Differential Equations

Herb Keller (co-director), Andrew White (co-director), Todd Arbogast, Clint Dawson, Joel Dendy, Eusebius Doedel, Donald Estep, David Forslund, David Harrar, Mac Hyman, Phil Keenan, Dan Meiron, Dale Pullin, Steve Roy Karmesin, Jeff Saltzman, Eric Van de Velde, Mary Wheeler, Roy Williams, and Nai-Ying Zhang

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.

Multilevel Methods

Algorithms involving nested grids have proven to be particularly useful in scientific computation. However, they have usually not performed optimally on distributed-memory parallel architectures because of problems with data structures and locality. The Differential Equations group is concentrating on multilevel algorithms in four areas: discretized algebraic equations, segment relaxation, nonlinear equations, and homogenization.

Domain Decomposition

Domain decomposition can be used to impose a structure on the code that advantageously restricts the amount and type of interprocessor communication required to solve the problem. The Subsurface Modeling Group (SMG) is developing codes for parallel domain decomposition and multigrid for large non-linear systems of coupled elliptic and parabolic partial differential equations. Other research areas include the development of operator-based averaging techniques for rough coefficient problems, local mesh refinements near localized heterogeneities or singularities (such as around horizontal and vertical wells), and extensions of the algorithms to nonstructured grids. Major difficulties that arise in these models include load balancing, time-step constraints, the definition of effective preconditioners for solving the interface problem that connects the subdomains, and the ability to follow interacting reactive and advective flows.

Homotopy and Continuation Methods

Research in continuation and homotopy methods for solving nonlinear problems will emphasize parallel algorithms and implementations. A joint collaboration effort with the RIACS group at NASA-Ames is examining the use of homotopy methods as an aid for generating accurate simulations in difficult parameter ranges. A basic class of model problems to which a new library should be applicable is the discretization of the steady-state Navier-Stokes equations in two or three dimensions. New procedures for stabilization of steady states in time-evolution methods will be extended to general fixed-point interactions, stiff systems, periodic solutions, and other areas, when possible.

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.

Discretization Techniques and Computational Fluid Dynamics

Computational fluid dynamics projects concerning vortex reconnection and chemically reacting hypersonic flows play an important part in the Differential Equations group's plans. These simulations are possible only on parallel machines. In addition, a joint effort with Argonne National Laboratory is examining the transition from stable laminar flow over a rear-facing step to unsteady, oscillating flow. Finally, the treatment of materials interfaces will be compared with the treatment in the three-dimensional multimaterial fluid dynamics PAGOSA code.

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.

Irregular-grid Codes

Using the Voxel database concept of Roy Williams, the group is exploring the application of dynamically-adaptive irregular grids to reaction-diffusion problems. The Voxel database is a concurrent, structured, and general interface for dynamically changing grids. To decide where grids need to be refined or coarsened, they use rigorous error-estimation techniques developed by Donald Estep,and to solve the resulting sparse matrix problems group researchers use parallel solvers from Eric Van de Velde.

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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