Research
Modelling and Scientific Computation
Application to Materials Science and Electrochemical Systems
Modelling and Scientific Computation
Application to Materials Science and Electrochemical Systems
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 interface problems including models of sea ice formation. I have explored equivalent formulations for the Oxygen Depletion problem (the simplest implicit free boundary value problem) and their corresponding numerical approximation to get insight into the more complicated models.
The expertise in scientific computing I have developed has found an interesting new application in neural network training. A particular type of network, a Neural Ordinary Differential Equation or NODE, may be amenable to efficient training with refinement. This is an area of current interest working with a new student.
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.
Some publications I am proud of for various reasons:
Wetton and Brooke, "One-way wave equations for seismo acoustic propagation in elastic wave guides," Journal of the Acoustical Society of America 87, 624-632 (1990).
Ascher, Ruuth, and Wetton, "Implicit-Explicit Methods for Time-Dependent PDE's," SIAM Journal on Numerical Analaysis 32, 797-823 (1995).
Bronsard and Wetton, "A Numerical Method for Tracking Curve Networks Moving with Mean Curvature Motion," Journal of Computational Physics 120, 66-87 (1995).
Wetton, "Error Analysis for Chorin's original fully discrete projection method and regularizations in space and time", SIAM Journal on Numerial Analysis 34, 1683-1697 (1997).
Stockie and Wetton, "Analysis of stiffness of the immersed boundary method and implications for time-stepping schemes," Journal of Computational Physics 154, 41-64 (1999).
Berg, Promislow, St-Pierre, Stumper, and Wetton, "Water Management in PEM fuel cells," Journal of the Electrochemical Society 151, A341-A353 (2004).
Donaldson and Wetton, "Solving Steady Interface Problems Using Residual Velocities," IMA Journal of Applied Mathematics 71, 877-897 (2006).
Chang, Kim, Promislow and Wetton, "Reduced Dimensional Computational Models of Polymer Electrolyte Membrane Fuel Cell Stacks," Journal of Computational Physics 223, 797-821 (2007).
Bridge and Wetton, "A mixture formulation for numerical capturing of a two-phase/vapour interface in a porous medium," Journal of Computational Physics 225, 2043-2068 (2007).
Pan and Wetton, "A numerical method for coupled surface and grain boundary motion," European Journal of Applied Mathematics 19, 311-327 (2008).
Promislow and Wetton, "PEM Fuel Cells: A Mathematical Overview" (invited review article) SIAM Journal of Applied Mathematics 70, 369 (2009).
Christlieb, Jones, Promislow, Wetton and Willoughby, "High accuracy solutions to energy gradient flows from material science models", Journal of computational physics 257, 193-215 (2014).
Lindstrom, Wetton and Kiefl, "Mathematical modelling of the effect of surface roughness on magnetic field profiles in type II superconductors," Journal of Engineering Mathematics 85, 149-177 (2014).
Moyles and Wetton, "A numerical framework for singular limits of a class of reaction diffusion problems," Journal of Computational Physics 300, 308-326 (2015).
Church, Guo, Jimack, Madzvamuse, Promislow, Wetton, Wise, and Yang, "High Accuracy Benchmark Problems for Allen-Cahn and Cahn-Hilliard Dynamics," Communications in Computational Physics 26, 947-972 (2019).
Cheng, Li, Promislow, and Wetton, "Asymptotic Behaviour of Time Stepping Methods for Phase Field Models," Journal of Scientific Computing 86, Article #32 (2021).
Cheng, Fu, and Wetton, "Equivalent Formulations of the Oxygen Depletion Problem, Other Implicit Moving Boundary Value Problems, and Implications for Numerical Approximation," SIAM Applied Mathematics 83, https://doi.org/10.1137/21M141916 (2023).