0

We show that problem I is not well posed, while problem II is stable and possess a uniquesolution. A new approach for solving problem II numerically was developed.

∗School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel∗School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel†This research was supported by the UNITED STATES-ISRAEL BINATIONAL SCIENCE FOUNDATION (grant

No. 2004099).‡Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA

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Propagation of Uncertainty in StructuredDynamical Systems

H.T. BanksCenter for Research in Scientific ComputationCenter for Quantitative Sciences in Biomedicine

N.C. State UniversityRaleigh, NC

htbanks@ncsu.edu

Abstract

After discussing the shortcomings of a crypto-deterministic for-mulation, we compare two other approaches for inclusion of uncer-tainty/variability in modeling growth in structured population mod-els. One entails imposing a probabilistic structure on growth rates inthe population that leads to random differential equations while theother involves formulating growth as a stochastic Markov diffusionprocess, resulting in forward Kolmogorov or Fokker-Planck equations.We present a theoretical analysis that allows one to include compa-rable levels of uncertainty in the two distinct formulations in makingcomparisons of the two approaches. Computational aspects of theapproaches are also discussed.

Another look at h-box methods for cut cells

Marsha Berger

Courant Institute, NYU

H-box methods are an approach to solving time dependent hyperbolic

problems on embedded boundary meshes using an explicit difference scheme.

Several schemes have previously been developed in the literature that address

the “small cell” problem”, namely that cut cells have cell volumes that can be

orders of magnitude smaller than the usual cell, and thus a straightforward

explicit method leads to unacceptable time step restrictions. In previous work

we have developed a rotated h-box scheme that retains second order accuracy

for a full timestep chosen according to the regular mesh cfl. However, it is

rather complicated. In this talk we examine alternatives that provide many

of the same properties and are easier to implement.

1

TBA

Wei Cai

University of North Carolina at Charlotte

1

Multilevel Preconditioning of Discontinuos Galerkin

Spectral Element Methods

Claudio Canuto

Politecnico di TorinoTORINO ITALY

1

TBA

Mark Carpenter

NASA Langley Research Center

1

Spectral domain embedding technique for elliptic PDE with irregulardomain

Wai Sun DonHong Kong Baptist University

Brown University

Spectral methods have been employed to solve elliptic PDE with twodimensional complex domain where the complex domain Ω is embedded in-side a rectangular computational domain R (also known as fictitious domainapproach in the literature). The computational domain R is discretized bythe Chebyshev collocation methods. The Dirichlet or Neumann boundaryconditions on the δΩ are imposed by enforcing the boundary condition ona chosen set of control nodes distributed along the δΩ as a set of addi-tional constraint equations. In this study, the forcing function satisfies theexact solution in the extended region R\Ω. The resulting large system ofequation, which is ill-condition due to the ill-condition of the Chebyshevdifferentiation matrix, is solved via the Schur complement method and thepseudoinverse for solving a non-square matrix equation when appropriate.We shall conduct the work along the line laid out by Lui [JCAM 225, 2009]and shows an improvement of the method by using the Tal-Ezer mappingwhen large number of collocation points is used. Also, we will discuss somerelated issues regarding the choice of distribution of control nodes alongthe δΩ that demonstrates its effect on the accuracy of the method. Somenumerical examples on different complex domains will also be shown.

This is a collaborative research with Shi Cheng, Yi Zhong and LongHuaWu, who are the undergraduate students at HKBU.

Scalable Algorithms for Convection-Dominated Flows

Paul Fischer

Mathematics and Computer Science DivisionArgonne National Laboratory

Accuracy and stability have long been essential pillars of numerical algo-

rithms for the simulation of fluid flow. With the advent of tera- and petascale

parallel computers comprising thousands and hundreds of thousands of pro-

cessors, scalability is emergent as another essential pillar. To first order,

scalability implies that the solution time be only weakly dependent on the

number of processors P, with n/P fixed, where n is the number of degrees of

freedom in the problem. Time-dependent transport problems having minimal

dissipation, such as electromagnetics and convection-dominated flow, face an

additional scalability challenge, namely, that dispersion errors accumulated

at small scales may become dominant when propagated through the large

domains that are afforded by petaflops computers.

This talk will cover several critical developments that make it possible to

use spectral element simulations in large-scale convection-dominated incom-

pressible flow simulations on tens and hundreds of thousands of processors.

Discretization advances that have made the spectral element viable for these

problems include stabilizing filters and spectral element dealiasing. Solver

advances include spectral element multigrid methods that employ robust

Schwarz-based smoothers and scalable parallel coarse-grid solvers. In addi-

tion to these fundamental elements, we touch upon a few technical details

required to exceed processor counts of ten thousand. We present the results

of several spectral element simulations, including heat transfer in reactor

core subchannels, turbulent MHD, and a detailed discussion of transitional

flow in arteriovenous grafts. We conclude with some perspectives on the

future of high-order methods and high performance computing applied to

computational fluid dynamics.

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Electromagnetic Radiations as a Fluid Flow

Daniele FunaroDepartment of Mathematics, University of Modena

Via Campi 213/B, 41125 Modena (Italy)E-mail: daniele.funaro@unimore.it

Abstract - Since the advent of the theory of electromagnetic fields,more than a century ago, waves have been described as a kind of energy flow,governed by suitable transport equations in vector form, namely Maxwell’sequations. In void, the electric and magnetic fields (E and B, respectively)are transversally oriented with respect to the direction of propagation, andtheir envelope produces a sequence of wave-fronts. This is in agreement withthe fact that the energy develops according to the evolution of the vectorproduct E× B, otherwise known as Poynting vector.

On the other hand, the dynamical behavior of a compressible non viscousfluid is well described by Euler’s equation, where, in principle, the velocityvector field (denoted by V) might not be necessarily related to a real materialfluid. In particular, one could replace the mass density by a sort of chargedensity. Therefore, the temptation to describe electromagnetic and velocityfields, through a combination of the respective modelling equations, is wellmotivated.

We are going to present a system of equations in the three independentvector unknowns: (E, B, V). In pure void, the electric and magnetic fieldsfollow the Faraday’s law together with the Ampere’s law, where a current,flowing at velocity V, is supposed to be naturally associated with the wave.In order to close the system, the third relation is the Euler’s equation for V,containing an added forcing term E + V × B, perfectly analogous to thatexpressing the Lorentz’s law. In this way, the three entities (E, B, V) turnout to be strictly entangled.

Despite the appearance, the new model allows for a very large spaceof solutions. Moreover, it displays numerous conservation and invarianceproperties, all deducible from a standard analysis. An interesting invariantsubspace of solutions is the one where the third equation is reduced toE + V × B = 0, which means that no acceleration is acting on the wave,

and the flow is somehow laminar. For this circumstance, the solutions,called free-waves, perfectly follow the laws of geometrical optics, ruled bythe eikonal equation. Together with usual known solutions, free-waves alsoinclude solitary electromagnetic waves with compact support almost of anyshape, intensity, frequency and polarization. Such a result, never achievedbefore, reopens the path to a serious discussion on photons, the dualitywave-particle and the quantum properties of matter.

Far more complicated solutions (not of the free-wave type) are howeverpossible. Since our electromagnetic radiations actually behave as a fluid,they can be constrained to evolve in bounded regions of space, similar forinstance to vortex rings. According to the model equations, rotating pho-tons in a vortex structure may carry a charge and deform, via Einstein’sequation, the local geometry of the space-time in order to create a gravita-tional environment assimilable to the presence of mass. The same metricspace is responsible for the stability of such a wave, obliged in this way todevelop along self-created geodesics.

This leaves us with the conjecture that some stable elementary parti-cles (such as the electron) could be made by rotating photons, an idea thathas been put forward by many authors in the past, although with not toomuch recognition, basically due to the lack of a sufficient theoretical descrip-tion of electromagnetic phenomena, able to go beyond the classical linearMaxwellian approach. Since now we exactly know what a photon is, addi-tional elements are available for a deeper investigation.

References

[1] Funaro D., Electromagnetism and the Structure of Matter, World Sci-entific, Singapore, 2008.

Reconstruction of Piecewise Smooth Functions from

Non-uniform Fourier Data

Anne Gelb

Arizona State University

We discuss the reconstruction of compactly supported piecewise- smooth

functions from non-uniform samples of their Fourier transform. This prob-

lem is relevant in applications such as magnetic resonance imaging (MRI).

We summarize two standard techniques, convolutional gridding and uniform

resampling, and address the issue of non-uniform sampling density and its

effect on reconstruction quality. We compare these classical reconstruction

approaches with alternative methods such as spectral re-projection and meth-

ods incorporating jump information.

1

International Conference on Advances in Scientific Computing

Providence, December 6-8, 2009

The Work of David Gottlieb: A Success Story

by

Bertil GustafssonDivision of Scientific Computing

Uppsala University

Sweden

Abstract

The scientific work of David Gottlieb is a true success story. He has been the key person in

the development of spectral methods over the last thirty years beginning with his book 1977

with Steve Orzag. His theoretical analysis and generalization to new types of basis functions

took the use of spectral methods to higher levels of applications. He also made significant

contributions to other areas of numerical analysis, as for example the development of high

order difference methods.

This talk is a survey of David Gottlieb’s work, and its impact on scientific computing.

1

TBA

Antony Jameson

Stanford University

1

On Computations of Nonconservative Products

Smadar Karni

University of Michigan

1

Open Issues in Polynomial Chaos

George Karniadakis

Brown University

We will review some of the progress in polynomial chaos (PC) methods

in the last ten years and discuss problems associated with application of PC

to solving stochastic differential equations. Specifically, while progress has

been made in solving stochastic problems in steady-state, long-time integra-

tion, high-dimensions and white noise modeling are still open problems. We

will discuss possible treatment based on adaptivity, functional ANOVA, and

Malliavin calculus.

1

A Gibbs phenomenon for finite difference schemes,

and how to beat it

Peter Lax

New York University (Courant)

1

TBA

Yvon Maday

University of Paris 6, France

1

TBA

Alfio Quarteroni

Ecole Polytechnique Federale de LausanneSwitzerland

1

Shock-Fitting Applied to a Simple Model of Elastoplasticity

Manuel D. Salas m.d.salas@nasa.gov

Langley Distinguished Research Associate NASA Langley Research Center

Hampton, VA 23681

Abstract Nonlinear Elasticity is a relatively new field of study. It has its roots in the works of Rivlin in the late 1940’s and of Truesdell a decade later. The complex nonlinear character of the mechanical and ther-mal properties of the subject matter provides a rich and challenging field for mathematical studies. It is, therefore, not surprising that the subject has attracted the attention of researchers from many other fields. In this study, we look, from the perspective of a fluid-dynamist, at the numerical treatment of shock waves occurring in a very simple model representative of elastoplastic behavior. The approach we use for the numerical treatment of shocks requires knowledge of the shock jump conditions, but because the governing equations cannot be written in divergence form the Rankine Hugoniot jumps cannot be obtained by the usual method. To get around this problem, we turn to Colombeau’s theory of generalized functions. Numerical examples are presented for both elastic and plastic shocks.

TBA

James Sethian

University of California at Berkeley

1

A High-Resolution Method Using Adaptive Polynomials for Local Refinement

J.S. Shang

Mechanical and Materials Engineering Department

Wright State University

Dayton, OH

Abstract

A numerical procedure of local resolution refinement using the Gauss quadrature is formulated for a reduced model equation for laminar flame. The developed scheme is implicit both in space and time by successive polynomial refinements. A comparison of the Gauss-Legendre and Gauss-Lobatto quadrature is first performed to bring out the outstanding characteristics of these spectral formulations. A detailed comparison with high-order compact difference schemes has also been carried out to assess the numerical resolution in an isolated high-gradient domain. The preliminary numerical solutions of the reduced model equation for counter flowing combustion are being verified by comparison with solution of conventional approach and will be validated by experiment observation in the near future.

Fast algorithms for elliptic systems with moving

interfaces

John Strain

University of California (Berkeley)

Mathematical models of material phenomena often involve complex mov-

ing interfaces, with velocities determined by elliptic systems of PDEs which

connect the material physics to interface geometry. Such interfaces are

evolved by a semi-Lagrangian contouring (SLC) algorithm which treats the

velocity as a black box, and separates model-dependent physics from inter-

face motion. SLC converts stiff moving interfaces to an implicit grid-free

contouring problem, and computes highly accurate solutions with merging,

anisotropy, faceting, curvature, dynamic topology and nonlocal interactions.

The interface velocity is computed by locally-corrected spectral (LCS)

methods which convert arbitrary elliptic problems to first-order overdeter-

mined systems, mollify a periodic fundamental solution for convergence, and

correct via Ewald summation. Local linear algebra and distribution theory

yield a simple boundary integral equation. With a new geometric nonuniform

fast Fourier transform, LCS methods provide efficient accurate solutions to

elliptic systems in complex domains.

1

TBA

Eitan Tadmor

University of Maryland

1

An hyperbolic boundary value problem:

The Zakharov-Kuznetsov equation in an interval

Roger TemamUniversity of Indiana

We consider the so-called Zakharov-Kuznetsov equation which describes the propagation of

non-linear ionic-sonic waves in a plasma submitted to a magnetic field directed along the x-axis.

The equation is considered in the interval 0 < x < 1. We provide a set of suitable boundary

conditions for the linearized equation and then the full nonlinear equation, and we show a result

of existence of solutions for the nonlinear equation in space dimension two and three, with

uniqueness in space dimension two.

1

Eli Turkel

Dept. of Applied Mathematics

School of Mathematical Sciences

Tel-Aviv UniversityTel-Aviv 69978, Israel

On Preconditioning

We consider preconditionings that are equation based, i.e. the preconditioninguses properties of the equation being solved rather than the more usual algebraicapproach. One application is to the steady state solution to the compressible Navier-Stokes equations. The preconditioning is based on an upwind first order approxi-mation which is solved by several Gauss-Seidel iterations. Results are presented forseveral complicated configurations. The second preconditioning is for a high order ac-curate scheme for the Helmholtz equation. The preconditioning is based on a modifiedHelmholtz equation with a complex wavenumber. This is then solved by multigrid.Results show that no deterioration occurs in spite of the high accuracy of the originalalgorithm.

A venerable family of DG schemes for diffusion

revisited

Bram van Leer

University of Michigan

1