uucp@ucbvax.BERKELEY.EDU (UNIX Copy) (01/27/86)
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NUMERICAL ANALYSIS TITLES
-- Volume 3, Number 2, Part 2 of 3
-- August 27, 1985
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Please note: Due to its length, this list is
being distributed in 3 parts, each
part is about 7 pages in length.
##### GMD, WEST GERMANY #####
All the papers of the GMD may be ordered from
Dr. Kurt Brand
Gesellschaft fuer Mathematik und
Datenverarbeitung (GMD)
F1/T
Postfach 1240
D-5205 St. Augustin 1
F.R.G.
or send a request by electronic mail to
gmap18%dbngmd21.bitnet@wiscvm.arpa
No abstracts were submitted.
###################
[20] K. Becker
Ein Mehrgitterverfahren zur Berechnung subsonischer Potential-
stroemungen um Tragflaechenprofile
Berichte der Gesellschaft fuer Mathematik und Datenverarbeitung,
Bericht Nr. 152, Oldenbourg-Verlag, 1985
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[21] K. Becker, U. Trottenberg
Development of Multigrid Algorithms for Problems from Fluid Dynamics
Arbeitspapiere der GMD no.111, Gesellschaft fuer Mathematik und
Datenverarbeitung, Bonn, 1984. (DM 6,-)
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[22] A. Brandt
Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics
GMD-Studien no. 85, Gesellschaft fuer Mathematikund Daten-
verarbeitung, Bonn, 1984. (DM 39,-)
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[23] O. Kolp
Parallelisierung eines Mehrgitterverfahrens fuer einen Baumrechner
Arbeitspapiere der GMD no. 82, Bonn, 1984. (DM 10,-)
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[24] R. Lorenz
Some New Periodic Chebycheff Spaces
Arbeitspapiere der GMD no. 79, Gesellschaft fuer Mathematik und
Datenverarbeitung, Bonn, 1984
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[25] J. Ruge, K. Stueben
Efficient Solution of Finite Difference and Finite Element
Equations by Algebraic Multigrid (AMG)
Arbeitspapiere der GMD no. 89, Gesellschaft fuer Mathematik
und Datenverarbeitung, Bonn, 1984. (DM 10,-)
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[26] B. Ruttmann, K. Solchenbach
A Multigrid Solver for the computation of in-cylinder turbulent
flows in engines
Efficient Solutions of Elliptic Systems,
Proceedings of GAMM-Seminar, Kiel, January 27 to 29, 1984,
(W. Hackbusch ed.).
Notes on Numerical Fluid Mechanics, 10, pp. 87-108, Vieweg,
Braunschweig, 1984.
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[27] K. Stueben, U. Trottenberg, K. Witsch
Software Development Based on Multigrid Techniques
Arbeitspapiere der GMD no. 84, Gesellschaft fuer Mathematik
und Datenverarbeitung, Bonn, 1984. (DM 10,-)
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[28] C.-A. Thole, U. Trottenberg
Basic Smoothing Procedures for the Multigrid Treatment of
Elliptic 3D-Operators
Arbeitspapiere der GMD no. 141, Bonn, 1985. (DM 10,-)
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[29] U. Trottenberg
Schnelle Loesung elliptischer Differentialgleichungen nach dem
Mehrgitterprinzip
GAMM-Mitteilungen, 1/84, February 1984.
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[30] U. Trottenberg, P. Wypior (Ed.)
Rechnerarchitekturen fuer die numerische Simulation auf der
Basis superschneller Loesungsverfahren I
Workshop "Rechnerarchitektur", Erlangen, 14.-15. Juni 1984.
GMD-Studien no. 88, Gesellschaft fuer Mathematik und Daten-
verarbeitung, Bonn, 1984. (DM 48,-)
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[31] U. Trottenberg, P. Wypior (Ed.)
Rechnerarchitekturen fuer die numerische Simulation auf der
Basis superschneller Loesungsverfahren II
Workshop "Simulation/Anwendungen und Numerik",
GMD-Studien no. 102, Gesellschaft fuer Mathematik und Daten-
verarbeitung, Bonn, 1985. (DM 48,-)
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[32] U. Trottenberg
Mehrgitterprinzip und Rechnerarchitektur
GAMM-Mitteilungen, 1/84, February 1984.
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##### ICASE #####
All ICASE reports listed here may be
obtained by writing to:
Barbara Kraft
ICASE, M/S 132C
NASA Langley Research Center
Hampton, VA 23665
###################
[33] Zang, T. A. and M. Y. Hussaini
A three-dimensional spectral algorithm for
simulations of transition and turbulence
ICASE Report No. 85-19, March 6, 1985, 39 pages.
AIAA Paper No. 85-0296 presented at the AIAA 23rd
Aerospace Sciences Meeting, January 14-17, 1985, Reno, Nevada.
A spectral algorithm for simulating three-dimensional, incompressible,
parallel shear flows is described. It applies to the channel, to the parallel
boundary layer, and to other shear flows with one wall-bounded and two
periodic directions. Representative applications to the channel and to the
heated boundary layer are presented.
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[34] Drummond, J. P., M. Y. Hussaini, and T. A. Zang
Spectral methods for modeling supersonic
chemically reacting flow fields
ICASE Report No. 85- 20,
March 8, 1985, 37 pages. AIAA J.
A numerical algorithm has been developed for solving the equations
describing chemically reacting supersonic flows. The algorithm employs a two-
stage Runge-Kutta method for integrating the equations in time and a Chebyshev
spectral method for integrating the equations in space. The accuracy and
efficiency of the technique have been assessed by comparison with an existing
implicit finite-difference procedure for modeling chemically reacting flows.
The comparison showed that the new procedure yielded equivalent accuracy on
much coarser grids as compared to the finite-difference procedure with
resultant significant gains in computational efficiency.
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[35] LeVeque, R. J
Intermediate boundary conditions for LOD, ADI,
and approximate factorization methods
ICASE Report No. 85-21, April 17, 1985, 24 pages.
Submitted to J. Comput. Phys.
A general approach to determining the correct intermediate boundary
conditions for dimensional splitting methods is presented and illustrated.
The intermediate solution U* is viewed as a second-order accurate
approximation to a modified equation. Deriving the modified equation and
using the relationship between this equation and the original equation allows
us to determine the correct boundary conditions for U*. To illustrate this
technique, we apply it to LOD and ADI methods for the heat equation in two and
three space dimensions. The approximate factorization method is considered in
slightly more generality.
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[36] Rosen, I. G.
Spline-based Rayleigh-Ritz methods for the approximation of the
natural modes of vibration for flexible beams with tip bodies
ICASE Report No. 85-22, March 18, 1985, 31 pages.
Submitted to Quarterly of Applied Mathematics.
Rayleigh-Ritz methods for the approximation of the natural modes for a
class of vibration problems involving flexible beams with tip bodies using
subspaces of piecewise polynomial spline functions are developed. An abstract
operator theoretic formulation of the eigenvalue problem is derived and
spectral properties investigated. The existing theory for spline-based
Rayleigh-Ritz methods applied to elliptic differential operators and the
approximation properties of interpolatory splines are used to argue
convergence and establish rates of convergence. An example and numerical
results are discussed.
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[37] Bayliss, A., L. Maestrello, P. Parikh, and E. Turkel
Numerical simulation of boundary layer excitation
by surface heating/cooling
ICASE Report No. 85-23, March 25, 1985, 22 pages.
AIAA Paper No. 85-0565, AIAA Shear Flow
Control Conference, March 12-14, 1985, Boulder, CO.
This paper is a numerical study of the concept of active control of
growing disturbances in an unstable compressible flow by using time periodic,
localized surface heating. The simulations are calculated by a fourth-order
accurate solution of the compressible, laminar Navier-Stokes equations.
Fourth-order accuracy is particularly important for this problem because the
solution must be computed over many wavelengths. The numerical results
demonstrate the growth of an initially small fluctuation into the nonlinear
regime where a local breakdown into smaller scale disturbances can be
observed. It is shown that periodic surface heating over a small strip can
reduce the level of the fluctuation provided that the phase of the heating
current is properly chosen.
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[38] Hossain, M., G. Vahala, and D. Montgomery
Forced MHD turbulence in a uniform external magnetic field
ICASE Report No. 85-24, March 28, 1985, 39 pages.
Submitted to Phys. Fluid.
Two-dimensional dissipative MHD turbulence is randomly driven at small
spatial scales and is studied by numerical simulation in the presence of a
strong uniform external magnetic field. A novel behavior is observed which is
apparently distinct from the inverse cascade which prevails in the absence of
an external magnetic field. The magnetic spectrum becomes dominated by the
three longest-wavelength Alfv'n waves in the system allowed by the boundary
conditions: those which, in a box size of edge 2 pi, have wave numbers
(kx, ky) = (1, 0), (1, 1), and (1, -1), where the external magnetic field is
in the x direction. At any given instant, one of these three modes
dominates the vector potential spectrum, but they do not constitute a
resonantly coupled triad. Rather, they are apparently coupled by the smaller-
scale turbulence.
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[39] Davis, S. F.
Shock capturing
ICASE Report No. 85-25, April 26, 1985, 23 pages.
To appear in Numerical Methods for Partial Differential Equations,
(S. I. Hariharan and T. H. Moulder, eds.), Pitman Press, 1986.
This chapter describes recent developments which have improved our
understanding of how finite difference methods resolve discontinuous solutions
to hyperbolic partial differential equations. As a result of this
understanding improved shock capturing methods are currently being developed
and tested. Some of these methods are described and numerical results are
presented showing their performance on problems containing shocks in one and
two dimensions.
We begin this discussion by defining what is meant by a conservative
difference scheme and showing that conservation implies that, except in very
special circumstances, shocks must be spread over at least two grid intervals.
These two interval shocks are actually attained in one dimension if the shock
is steady and an upwind scheme is used. By analyzing this case, we determine
the reason for this excellent shock resolution and use this result to provide
a mechanism for improving the resolution of two-dimensional steady shocks.
Unfortunately, this same analysis shows that these results cannot be extended
to shocks which move relative to the computing grid.
To deal with moving shocks and contact discontinuities we introduce total
variation diminishing (TVD) finite difference schemes and flux limiters. We
show that TVD schemes are not necessarily upwind, but that upwind TVD schemes
perform better because they permit a wider choice of flux limiters. The
advantage of non-upwind TVD schemes is that they are easy to implement.
Indeed, it is possible to add an appropriately chosen artificial viscosity to
a conventional scheme such as MacCormack's method and make it TVD. We
conclude by presenting some theoretical results on flux limiters and some
numerical computations to illustrate the theory.
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[40] Brandt, A. and S. Ta'asan
Multigrid solutions to quasi-elliptic schemes.
ICASE Report No. 85-26, May 3, 1985, 21 pages.
Progress and Supercomputing in Computational Fluid Dynamics,
(Earl. S. Murman and Saul Abarbanel, eds.),
Birkhauser Boston, Inc., (tentative publication date:
August 20, 1985).
Quasi-elliptic schemes arise from central differencing or finite element
discretization of elliptic systems with odd order derivatives on non-staggered
grids. They are somewhat unstable and less accurate then corresponding
staggered-grid schemes. When usual multigrid solvers are applied to them, the
asymptotic algebraic convergence is necessarily slow. Nevertheless, it is
shown by mode analyses and numerical experiments that the usual FMG algorithm
is very efficient in solving quasi-elliptic equations to the level of
truncation errors. Also, a new type of multigrid algorithm is presented, mode
analyzed and tested, for which even the asymptotic algebraic convergence is
fast. The essence of that algorithm is applicable to other kinds of problems,
including highly indefinite ones.
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[41] Zang, T. A. and M. Y. Hussaini
On spectral multigrid methods for the
time-dependent Navier-Stokes equations
ICASE Report No. 85-27, May 13, 1985, 24 pages.
Presented at the 2nd Copper Mountain Conference on Multigrid
Methods, April 1-3, 1985, Copper Mountain, CO.
A new splitting scheme is proposed for the numerical solution of the
time-dependent, incompressible Navier-Stokes equations by spectral methods. A
staggered grid is used for the pressure, improved intermediate boundary
conditions are employed in the split step for the velocity, and spectral
multigrid techniques are used for the solution of the implicit equations.
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[42] Osher, S. and E. Tadmor
On the convergence of difference approximations
to scalar conservation laws
ICASE Report No. 85-28, May 14, 1985, 70 pages.
Submitted to Math. Comp.
We present a unified treatment of explicit in time, two level, second
order resolution, total variation diminishing, approximations to scalar
conservation laws. The schemes are assumed only to have conservation form and
incremental form. We introduce a modified flux and a viscosity coefficient and
obtain results in terms of the latter. The existence of a cell entropy
inequality is discussed and such an equality for all entropies is shown to
imply that the scheme is an E scheme on monotone (actually more general)
data, hence at most only first order accurate in general. Convergence for
TVD-SOR schemes approximating convex or concave conservation laws is shown by
enforcing a single discrete entropy inequality.
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[43] Mehrotra, P. and J. Van Rosendale
The BLAZE language: A parallel language
for scientific programming
ICASE Report No. 85-29, May 15, 1985, 57
pages. Submitted to Parallel Computing.
Programming multiprocessor parallel architectures is a complex task.
This paper describes a Pascal-like scientific programming language, Blaze,
designed to simplify this task. Blaze contains array arithmetic, "forall"
loops, and APL-style accumulation operators, which allow natural expression of
fine grained parallelism. It also employs an applicative or functional
procedure invocation mechanism, which makes it easy for compilers to extract
coarse grained parallelism using machine specific program restructuring. Thus
Blaze should allow one to achieve highly parallel execution on multiprocessor
architectures, while still providing the user with conceptually sequential
control flow.
A central goal in the design of Blaze is portability across a broad range
of parallel architectures. The multiple levels of parallelism present in
Blaze code, in principle, allows a compiler to extract the types of
parallelism appropriate for the given architecture, while neglecting the
remainder. This paper describes the features of Blaze, and shows how this
language would be used in typical scientific programming.
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[44] Trefethen, L. N. and L. Halpern
Well-Posedness of one-way wave equations and
absorbing boundary conditions
ICASE Report No. 85-30, June 10, 1985, 23 pages.
Submitted to Math. Comp.
A one-way wave equation is a partial differential equation which, in some
approximate sense, behaves like the wave equation in one direction but permits
no propagation in the opposite one. The construction of such equations can be
reduced to the approximation of the square root of 1 - s2 on [-1,1] by a
rational function r(s) = Pm(s)/qn(s). This paper characterizes those
rational functions r for which the corresponding one-way wave equation is
well-posed, both as a partial differential equation and as an absorbing
boundary condition for the wave equation. We find that if r(s) interpolates
the square root of 1 - s2 at sufficiently many points in (-1,1), then well-
posedness is assured. It follows that absorbing boundary conditions based on
Pade approximation are well-posed if and only if (m,n) lies in one of two
distinct diagonals in the Pade table, the two proposed by Engquist and
Majda. Analogous results also hold for one-way wave equations derived from
Chebyshev or least-squares approximation.
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[45] Majda, G.
A new theory for multistep discretizations of stiff ordinary
differential equations: Stability with large step sizes
ICASE Report No. 85-31, 70 pages.
Submitted to Math. Comp.
In this paper we consider a large set of variable coefficient linear
systems of ordinary differential equations which possess two different time
scales, a slow one and a fast one. A small parameter epsilon characterizes
the stiffness of these systems. We approximate a system of o.d.e.s in this
set by a general class of multistep discretizations which includes both one-
leg and linear multistep methods. We determine sufficient conditions under
which each solution of a multistep method is uniformly bounded, with a bound
which is independent of the stiffness of the system of o.d.e.s, when the step
size resolves the slow time scale but not the fast one. We call this property
stability with large step sizes.
The theory presented in this paper lets us compare properties of one-leg
methods and linear multistep methods when they approximate variable
coefficient systems of stiff o.d.e.s. In particular, we show that one-leg
methods have better stability properties with large step sizes than their
linear multistep counterparts. This observation is consistent with results
obtained by Dahlquist and Lindberg {11}, Nevanlinna and Liniger {32} and van
Veldhuizen {41}. Our theory also allows us to relate the concept of D-
stability (van Veldhuizen {41}) to the usual notions of stability and
stability domains and to the propagation of errors for multistep methods which
use large step sizes.
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[46] Banks, H. T.
On a variational approach to some
parameter estimation problems.
ICASE Report No. 85-32, 38 pages.
Invited lecture, Internationl Conference on
Control Theory for Distributed Parameter
Systems and Applications, July 9 - 14, 1984, Voran, Austria.
We consider examples (1-D seismic, large flexible structures,
bioturbation, nonlinear population dispersal) in which a variational setting
can provide a convenient framework for convergence and stability arguments in
parameter estimation problems.
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[47] Hariharan, S. I.
Absorbing boundary conditions for exterior problems
ICASE Report No. 85-33, July 18, 1985, 33 pages.
To appear in Numerical Methods for Partial Differential
Equations, (S. I Hariharan and T. H. Moulden, eds.),
Pitman Press, 1986.
In this paper we consider elliptic and hyperbolic problems in unbounded
regions. These problems, when one wants to solve them numerically, have the
difficulty of prescribing boundary conditions at infinity. Computationally,
one needs a finite region in which to solve these problems. The corresponding
conditions at infinity imposed on the finite distance boundaries should
dictate the boundary condition at infinity and be accurate with respect to the
interior numerical scheme. Such boundary conditions are commonly referred to
as absorbing boundary conditions. This paper presents a survey and covers our
own treatment on these boundary conditions for wave-like equations.
---------------