Brian Wetton

Implicit-explicit methods for time-dependent partial differential equations

Implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized partial differential equations (PDEs) of diffusion-convection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reaction-diffusion problems can also be approximated in this manner. In this work we systematically analyze the performance of such schemes, propose improved new schemes, and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods.For the prototype linear advection-diffusion equation, a stability analysis for first-, second-, third-, and fourth-order multistep IMEX schemes is performed. Stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified. Numerical experiments demonstrate that weak decay...

Water Management in PEM Fuel Cells

A simplified model for water management in a polymer electrolyte membrane (PEM) fuel cell operating under prescribed current with iso-potential plates is presented. The consumption of gases in the flow field channels, coupled to the electric potential and water content in the polymer membrane, is modeled in a two-dimensional slice from inlet to outlet and through the membrane. Both co- and counter-flowing air and fuel streams are considered, with attention paid to sensitivity of along-the-channel current density to inlet humidities, gas stream composition, and fuel and oxygen stoichiometries. The parameters describing the nonequilibrium kinetics of the membrane/catalyst interface are found to be fundamental to accurate fuel cell modeling. A new parameter which models nonequilibrium membrane water uptake rates is introduced. Four parameters, the exchange current, a membrane water transfer coefficient, an effective oxygen diffusivity, and an average membrane resistance, are fit to a subset of data and then held constant in subsequent runs which compare polarization curves, current density and membrane hydration distributions, water transfer, and stoichiometric sensitivity to the balance of the experimental data.

Implicit-Explicit Methods for Time-Dependent PDE''s

Implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusion-convection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reaction-diffusion problems can also be approximated in this manner. In this work we systematically analyze the performance of such schemes, propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods. For the prototype linear advection-diffusion equation, a stability analysis for first, second, third and fourth order multistep IMEX schemes is performed. Stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified. Numerical experiments demonstrate that weak decay of high frequency modes can lead to extra iterations on the finest grid when using multigrid computations with finite difference spatial discretization, and to aliasing when using spectral collocation for spatial discretization. When this behaviour occurs, use of weakly damping schemes such as the popular combination of Crank-Nicolson with second order Adams-Bashforth is discouraged and better alternatives are proposed. Our findings are demonstrated on several examples.

Analysis and computation of immersed boundaries, with application to pulp fibres

Immersed fibres are a very useful tool for modeling moving, elastic interfaces that interact with a surrounding fluid. The Immersed Boundary Method is a computational tool based on the immersed fibre model which has been used successfully to study a wide range of applications including blood flow in the heart and arteries and motion of suspended particles. This work centres around a linear analysis of an isolated fibre in two dimensions, which pinpoints a discrete set of solution modes associated solely with the fibre. We investigate the stability and stiffness characteristics of the fibre modes and then relate the results to the severe time step restrictions experienced in explicit and semi-implicit immersed boundary computations. A subset of the modes corresponding to tangential oscillations of the fibre are the main source of stiffness, which intensifies when the fibre force is increased or fluid viscosity is decreased--this explains why computations are limited to unrealistically small Reynolds numbers. We also investigate the effects of smoothing the fibre over a given thickness, which corresponds to the delta function approximation that is central to the discrete scheme. The results can be applied to explore the accuracy of various interpolation methods in an idealised setting. The analysis is next extended to predict time step restrictions and convergence rates for various explicit and semi-implicit discretisations. The results are verified in numerical experiments. Finally, we introduce a novel application of the Immersed Boundary Method to the motion of pulp fibres in a two-dimensional shear flow. The method is shown to reproduce both theoretical results and experimentally observed behaviour over a wide range of parameter values.

Stability analysis for the immersed fiber problem

A linear stability analysis is performed on a two-dimensional version of the “immersed fiber problem,” formulated by C. Peskin to model the flow of fluid in the presence of a mesh of moving, elastic fibers. The purpose of the analysis is to isolate the modes in the solution which are associated with the fiber and thereby determine the effect of the presence of a fiber on the fluid. The results are used not only to make conclusions about the stability of the problem but also to suggest guidelines for developing numerical methods for flows with immersed fibers.

TRANSPORT PHENOMENA IN THE POROUS CATHODE OF A PROTON EXCHANGE MEMBRANE FUEL CELL

Simultaneous heat and mass transfer in the porous cathode of a proton exchange membrane (PEM) fuel cell is generated by the exothermic chemical reaction at the catalyst layer. A two-dimensional steady state model in a cross section of the porous electrode taken normal to the gas flow in the channels is presented. We consider a multicomponent inlet feed composed of oxygen, nitrogen, and water vapor on the cathode side. The mathematical formulation reduces to a system of four nonlinear second-order elliptic partial differential equations subject to appropriate nonlinear boundary conditions. Numerical solutions are obtained using a finite-difference method. Results are presented for various operating conditions and design parameters in order to identify the important factors in the performance of the fuel cell. From the calculated values of the vapour pressure and temperature, the regions of vapor oversaturation are identified.

Convergence of a finite difference scheme for the Navier-Stokes equations using vorticity boundary conditions

A rigorous convergence result is presented for a finite difference scheme for the Navier-Stokes equations which uses vorticity boundary conditions. The approximating scheme is based on the vorticity-stream function formulation of the Navier-Stokes equations. The no-slip boundary condition is satisfied approximately by using a boundary condition of vorticity creation type. Convergence with second-order accuracy in vorticity and velocity is established for general domains in two space dimensions. Generalization to three space dimensions is also considered.

PEM Fuel Cells: A Mathematical Overview

We present an overview of the mathematical issues that arise in the modeling of polymer electrolyte membrane fuel cells. These issues range from nanoscale modeling of network structures arising in pore formation within the polymer and the formation of nanostructured agglomerates within the catalyst layer, to macroscale models of multiphase flow and water management, degradation of catalyst layers and membrane, and development of stack level codes. The dominant themes are the development and analysis of multiscale models and their reduction to simplified forms that are implementable in stack-level computations.

Reduced dimensional computational models of polymer electrolyte membrane fuel cell stacks

A model of steady state operation of polymer electrolyte membrane fuel cell (PEMFC) stacks with straight gas channels is presented. The model is based on a decoupling of transport in the down-channel direction from transport in the cross-channel plane. Further, cross-channel transport is approximated heuristically using one-dimensional processes. The model takes into account the consumption of reactants down the channel, the effect of membrane hydration on its conductivity, water crossover through the membrane, the electrochemistry of the oxygen reduction reaction, thermal transport within the membrane electrode assembly (MEA) and bipolar plates to the coolant, heat due to reaction and condensation and membrane resistance, electrical interaction between unit cells due to in-plane currents in the bipolar plates, and thermal coupling of unit cells through shared bipolar plates. The model corresponds to the typical operation with counter-flowing reactant gas streams. The model is a nonstandard system of non-smooth boundary value differential algebraic equations (DAEs) with strong, nonlocal coupling. A discretization of the system and a successful iterative strategy are described. Some preliminary analysis of the system and iterative strategy is given, using simple, constant coefficient, linear versions of the key components of the model. Representative computational results, validation against existing experimental data and a numerical convergence study are shown.

A Simple Scheme for Volume-Preserving Motion by Mean Curvature

In this article, we present a diffusion-generated approach for evolving volume-preserving motion by mean curvature. Our algorithm alternately diffuses and sharpens characteristic functions to produce a normal velocity which equals the mean curvature minus the average mean curvature. This simple algorithm naturally treats topological mergings and breakings and can be made very fast even when the volume constraint is enforced to double precision (or more). Two dimensional numerical studies are provided to demonstrate the convergence of the method for smooth and nonsmooth problems.