Calculating larger systems in less time

Adrianna Gillman is a professor at the University of Colorado at Boulder, USA. Thanks to a grant from the Knut and Alice Wallenberg Foundation, she will be a visiting professor at the Department of Mathematics, KTH Royal Institute of Technology, Stockholm.

Increasing numbers of physical experiments are now being carried out, entirely or partly, using computer simulations. For example, numerical simulations of fluid flows with very small water drops dispersed in oil are used to better understand biological systems or develop industrial processes. Each individual droplet, smaller than one thirty-thousandths of a millimetre in diameter, can contain a unique combination of substances or biological materials that is analysed in a micro-system.

The difficulty is that the droplets in the liquid move and deform, so their dynamics – the system’s time evolution – are difficult to simulate accurately. However, the main limitation is that simulations of systems with many particles are extremely expensive, as they require long computation times. The aim of the project is to reduce the computational cost, so that accurate simulations can be made with more particles over a longer period of time.

When the particles are small, the physics is well described by linear partial differential equations for viscous fluids – the Navier-Stokes equations. These were formulated in the early nineteenth century, but still pose a challenge to mathematicians because the
equations lack general solutions. However, numerical methods can be developed to solve them; for the relevant fluid flows simulations are performed using boundary integral equations, which are variants of the Navier-Stokes equations.

The use of boundary integral methods for fluid flows and the development and analysis of new algorithms and numerical techniques permits increased accuracy in the approximate solution. The aim is to reduce computing times and push the boundaries of what is considered possible in computer simulations.