Abstract

April 30, 2007

Title: High-Resolution and Adaptive Numerical Methods and Software for Partial Differential Equations
Presented by: Phil Colella
Lawrence Berkeley Lab, Berkeley, CA

Abstract

Problems in areas such as astrophysics, combustion, magnetic fusion, systems biology, and climate change are described in terms of various generalizations of the classical elliptic, parabolic, and hyperbolic partial differential equations (PDE) of mathematical physics. The enormous variety and subtlety in the behavior of these problems comes from the ways in which these classical PDEs are coupled, generalized, and combined with other models to describe multiple physical processes operating on multiple space and time scales. For that reason, the development of simulation software in these areas is an end-to-end activity, with the choices regarding mathematical models, discretization methods, and software design and implementation all strongly interacting with one another and with fundamental mathematical questions that underlie these choices. This places a premium on the availability of a diverse and agile software toolset that enables the use of computational experimentation to explore the design space.

The Applied Partial Differential Equations Center for Enabling Technologies (APDEC), funded by the U.S. Department of Energy SciDAC program, has been developing a collection of algorithmic and software components that can be assembled to simulate a broad range of complex multicomponent physical systems in which partial differential equations play a central role. Specifically, our goal is to enable the agile development of high-performance simulation codes for complex multiphysics and multiscale applications, by providing a flexible toolset that meets the following requirements:

• Capability: a sufficient variety of tools to support the broadest possible range of applications and the ability to combine the tools in as many ways as possible;
• Expressiveness: a factoring of the tools so that non-essential details are hidden from the user who doesn’t need them, while providing access to those details to a user who wants to customize the capabilities of the software; and
• Performance: the implementation of the tools to obtain the highest possible performance on high-end platforms.

In this talk, we will give an overview of the approach taken in the development of the APDEC software so that it meets these requirements; examples of the use of the tools on several applications; and the plans for developing the tools over the next five years.