Research

My original background is in Numerical Analysis of methods used to compute approximate solutions of partial differential equations. I have since become more motivated by industrial applications and have developed a skill in mathematical modelling.

I am still interested in some theoretical questions in Numerical Analysis and have some recent results in the analysis of errors from piecewise uniform grids used to approximate smooth solutions. I have also found a particularly simple setting to describe known results on the errors from a large class of projection methods for incompressible flow. I am considering writing a longer review on the subject of Asymptotic Error Analysis, the technique used to obtain these results and many other results from early in my career.

I have an ongoing interest in the development of numerical methods for geometric motion (curvature motion and surface diffusion for example) of curves in 2D and surfaces in 3D. These geometric problems are idealizations of some Material Science and Chemical Reaction models. They become more difficult to solve numerically when junctions are present in the curve (or surface) networks or there are nonlocal terms in the geometric motion. Much of my work in this area has been in an idealized setting, although some recent models developed by Keith Promislow show that geometric motion can also arise in models of the pore structure in fuel cell membranes (or more generally, in functionalized polymer materials). I have recently become interested in adaptive time and space methods for phase field versions of these geometric models working with Keith Promislow and others at Michigan State University.

Another problem that is of interest to me is the computational capturing of two phase zones in porous media. There are underyling questions of how to handle the degenerate parabolic features of such models numerically and analytically. Interest in this problem has also led to the development of some novel methods to compute a general class of steady free boundary value problems.

In the period 1998-2008, my main research activity was fuel cell modelling. In general terms, the work focussed on the development of computational design tools for the fuel cell industry. We developed models describing various aspects of unit cell operation (transport and chemical processes) and investigated coupling effects of the unit cells in stack operation. Corresponding computational models were developed. This was a larger project done in collaboration with the company, Ballard Power Systems, and the MITACS NCE. I am still interested in fuel cell modelling although I am also interested in pursuing other industrial projects with local industries. My expertise in electrochemical system modelling has found recent outlets in generalized dialysis systems and lithium ion battery packs. These latter projects are driven by collaboration with researchers in the Chemical and Biological Engineering Department.

Slides from some of my latest talks are listed at the bottom of the page.

Some recent preprints

Additional material for publications

Former Students

I value the achievements of graduate students, post-doctoral fellows, research associates, and undergraduate summer students I supervised. It was a pleasure working with you all!

Selected Publications

Some selected publications I'm proud of for various reasons:

Some recent talks

  • Asymptotic Analysis of Implicit Time Stepping for Allen Cahn Dynamics, in the Computational Applied Mathematics Seminar at MSU, April, 2018. The material has a different focus, but the videos are the same as the ones linked below (in a different time scale).
  • Time Stepping for Energy Gradient Flows, in the Computational Applied Mathematics Seminar at UT Knoxville, March, 2018.
  • Asymptotic Error Analysis, in the student collquium at UT Knoxville, March, 2018. Different versions were given at the SIAM student day in Limerick, Ireland, December, 2016, the PIMS Young Researcher Conference at UBC, June 2014, the PIMS-Shell Lunchbox series in Calgary, April 25, 2013 and University of Hawaii at Manoa Mathematics Department Seminar, February 24, 2012.
  • Modelling Lithium Ion Batteries SFU CSC Seminar, March, 2018. Previously given at the UBC SCAIM Seminar, November, 2017. Accompanying video:
  • Numerical Methods for Interface Motion PIMS Numerical Methods for PDEs on Surfaces Workshop, June, 2017.
  • A general framework for high accuracy solutions to energy gradient flows from material science models Southern China University of Science and Techonlogy May, 2017.
  • Mathematical Models of Electrochemical Systems, given at the Mathematics Applications Consortium for Science and Industry tenth anniversary workshop, Limerick, Ireland, December, 2016. Some additional detail on the basic fuel cell model described in the main part of the talk, and its computational approximation, can be found in the hand written notes used at a workshop given at the Polytechnic University of Catalunya, October, 2011: part I, part II, part III. Matlab code: stack.m and solve_stack.m.
  • PIMS Graduate Mathematical Modelling in Industry Workshop, Vancouver, August 2016:
  • Computational approaches to a model problem of two phase flow in porous media with phase change MSU applied mathematics seminar May 27, 2016. Accompanying videos:
  • Modelling a Generalized Dialysis Device at the Universitat Polytechnica de Catalunya, November 2015. This was given to a audience in a capstone undergraduate applied mathematics class. The computations discussed in the talk are done with the MATLAB code nitric_iv.m with function calls to nitric_solve.m. Sorry, these are not pretty codes.
  • A numerical framework for tracking interfaces in generalized Mullins-Sekerka dynamics MSU applied mathematics seminar May 29, 2015. A shorter version was given at the CAIMS annual meeting in Waterloo in June, 2015.
  • Computational Models of Electrochemical Systems in the UBC SCAIM seminar January 2015.
  • Modelling fuel cells in start-up and reactant starvation conditions at the IPAM workshop on Batteries and Fuel Cells, November 7, 2013 and previously in the Industrial Mathematics Session, CAIMS annual meeting, June 20, 2013.
  • Successes and Challenges of Industrial Mathematics, in the IAM seminar, September 16, 2013 and also a Public Lecture at the CAIMS annual meeting, June 16, 2013.
  • Lecture notes, for the summer school on Microstructure Evolution and Dynamics at the Technion in Israel, August 25-29, 2013. The MATLAB codes referred to in the notes can be found here: 2D Cahn Hilliard code (preconditioned conjugate gradient solve).