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Analyzing Traveling Waves in a Viscoelastic Generalization of Burgers' EquationCamacho, Victor 01 May 2007 (has links)
We analyze a pair of nonlinear PDEs describing viscoelastic fluid flow in one dimension. We give a summary of the physical derivation and nondimensionlize the PDE system. Based on the boundary conditions and parameters, we are able to classify three different categories of traveling wave solutions, consistent with the results in [?]. We extend this work by analyzing the stability of the traveling waves. We thoroughly describe the numerical schemes and software program, VISCO, that were designed specifically to analyze the model we study in this paper. Our simulations lead us to conjecture that the traveling wave solutions found in [?] are globally stable for all sets of initial conditions with the appropriate asymptotic boundary conditions. We are able give some analytical evidence in support of this hypothesis but are unsuccessful in providing a complete proof.
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A control problem for Burgers' equationKang, Sungkwon 01 February 2006 (has links)
Burgers' equation is a one-dimensional simple model for convection-diffusion phenomena such as shock waves, supersonic flow about airfoils, traffic flows, acoustic transmission, etc. For high Reynolds number, the open-loop system (no control) produces steep gradients due to the nonlinear nature of the convection.
The steep gradients are stabilized by feedback control laws. In this phase, sufficient conditions for the control input functions and the location of sensors are obtained. Also, explicit exponential decay rates for open-loop and closed-loop systems are obtained.
Numerical experiments are given to illustrate some of typical results on convergence and stability. / Ph. D.
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Model Reduction of the Coupled Burgers Equation in Conservation FormKramer, Boris 30 August 2011 (has links)
This thesis is a numerical study of the coupled Burgers equation. The coupled Burgers equation is motivated by the Boussinesq equations that are often used to model the thermal-fluid dynamics of air in buildings. We apply Finite Element Methods to the coupled Burgers equation and conduct several numerical experiments. Based on these results, the Group Finite Element method (GFE) appears to be more stable than the standard Finite Element Method. The design and implementation of controllers heavily relies on rapid solutions to complex models such as the Boussinesq equations. Thus, we further examine the feasibility and efficiency of the Proper Orthogonal Decomposition (POD) for the coupled Burgers equation. Using POD, we reduce the system to a "minimal" number of ODE's and conduct numerous numerical studies comparing the POD and GFE method. Further numerical experiments consider an application where the dynamics are projected on a POD basis and then the governing parameters of the system are varied. / Master of Science
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On the computational algorithms for optimal control problems with general constraints.Kaji, Keiichi January 1992 (has links)
A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy / In this thesis we used the following four types of optimal control problems:
(i) Problems governed by systems of ordinary differential equations;
(ii) Problems governed by systems of ordinary differential equations with
time-delayed arguments appearing in both the state and the control
variables;
(iii) Problems governed by linear systems subject to sudden jumps in parameter
values;
(iv) A chemical reactor problem governed by a couple of nonlinear diffusion
equations.
• The aim of this thesis is to devise computational algorithms for solving the optimal control
problems under consideration. However, our main emphasis are on the mathematical
theory underlying the techniques, the convergence properties of the algorithms and the
efficiency of the algorithms.
Chapters II and III deal with problems of the first type, Chapters IV and V deal with
problems of the second type, and Chapters VI and VII deal with problems of the third and
fourth type respectively. A few numerical problems have been included in each of these
Chapters to demonstrate the efficiency of the algorithms involved.
The class of optimal control problems considered in Chapter II consists of a nonlinear
system, a nonlinear cost functional, initial equality constraints, and terminal equality
constraints. A Sequential Gradient-Restoration Algorithm is used to devise an iterative
algorithm for solving this class of problems. 'I'he convergence properties of the algorithm
are investigated.
The class of optimal control problems considered in Chapter III consists of a nonlinear
system, a nonlinear cost functional, and terminal as well as interior points equality
constraints. The technique of control parameterization and Liapunov concepts are used to
solve this class of problems,
A computational algorithm for solving a class of optimal control problems involving
terminal and continuous state constraints or inequality type was developed by Rei. 103 in
1989. In Chapter IV, we extend the results of Ref. 103 to a more general class of
constrained time-delayed optimal control problems, which involves terminal state equality
constraints, as well as terminal state inequality constraints and continuous state inequality
constraints.
In Ref. 104, a computational scheme using the technique of control parameterization was
developed for solving a class of optimal control problems in which the cost functional includes the full variation of control. Chapter V is a straightforward extension of Ref. 104
to the time-delayed case. However the main contribution of this chapter is that many
numerical examples have been solved.
In Chapter VI, a class of linear systems subject to sudden jumps in parameter values is
considered. To solve this class of stochastic control problem, we try to seek for the best
feedback control law depending only on the measurable output. Based on this idea, we
convert the original problem into an approximate constrained deterministic optimization
problem, which can be easily solved by any existing nonlinear programming technique.
In Chapter VII, a chemical reactor problem and its control to achieve a desired output
temperature is considered. A finite element Galerkin method is used to convert the
original distributed optimal control problem into a quadratic programming problem with
linear constraints, which can he solved by any standard quadratic programming software . / Andrew Chakane 2018
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Adaptive mesh methods for numerical weather predictionCook, Stephen January 2016 (has links)
This thesis considers one-dimensional moving mesh (MM) methods coupled with semi-Lagrangian (SL) discretisations of partial differential equations (PDEs) for meteorological applications. We analyse a semi-Lagrangian numerical solution to the viscous Burgers’ equation when using linear interpolation. This gives expressions for the phase and shape errors of travelling wave solutions which decay slowly with increasing spatial and temporal resolution. These results are verified numerically and demonstrate qualitative agreement for high order interpolants. The semi-Lagrangian discretisation is coupled with a 1D moving mesh, resulting in a moving mesh semi-Lagrangian (MMSL) method. This is compared against two moving mesh Eulerian methods, a two-step remeshing approach, solved with the theta-method, and a coupled moving mesh PDE approach, which is solved using the MATLAB solver ODE45. At each time step of the SL method, the mesh is updated using a curvature based monitor function in order to reduce the interpolation error, and hence numerical viscosity. This MMSL method exhibits good stability properties, and captures the shape and speed of the travelling wave well. A meteorologically based 1D vertical column model is described with its SL solution procedure. Some potential benefits of adaptivity are demonstrated, with static meshes adapted to initial conditions. A moisture species is introduced into the model, although the effects are limited.
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Modelling and analysis of geophysical turbulence : use of optimal transforms and basis setsGamage, Nimal K. K. 06 August 1990 (has links)
The use of efficient basis functions to model and represent flows with
internal sharp velocity gradients, such as shocks or eddy microfronts, are
investigated. This is achieved by analysing artificial data, observed atmospheric
turbulence data and by the use of a Burgers' equation based spectral
model. The concept of an efficient decomposition of a function into a basis
set is presented and alternative analysis methods are investigated. The
development of a spectral model using a generalized basis for the Burgers'
equation is presented and simulations are performed using a modified Walsh
basis and compared with the Fourier (trigonometric) basis and finite difference
techniques.
The wavelet transform is shown to be superior to the Fourier transform
or the windowed Fourier transform in terms of defining the predominant
scales in time series of turbulent shear flows and in 'zooming in' on local
coherent structures associated with sharp edges. Disadvantages are found
to be its inability to provide clear information on the scale of periodicity of
events. Artificial time series of varying amounts of noise added to structures
of different scales are analyzed using different wavelets to show that the
technique is robust and capable of detecting sharp edged coherent structures
such as those found in shear driven turbulence.
The Haar function is used as a wavelet to detect ubiquitous zones of
concentrated shear in turbulent flows sometimes referred to as microfronts.
The location and organization of these shear zones suggest that they may be
edges of larger scale eddies. A wavelet variance of the wavelet phase plane is
defined to detect and highlight events and obtain measures of predominant
scales of coherent structures. Wavelet skewness is computed as an indicator
of the systematic sign preference of the gradient of the transition zone. Inverse
wavelet transforms computed at the dilation corresponding to the peak
wavelet variance are computed and shown to contain a significant fraction of
the total energy contained in the record. The analysis of data and the numerical
simulation results are combined to propose that the sharp gradients
normally found in shear induced turbulence significantly affect the nature of
the turbulence and hence the choice of the basis set used for the simulation
of turbulence. / Graduation date: 1991
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The viscosity of fiber suspensionsBlakeney, William Roy, January 1965 (has links) (PDF)
Thesis (Ph. D.)--Institute of Paper Chemistry, 1965. / Includes bibliographical references (p. 107-109).
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Dynamic compensators for a nonlinear conservation law /Marrekchi, Hamadi, January 1993 (has links)
Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1993. / Vita. Abstract. Includes bibliographical references (leaves 106-108). Also available via the Internet.
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Analysis and implementation of high-order compact finite difference schemes /Tyler, Jonathan, January 2007 (has links) (PDF)
Thesis (M.S.)--Brigham Young University. Dept of Mathematics, 2007. / Includes bibliographical references (p. 100-102).
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Impact of Discretization Techniques on Nonlinear Model Reduction and Analysis of the Structure of the POD BasisUnger, Benjamin 19 November 2013 (has links)
In this thesis a numerical study of the one dimensional viscous Burgers equation is conducted. The discretization techniques Finite Differences, Finite Element Method and Group Finite Elements are applied and their impact on model reduction techniques, namely Proper Orthogonal Decomposition (POD), Group POD and the Discrete Empirical Interpolation Method (DEIM), is studied. This study is facilitated by examination of several common ODE solvers. Embedded in this process, some results on the structure of the POD basis and an alternative algorithm to compute the POD subspace are presented. Various numerical studies are conducted to compare the different methods and the to study the interaction of the spatial discretization on the ROM through the basis functions. Moreover, the results are used to investigate the impact of Reduced Order Models (ROM) on Optimal Control Problems. To this end, the ROM is embedded in a Trust Region Framework and the convergence results of Arian et al. (2000) is extended to POD-DEIM. Based on the convergence theorem and the results of the numerical studies, the emphasis is on implementation strategies for numerical speedup. / Master of Science
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