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

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Mathematics and Computer Science Division

Argonne National Laboratory

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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