**Date & Time:** Monday, August 2, 2010, 16:00-17:00.

**Venue:** Ramanujan Hall

**Title:**Multilevel Solution Algorithms on Adaptively
Refined Grids: Analysis and Performance

**Speaker: ** Bobby Philip

Computer Science and Mathematics Division,
Oak Ridge National Laboratory

**Abstract:** Continuum partial differential equation (PDE) descriptions of complex physical
phenomena often exhibit multiscale solution features that span extreme spatio-
temporal scales. Examples include magnetohydrodynamics (MHD), multiphase
multicomponent reactive groundwater flows, hurricane system models, and
astrophysical jets, where the spatial and temporal scales often easily span more
than six orders in magnitude. Accuracy considerations often dictate that fine
scale solution features be resolved. However, uniform grid simulation at the
finest resolution required are infeasible even on today’s largest supercomputers
and will continue to be so for the foreseeable future. Adaptive mesh refinement
(AMR) is a numerical technique to introduce local grid resolution only where
required, significantly reducing the memory footprint and computational cost
associated with such simulations.
This talk will concentrate on two aspects for the accurate, efficient, and robust
long term time integration of stiff multiscale, multiphysics systems on AMR
grids. Firstly, the mathematical analysis and performance of an asynchronous
multilevel iterative solution algorithm, AFACx, for elliptic PDE systems will be
described. AMR level independent condition number estimates for the AFACx
error propagation operator will be presented and the performance of the solver
on First Order System Least Squares (FOSLS) formulations of scalar elliptic and
Stokes systems on curvilinear AMR grids will be presented. Secondly, examples
of accurate and efficient implicit time integration of nonlinear 2D reduced
resistive MHD, equilibrium, and non-equilibrium radiation diffusion on AMR
grids will be presented.
Time permitting, ongoing research on massively parallel groundwater flow
simulations on AMR grids and the design of scientific computing frameworks will
be briefly presented.