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From
•>>January 2006
John D. Towers answers
a few questions about this month's fast moving front in the
field of Mathematics.
Field: Mathematics
Article: A difference scheme for conservation laws with a discontinuous flux: The nonconvex case
Authors: Towers, JD
Journal: SIAM J NUMER ANAL, 39 (4): 1197-1218, DEC 4 2001
MiraCosta Coll, Dept Math, 3333 Manchester Ave, Cardiff By The Sea, CA 92007 USA.
MiraCosta Coll, Dept Math, Cardiff By The Sea, CA 92007 USA.
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Why do you think your
paper is highly cited?
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“The significance of this paper is that it presents a simple method that can be programmed on a computer for a difficult mathematical problem, and that problem has real world applications.”
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I can only guess why the paper is highly cited. Perhaps it is
because the paper presents a very simple algorithm for a difficult
mathematical problem, and that problem has some important practical
applications.
Does it describe a new discovery or a new methodology that’s
useful to others?
From a mathematical point of view, the algorithm and some of the
analytical techniques in the paper have turned out to be useful for
a more general class of problems, as demonstrated by later work done
jointly with Raimund Bürger,
Kenneth Hvistendahl Karlsen, and Nils Henrik Risebro. On the more
practical side, the algorithm analyzed in this paper has turned out
to be useful for modeling sedimentation in clarifier-thickener
units, which are found in many industrial applications.
Could you summarize the significance of your paper in layman’s
terms?
The significance of this paper is that it presents a simple
method that can be programmed on a computer for a difficult
mathematical problem, and that problem has real-world applications.
The paper also explains why one can be confident that the results
that the computer gives will be accurate and valid.
How did you become involved in this research, and were there
successes or failures along the way?
I wrote my Ph.D. thesis on a topic related to this general area,
and then didn’t think about this branch of math for many years. In
1999, I stumbled onto a preprint server concerning conservation laws
that is maintained by Norwegian mathematicians. Through it I became
aware of the problem that this paper talks about, specifically
through the work of Klingenberg and Risebro, who developed a very
effective front-tracking algorithm for solving this type of problem.
I was curious to see if it would be possible to solve these problems
with a different method, specifically a difference scheme. The paper
documents the extent to which this effort was successful. Many
difficult questions remain unanswered concerning these equations and
their efficient solution on a computer.
John D. Towers
Professor of Mathematics
MiraCosta College
Cardiff, CA
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