Applied Mathematics

Applied mathematics research at Argonne involves the development of appropriate numerical approximations that can provide sufficiently accurate solutions more efficiently than existing methods. We expect that these gains will come from several sources:

1. approximations that are tuned to the behavior of the solution,
2. parallel methods that use the structure of the solution to achieve higher efficiencies, and
3. better mathematical models of the original problems.

We are working with computational scientists to evaluate these new models and assess their domain of validity by means of numerical simulations

• Asymptotic Analysis and Domain Decomposition

This project addresses the development of asymptotic-enhanced numerical methods for boundary-value problems whose solutions show rapid variations over short distances, as in boundary layers and transition layers (shocks), or over short time intervals (initial layers).

• Scientific Sonification

This project addresses the use of sound, alone or in combination with visual images, for the exploration and analysis of complex data sets. Research focuses on finding coordinates in sound space that optimize the aural perception of given data sets and their salient features. Sonification (the rendering of data in sounds) is done with DIASS, a Digital Instrument for Additive Sound Synthesis. Sound objects are visualized in a VR environment to enhance their aural perception.

• Superconductivity

This project focuses on the numerical simulation of vortex dynamics in type-II superconductors. The simulations are based on macroscopic models, such as the time-dependent Ginzburg-Landau model of superconductivity and the elastic-string model of a vortex. Large-scale simulations involving hundreds or even thousands of vortices yield information about their collective behavior in the presence of external forces, material defects, thermal agitation, etc.

• Ginzburg-Landau Equations

This project focuses on nonlinear equations such as the Ginzburg-Landau equations, which model a variety of nonlinear physical phenomena near transition points. Emphasis is on finding characteristic behavior and universal features of their solutions.

• Incompressible Flow Simulations

This project addresses high-order spectral element discretizations and solution algorithms for time advancement of the incompressible Navier-Stokes equations. Several three-dimensional simulation examples are illustrated.

• Paul Fischer
• Hans Kaper
• Gary Leaf
• Sever Tipei