Mathematical Analysis of Large Data Sets

Monday, April 17, 2006, 3:40 p.m. PSA 113

Paul F. Fischer

Mathematics and Computer Science Division
Argonne National Laboratory

High-Order Methods for Petascale Computation

Abstract In many scientific computing applications, high-order methods (HOMs) provide the most effective pathway to highly accurate solution of the governing partial differential equations (PDEs). Such levels of accuracy have become increasingly important as tera- and petascale computations seek to elucidate subtle interscale and multiphysics interactions in applications such as plasma physics, climate modeling, accelerator design, MHD, and turbulence. For problems featuring a broad range of scales, it is vital that errors associated with marginally resolved high wavenumber components not overwhelm the physical interactions central to the simulation. Under such stringent error constraints, HOMs can be particularly cost effective. HOMs have also proven to be effective when physical dissipation is insufficient to eliminate small scale error, even when the numerical solution is not smooth (e.g., is only marginally resolved, as often occurs in cutting-edge simulations), despite well-known theoretical limitations on asymptotic convergence rates in such cases.

In the first half of this talk, we present some motivation, recent developments, and results for the spectral element method (SEM), which is a high-order weighted residual technique for numerical solution of PDEs. Several recent advances in the SEM, including multigrid solvers, discontinuous Galerkin formulations, stabilizing filters, and dealiasing have significantly expanded its applicability. Among these, we discuss high-order filtering, which has enabled SE-based flow simulations at Reynolds numbers that were intractable just a few years ago. We illustrate the potential of stabilized high-order methods through several application examples, including simulations of turbulence in vascular flows. In the second half of the talk, we present some important considerations in scaling PDE computations in general, and SEM-based computations in particular, to petascale architectures. Computational complexity analysis reveals some surprising and unexpected predictions for algorithms capable of scaling to more than 100,000 processors.

For further information please contact: Anne Gelb