In this study, a simple two-dimensional, unsteady Proton Exchange Membrane Fuel Cell (PEMFC) model is developed and validated by experimental results. The numerical model
considers fluid flow, mass transport and electrochemical reactions in the PEMFC cathode. The
vorticity-stream function method and Alternating Direction Implicit (ADI) scheme are employed
to solve the coupled fluid flow equations effciently. The I-V characteristics obtained from the
numerical model are in good agreement with the experimental results. The simulation results show
that the gas flow velocity, concentration of oxygen and porosity of gas diffusion layer significantly
uence the cell performance. Moreover, it could be inferred that, despite the real flow is three-
dimensional, a two-dimensional numerical model is time-effcient to predict the location of liquid
water formation and the fuel cell performance satisfactorily in some circumstances.
Lyapunov exponents (LEs) play a central role in the study of stability properties and asymptotic behavior of dynamical systems. However, explicit formulas for them can be derived for
very few systems, therefore numerical methods are required. Such is the case of random dynamical
systems described by stochastic differential equations (SDEs), for which there have been reported
just a few numerical methods. The first attempts were restricted to linear equations, which have
obvious limitations from the applications point of view. A more successful approach deals with
nonlinear equation defined over manifolds but is effective for the computation of only the top LE.
In this paper, two numerical methods for the efficient computation of all LEs of nonlinear SDEs
are introduced. They are, essentially, a generalization to the stochastic case of the well known
QR-based methods developed for ordinary differential equations. Specifically, a discrete and a
continuous QR method are derived by combining the basic ideas of the deterministic QR methods
with the classical rules of the differential calculus for the Stratanovich representation of SDEs.
Additionally, bounds for the approximation errors are given and the performance of the methods
is illustrated by means of numerical simulations.
The solution for the Navier-Stokes equations for incompressible steady state flow is presented using cubic spline (C2) continuous interpolation functions for the primary variables
(velocities and pressure) on rectangular domains. The solution was explored for laminar
low, intermediate and high inertia effects. Two problems (Fluid squeezed between two plates and
Wall-driven 2-D cavity flow) were solved using the presented scheme. Trial functions for velocities
and pressure were chosen with cubic spline continuous interpolation functions on a rectangular grid
that also satisfied the essential boundary conditions. The Galerkin weighted residual integrals were
evaluated for the continuity and momentum equations. Using interpolation functions that satisfy
the essential boundary conditions enabled the vanishing of any unknown boundary stress terms
in the developed equations. The nonlinear equations were solved using an iterative technique.
For low Reynolds number flows, coarse meshes were suffcient to reach convergence with very few
iterations. For higher Reynolds number flows, a relatively finer mesh was necessary to reach a
solution. The results show that cubic spline interpolation functions are suitable for solving the
incompressible steady state flow Navier-Stokes equations using the Galerkin weighted residuals
method. The chosen interpolation functions produced smooth continuous and differentiable results
with relatively coarse meshes.
In this paper, the system of the semiconductor device equations with heat effect is considered. An approximation to the system that makes use of a mixed finite element method
for the electrostatic potential equation combined with a finite element method for the densities
equations and the temperature equation is proposed. Existence and uniqueness of the approximate
solution are proved. A convergence analysis is also given.
We present an energy, cross-helicity and magnetic helicity preserving method for solving incompressible magnetohydrodynamic equations with strong enforcement of solenoidal
constraints. The method is a semi-implicit Galerkin finite element discretization, that enforces
pointwise solenoidal constraints by employing the Scott-Vogelius finite elements. We prove the
unconditional stability of the method and the optimal convergence rate. We also perform several
numerical tests verifying the effectiveness of our scheme and, in particular, its clear advantage
over using the Taylor-Hood finite elements.
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