Better Numerical Solutions of Flow Equations 

Ludvig af Klinteberg received his Ph.D. in Mathematics from KTH Royal Institute of Technology in 2016. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Mary-Catherine Kropinski at the Department of Mathematics, Simon Fraser University, British Columbia, Canada. 

In the field of fluid dynamics, Navier-Stokes partial differential equations are used to describe the flow of fluids or gases. The equations have wide ranging applications in areas of study as diverse as aerodynamics, meteorology, and medicine. Even though the equations were first formulated over 200 years ago, no analytic solutions have been found except in a few special cases. However, with the help of today’s powerful computers, researchers have been able to obtain numerical solutions to many fluid dynamics problems, even though some have still remained beyond reach. 

The goal of the project is to extend the boundaries of what can be solved with computers, as well as improving computation speed by developing better and more efficient algorithms, and improving the accuracy of the numerical solutions. 

Fast integral equation methods have been successfully used to solve some equations in mathematical physics. The advantage of transforming differential equations into integral equations is that it results in a reduction in the number of dimensions, which can simplify certain geometric representations of the problem. It is possible to arrive at a numerical solution to an integral equation in a reasonable amount of computer time by applying already known algorithms. 

New ways of approaching fluid flow equations could build upon fast integral methods. However, this is not sufficient in itself to solve Navier-Stokes equations, as it requires a technically advanced process with both mathematical difficulties and new algorithms. Researchers in the field have postulated this line of attack for a long time. Mary-Catherine Kropinski was one of the first mathematicians to introduce such methods, twenty years ago. However, the field has only matured enough in the last few years to put solving the problem within researchers’ reach. Developing this approach further is the goal of the project.  

Photo: KTH Royal Institute of Technology