John Stockie: Research

Research Interests:

My research focuses on solving problems in fluid dynamics that are characterized by multiphysics and multiple scales (in space and time). Mathematically, these problems are described by coupled nonlinear systems of partial differential equations to which I apply a combination of analytical methods including asymptotic analysis, periodic homogenization, Fourier analysis and linear stability theory. Computationally, my ultimate goal is to design accurate and efficient solution algorithms by solving the underlying PDEs using finite volume methods combined with stiff time-stepping schemes.

Most of the fluid flow problems I study are inspired by applications arising in either engineering or biology. My current projects can be classified into four general areas:

Industrial mathematics: Mathematical modelling of real problems from industry using PDEs and then employing analytical and computational approaches to gain insight into their solution. Current projects include studies of sap flow in maple trees, atmospheric pollutant transport and traffic flow. I also include in this category a new project on modelling of athlete performance in a cross-country ski race (recognizing that organized sport is a pretty major industry).
Fluid-structure interaction: Applying the immersed boundary method to simulate the interaction of an elastic deformable interface or structure with an incompressible fluid flow. I am motivated by applications from biology and engineering such as swimming marine organisms, sound wave propagation within the mammalian cochlea (or inner ear), and active suspensions.
Porous media flow: Studying moisture transport through porous materials that gives rise to nonlinear diffusion equations from the application of Darcy's Law. I am working on a diverse range of applications including tree sap hydraulics, gravity-driven fingering instabilities in soils, and osmotic transport through bio-membranes.
Scientific computation: A major component of all projects mentioned above is the development of accurate and efficient algorithms for solving systems of nonlinear (and mostly parabolic) PDEs. My expertise is primarily in finite volume schemes, although I do dabble in hyperbolic (or mixed) PDEs and other numerical methods . . .