285 research outputs found
Thermal explosion analysis of a strong exothermic chemical reaction with variable pre-exponential factor in a spherical vessel
This study is devoted to investigate the analysis of thermal explosion of a strong exothermic
chemical reaction with variable pre-exponential factor in a spherical vessel. The steady state
solutions for strong exothermic decomposition of a combustible material uniformly distributed in a
heated spherical vessel under Bimolecular, Arrhenius and Sensitised reaction rates. Analytical
solutions are constructed for the governing nonlinear boundary-value problem using perturbation
technique together with a special type of Hermite-Padé approximants and important properties of
the temperature field including bifurcations and thermal criticality are discussed
Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics
Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit
Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations. I. The Dynamics of Time Discretization and Its Implications for Algorithm Development in Computational Fluid Dynamics
The goal of this paper is to utilize the theory of nonlinear dynamics approach to investigate the possible sources of errors and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic and parabolic partial differential equations terms. This interdisciplinary research belongs to a subset of a new field of study in numerical analysis sometimes referred to as the dynamics of numerics and the numerics of dynamics. At the present time, this new interdisciplinary topic is still the property of an isolated discipline with all too little effort spent in pointing out an underlying generality that could make it adaptable to diverse fields of applications. This is the first of a series of research papers under the same topic. Our hope is to reach researchers in the fields of computational fluid dynamics (CFD) and, in particular, hypersonic and combustion related CFD. By simple examples (in which the exact solutions of the governing equations are known), the application of the apparently straightforward numerical technique to genuinely nonlinear problems can be shown to lead to incorrect or misleading results. One striking phenomenon is that with the same initial data, the continuum and its discretized counterpart can asymptotically approach different stable solutions. This behavior is especially important for employing a time-dependent approach to the steady state since the initial data are usually not known and a freest ream condition or an intelligent guess for the initial conditions is often used. With the unique property of the different dependence of the solution on initial data for the partial differential equation and the discretized counterpart, it is not easy to delineate the true physics from numerical artifacts when numerical methods are the sole source of solution procedure for the continuum. Part I concentrates on the dynamical behavior of time discretization for scalar nonlinear ordinary differential equations in order to motivate this new yet unconventional approach to algorithm development in CFD and to serve as an introduction for parts II and III of the same series of research papers
Applications of dynamical systems with symmetry
This thesis examines the application of symmetric dynamical systems theory to
two areas in applied mathematics: weakly coupled oscillators with symmetry, and
bifurcations in flame front equations.
After a general introduction in the first chapter, chapter 2 develops a theoretical
framework for the study of identical oscillators with arbitrary symmetry group under an
assumption of weak coupling. It focusses on networks with 'all to all' Sn coupling. The
structure imposed by the symmetry on the phase space for weakly coupled oscillators
with Sn, Zn or Dn symmetries is discussed, and the interaction of internal symmetries
and network symmetries is shown to cause decoupling under certain conditions.
Chapter 3 discusses what this implies for generic dynamical behaviour of coupled
oscillator systems, and concentrates on application to small numbers of oscillators (three
or four). We find strong restrictions on bifurcations, and structurally stable heteroclinic
cycles.
Following this, chapter 4 reports on experimental results from electronic oscillator
systems and relates it to results in chapter 3. In a forced oscillator system, breakdown
of regular motion is observed to occur through break up of tori followed by a symmetric
bifurcation of chaotic attractors to fully symmetric chaos.
Chapter 5 discusses reduction of a system of identical coupled oscillators to phase
equations in a weakly coupled limit, considering them as weakly dissipative Hamiltonian
oscillators with very weakly coupling. This provides a derivation of example phase
equations discussed in chapter 2. Applications are shown for two van der Pol-Duffing
oscillators in the case of a twin-well potential.
Finally, we turn our attention to the Kuramoto-Sivashinsky equation. Chapter 6
starts by discussing flame front equations in general, and non-linear models in particular.
The Kuramoto-Sivashinsky equation on a rectangular domain with simple
boundary conditions is found to be an example of a large class of systems whose linear
behaviour gives rise to arbitrarily high order mode interactions.
Chapter 7 presents computation of some of these mode interactions using competerised
Liapunov-Schmidt reduction onto the kernel of the linearisation, and investigates
the bifurcation diagrams in two parameters
Dynamical problems and phase transitions
Issued as Financial status report, Technical reports [nos. 1-12], and Final report, Project B-06-68
A PHYSICS-BASED APPROACH TO MODELING WILDLAND FIRE SPREAD THROUGH POROUS FUEL BEDS
Wildfires are becoming increasingly erratic nowadays at least in part because of climate change. CFD (computational fluid dynamics)-based models with the potential of simulating extreme behaviors are gaining increasing attention as a means to predict such behavior in order to aid firefighting efforts. This dissertation describes a wildfire model based on the current understanding of wildfire physics. The model includes physics of turbulence, inhomogeneous porous fuel beds, heat release, ignition, and firebrands. A discrete dynamical system for flow in porous media is derived and incorporated into the subgrid-scale model for synthetic-velocity large-eddy simulation (LES), and a general porosity-permeability model is derived and implemented to investigate transport properties of flow through porous fuel beds. Note that these two developed models can also be applied to other situations for flow through porous media. Simulations of both grassland and forest fire spread are performed via an implicit LES code parallelized with OpenMP; the parallel performance of the algorithms are presented and discussed. The current model and numerical scheme produce reasonably correct wildfire results compared with previous wildfire experiments and simulations, but using coarser grids, and presenting complicated subgrid-scale behaviors. It is concluded that this physics-based wildfire model can be a good learning tool to examine some of the more complex wildfire behaviors, and may be predictive in the near future
Coupled complex Ginzburg-Landau systems with saturable nonlinearity and asymmetric cross-phase modulation
We formulate and study dynamics from a complex Ginzburg-Landau system with
saturable nonlinearity, including asymmetric cross-phase modulation (XPM)
parameters. Such equations can model phenomena described by complex
Ginzburg-Landau systems under the added assumption of saturable media. When the
saturation parameter is set to zero, we recover a general complex cubic
Ginzburg-Landau system with XPM. We first derive conditions for the existence
of bounded dynamics, approximating the absorbing set for solutions. We use this
to then determine conditions for amplitude death of a single wavefunction. We
also construct exact plane wave solutions, and determine conditions for their
modulational instability. In a degenerate limit where dispersion and
nonlinearity balance, we reduce our system to a saturable nonlinear
Schr\"odinger system with XPM parameters, and we demonstrate the existence and
behavior of spatially heterogeneous stationary solutions in this limit. Using
numerical simulations we verify the aforementioned analytical results, while
also demonstrating other interesting emergent features of the dynamics, such as
spatiotemporal chaos in the presence of modulational instability. In other
regimes, coherent patterns including uniform states or banded structures arise,
corresponding to certain stable stationary states. For sufficiently large yet
equal XPM parameters, we observe a segregation of wavefunctions into different
regions of the spatial domain, while when XPM parameters are large and take
different values, one wavefunction may decay to zero in finite time over the
spatial domain (in agreement with the amplitude death predicted analytically).
While saturation will often regularize the dynamics, such transient dynamics
can still be observed - and in some cases even prolonged - as the saturability
of the media is increased, as the saturation may act to slow the timescale.Comment: 36 page
Dynamics of Numerics & Spurious Behaviors in CFD Computations
The global nonlinear behavior of finite discretizations for constant time steps and fixed or adaptive grid spacings is studied using tools from dynamical systems theory. Detailed analysis of commonly used temporal and spatial discretizations for simple model problems is presented. The role of dynamics in the understanding of long time behavior of numerical integration and the nonlinear stability, convergence, and reliability of using time-marching approaches for obtaining steady-state numerical solutions in computational fluid dynamics (CFD) is explored. The study is complemented with examples of spurious behavior observed in steady and unsteady CFD computations. The CFD examples were chosen to illustrate non-apparent spurious behavior that was difficult to detect without extensive grid and temporal refinement studies and some knowledge from dynamical systems theory. Studies revealed the various possible dangers of misinterpreting numerical simulation of realistic complex flows that are constrained by available computing power. In large scale computations where the physics of the problem under study is not well understood and numerical simulations are the only viable means of solution, extreme care must be taken in both computation and interpretation of the numerical data. The goal of this paper is to explore the important role that dynamical systems theory can play in the understanding of the global nonlinear behavior of numerical algorithms and to aid the identification of the sources of numerical uncertainties in CFD
- …