Modern numerical methods for problems in physics

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The primary theme in this research program is to bring together state-of-the-art methods in scientific computing, physical modeling, and classical methods in applied mathematical analysis for studying problems that arise in fluid dynamics and material science. Mathematically, the equations to be solved are high-order, nonlinear partial differential equations, and the domains are often extremely complex. The underlying computational framework is to employ, whenever indicated, ``designer'' algorithms, i.e. those that target specific problems. These modern and powerful computational methods are able to accurately capture the solution of a problem by fully exploiting its underlying analytical structure. This is in contrast to more standard numerical methods that have been developed to have the widest possible application.