My PhD topic was within geometric numerical analysis of differential equations. This is a field of numerical analysis that deals with the numerical solution of differential equations  while simultaneously preserving the physics and geometry behind the equations. This is in contrast to conventional numerical approaches that often break these physical laws.

Being able to accurately calculate the trajectories of small particles in fluid flows is important when understanding real-world phenomena involving particle suspensions. For example: pollutants in the atmosphere from combustion processes, micro-plastics or plankton drifting in the ocean,  droplets circulating through ventilation systems and so forth. 

The dynamics of many systems are often governed by physical laws, such as the conservation of energy or momentum. However, these conservation laws are usually broken when the equations are solved using standard numerical methods, which yield physically irrelevant solutions (e.g., the non-conservation of energy). 

Runge-Kutta methods are one of the most popular methods used in the computational sciences and have been studied  for over 120 years. However, leveraging the recently discovered theory of Discrete Darboux Polynomials (AKA second integrals, weak integrals) we can uncover new theorems about Runge-Kutta methods.