Applied Mathematics Colloquium

While traditional numerical methods for PDEs are constructed using a basis representation used to discretize a differential operator (Finite-Differences, Finite-Elements, Spectral Collocation,...), only a few methods leverage knowledge about the structure of the solution, i.e. Green's function, characteristics, symplecticity etc.

I will show that using knowledge of the solution structure produces a framework for arbitrary order, optimally compact, and efficient schemes. I will illustrate this for two very different PDEs: the linear transport equation (hyperbolic), and Poisson's equation with co-dimension 1 jump discontinuities (elliptic).
When the structure of the solution is not known, we turn to the geometry of the operator. By determining the maximal Lie invariance group of the differential operator, we can build numerical discretizations that preserve a subgroup. Using this, I will show how to systematically construct schemes that lie within the structural framework described above. Furthermore, I will show this construction preserves order of convergence, compactness, and efficiency. This will be illustrated using simple examples.

Throughout this talk, I will motivate and illustrate these techniques using examples such as the dynamics of bubbles, drops, soap film, fluid/elastic-structure interaction, and topological control in multi-body problems.