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.
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.
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.
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.
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.
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.