This is simpler and more robust that finite difference or finite element methods as it doesn't depend explicitly on either distances or directions. The new method is especially good for solving vector problems such as the full Maxwell equations. It is interesting to do this in inhomogeneous random structures in order to treat photonic or acoustic localization in amorphous materials, and so on.
Applications exist in device physics and materials science.
Another problem of interest is the Navier stokes family of fluid dynamical equations. There are many chemical engineering applications to gas separation and catalysis in porous media. The student will be involved in coding the algorithms for computer simulations, primarily to demonstrate the methods and explore numerical analysis aspects. There are also several practical problems in transport and catalysis that can be addressed immediately. A student would be expected to have finished the equivalent of the first two courses of the introductory calculus sequence and be willing to learn some computer language.