Differential equations can be used to model almost any scientific phenomenon. However, to obtain accurate approximation to the solutions of complex problems, simulation algorithms must be scaled to large numbers of processors. Research on algorithms for the solution of differential equations serves both as an intermediate testbed for work on software and tools and as a toolkit for implementing the specific "feedback" applications of interest to the CRPC. These algorithms have applications to problems in combustion, enhanced oil recovery, ocean and atmospheric circulation, and plasma physics. Work on numerical methods in computational fluid dynamics is particularly relevant to simulations in these application areas.

The group emphasizes the solution of three-dimensional problems and the effects of multi-scale and subgrid-scale phenomena in the areas of linear and nonlinear equations, domain decomposition techniques, continuation methods, and discretization methods, particularly those tailored for computational fluid dynamics. In collaboration with the Parallel Paradigm Integration project, many of the algorithms developed by the Differential Equations group are being incorporated into programming templates.

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Herb Keller is an internationally recognized numerical analyst who has made
important contributions to large-scale scientific computing and
computational fluid dynamics. He has written several texts, research
monographs, and more than 140 research papers and he has directed the
dissertations of 25 Ph.D. students. He has been at Caltech since 1967, when
he departed the Courant Institute where he had been the associate director
of the AEC Computing and Applied Mathematics Center. He is a past president
of SIAM and a Fellow of the American Academy of Arts and Science and the
Guggenheim Foundation. He is an editor of numerous journals and a monograph
series.
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Andrew White's research interests are in adaptive and moving grid
techniques and schemes, accurate finite difference (and element and volume)
discretizations on irregular grids, high-performance computation and
networking, and theory and simulation of nonlinear diffusive phenomena
including diffusion in polymer entanglement networks and flow in porous
media. White received his Ph.D. in 1974 in applied mathematics from the
California Institute of Technology. He is currently the Deputy Division
Leader of the Computing and Communications Division at Los Alamos National
Laboratory (LANL). He is also the director of LANL's Advanced Computing
Laboratory, a member of the Basic Energy Sciences Advisory Committee,
co-director of the CRPC Differential Equations group, manager of LANL's
Applied Mathematical Sciences program, and on the editorial board of
Concurrent Computation: Theory and Practice.
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The group is planning to reduce the communications costs in adaptive mesh refinement (AMR) by using a data structure that requires only fast communications. AMR techniques for finite difference methods have resolved approximate solutions of partial differential equations without requiring a fine lattice or small time step in every part of the field. As with front reconstruction, however, AMR can lead to load imbalances and large communications overhead if not carefully implemented on SIMD architectures.

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Dan Meiron works in the area of scientific computations with particular
emphasis on computational fluid dynamics. Current active areas include
vortex reconnection, pattern selection in solidifying systems, and
Richtmyer-Meshkov instability. An active collaboration with Mani Chandy is
devoted to building a library of templates for scientific computations.
These will hide the details of the parallel communication and make the
transformation from a sequential to a parallel program straightforward.
They are designed to aid those working in application areas who make use of
spectral codes and linear algebra.
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Jeffrey Saltzman received his B.S. in applied mathematics, physics, and
engineering (1977) from the University of Wisconsin, Madison, and his M.S.
and Ph.D. in mathematics from the Courant Institute in 1981. After his
Ph.D. he worked in the Applied Theoretical Physics Division at Los Alamos
National Laboratory (LANL) on laser fusion simulations. For the last
several years he been working in the Computer Research Group (C-3) of the
Computing Division on the numerical solution of partial differential
equations. Saltzman is currently the section leader of the Applied Math
Section in the C-3 group at LANL.
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