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 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 and my current projects can be broadly classified into four 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.
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 diverse applications from biology and engineering such as swimming marine organisms, biofilm dynamics and pulp fiber 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 water uptake in concrete, gravity-driven fingering instability in soils, tree sap hydraulics, 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 . . .