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1 
Solitary wave solutions for the magma equation: symmetry methods and conservation lawsMindu, Nkululeko 30 January 2015 (has links)
A dissertation submitted for the degree of Masters of Science, School of Computational and Applied Mathematics, University of Witwatersrand, Johannesburg, 2014. / The magma equation which models the migration of melt upwards through
the Earth’s mantle is considered. The magma equation depends on the permeability
and viscosity of the solid mantle which are assumed to be a function
of the voidage . It is shown using Lie group analysis that the magma equation
admits Lie point symmetries provided the permeability and viscosity satisfy
either a power law, or an exponential law for the voidage or are constant. The
conservation laws for the magma equation for both power law and exponential
law permeability and viscosity are derived using the multiplier method.
The conserved vectors are then associated with Lie point symmetries of the
magma equation. A rarefactive solitary wave solution for the magma equation
is derived in the form of a quadrature for exponential law permeability and viscosity.
Finally small amplitude and large amplitude approximate solutions are
derived for the magma equation when the permeability and viscosity satisfy
exponential laws.

2 
Efficient upwind algorithms for solution of the Euler and NavierStokes equationsMcNeil, C. Y. January 1995 (has links)
An efficient threedimensional structured solver for the Euler and NavierStokes equations is developed based on a finite volume upwind algorithm using Roe fluxes. Multigrid and optimal smoothing multistage time stepping accelerate convergence. The accuracy of the new solver is demonstrated for inviscid flows in the range 0.675 :5M :5 25. A comparative grid convergence study for transonic turbulent flow about a wing is conducted with the present solver and a scalar dissipation central difference industrial design solver. The upwind solver demonstrates faster grid convergence than the central scheme, producing more consistent estimates of lift, drag and boundary layer parameters. In transonic viscous computations, the upwind scheme with convergence acceleration is over 20 times more efficient than without it. The ability of the upwind solver to compute viscous flows of comparable accuracy to scalar dissipation central schemes on grids of onequarter the density make it a more accurate, cost effective alternative. In addition, an original convergencea cceleration method termed shock acceleration is proposed. The method is designed to reduce the errors caused by the shock wave singularity M + 1, based on a localized treatment of discontinuities. Acceleration models are formulated for an inhomogeneous PDE in one variable. Results for the Roe and EngquistOsher schemes demonstrate an order of magnitude improvement in the rate of convergence. One of the acceleration models is extended to the quasi onedimensiona Euler equations for duct flow. Results for this case d monstrate a marked increase in convergence with negligible loss in accuracy when the acceleration procedure is applied after the shock has settled in its final cell. Typically, the method saves up to 60% in computational expense. Significantly, the performance gain is entirely at the expense of the error modes associated with discrete shock structure. In view of the success achieved, further development of the method is proposed.

3 
Turbulent wake flows: lie group analysis and conservation lawsHutchinson, Ashleigh Jane January 2016 (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. March 2016. / We investigate the twodimensional turbulent wake and derive the governing equations
for the mean velocity components using both the eddy viscosity and the Prandtl
mixing length closure models to complete the system of equations. Prandtl’s mixing
length model is a special case of the eddy viscosity closure model. We consider an
eddy viscosity as a function of the distance along the wake, the perpendicular distance
from the axis of the wake and the mean velocity gradient perpendicular to the
axis of thewake. We calculate the conservation laws for the system of equations using
both closure models. Three main types of wakes arise from this study: the classical
wake, the wake of a selfpropelled body and a new wake is discovered which we call
the combination wake. For the classical wake, we first consider the case where the
eddy viscosity depends solely on the distance along the wake. We then relax this condition
to include the dependence of the eddy viscosity on the perpendicular distance
from the axis of the wake. The Lie point symmetry associated with the elementary
conserved vector is used to generate the invariant solution. The profiles of the mean
velocity show that the role of the eddy viscosity is to increase the effective width of
the wake and decrease the magnitude of the maximum mean velocity deficit. An infinite
wake boundary is predicted fromthis model. We then consider the application
of Prandtl’s mixing length closure model to the classical wake. Previous applications
of Prandtl’s mixing length model to turbulent wake flows, which neglected the kinematic
viscosity of the fluid, have underestimated the width of the boundary layer. In
this model, a finite wake boundary is predicted. We propose a revised Prandtl mixing
length model by including the kinematic viscosity of the fluid. We show that this
model predicts a boundary that lies outside the one predicted by Prandtl. We also
prove that the results for the two models converge for very large Reynolds number
wake flows. We also investigate the turbulentwake of a selfpropelled body. The eddy
viscosity closure model is used to complete the system of equations. The Lie point
symmetry associated with the conserved vector is derived in order to generate the
invariant solution. We consider the cases where the eddy viscosity depends only on
the distance along the wake in the formof a power law and when a modified version
of Prandtl’s hypothesis is satisfied. We examine the effect of neglecting the kinematic
viscosity. We then discuss the issues that arisewhenwe consider the eddy viscosity to
also depend on the perpendicular distance from the axis of the wake. Mean velocity
profiles reveal that the eddy viscosity increases the boundary layer thickness of the
wake and decreases the magnitude of the maximum mean velocity. An infinite wake
boundary is predicted for this model. Lastly, we revisit the discovery of the combination
wake. We show that for an eddy viscosity depending on only the distance along
the axis of the wake, a mathematical relationship exists between the classical wake,
the wake of a selfpropelled body and the combination wake. We explain how the
solutions for the combination wake and the wake of a selfpropelled body can be
generated directly from the solution to the classical wake. / GR 2016

4 
Glimm type functional and one dimensional systems of hyperbolic conservation laws /Hua, Jiale. January 2009 (has links) (PDF)
Thesis (Ph.D.)City University of Hong Kong, 2009. / "Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves 8895)

5 
Conservation laws in optimal control theory / Aaron Baetsane TauTau, Baetsane Aaron January 2005 (has links)
Abstract: We study in optimal control the important relation between invariance
of the problem under a family of transformations, and the existence
of preserved quantities along the Pontryagin extremals. Several extensions of
Noether's theorem are given, in the sense which enlarges the scope of its application.
The dissertation looks at extending the second Noether's theorem
to optimal control problems which are invariant under symmetry depending
upon k arbitrary functions of the independent variable and their derivatives
up to some order m. Furthermore, we look at the Conservation Laws, i.e.
conserved quantities along EulerLagrange extremals, which are obtained on
the basis of Noether's theorem.
And finally we obtain a generalization of Noether's theorem for optimal control
problems. The generalization involves a oneparameter family of smooth
maps which may depend also on the control and a Lagrangian which is invariant
up to an addition of an exact differential. / (M.Sc.) NorthWest University, Mafikeng Campus, 2005

6 
Group invariant solutions and conservation laws for jet flow models of nonNewtownian powerlaw fluidsMagan, Avnish Bhowan 18 July 2014 (has links)
The nonNewtonian incompressible powerlaw
uid in jet
ow models is investigated.
An important feature of the model is the de nition of a suitable
Reynolds number, and this is achieved using the standard de nition of a
Reynolds number and ascertaining the magnitude of the e ective viscosity.
The jets under examination are the twodimensional free, liquid and wall
jets. The twodimensional free and wall jets satisfy a di erent partial di erential
equation to the twodimensional liquid jet. Further, the jets are reformulated
in terms of a third order partial di erential equation for the stream
function. The boundary conditions for each jet are unique, but more signi 
cantly these boundary conditions are homogeneous. Due to this homogeneity
the conserved quantities are critical in the solution process.
The conserved quantities for the twodimensional free and liquid jet are
constructed by rst deriving the conservation laws using the multiplier approach.
The conserved quantity for the twodimensional free jet is also derived
in terms of the stream function. For a Newtonian
uid with n = 1 the twodimensional
wall jet gives a conservation law. However, this is not the case for
the twodimensional wall jet for a nonNewtonian powerlaw
uid.
The various approaches that have been applied in an attempt to derive a
conservation law for the twodimensional wall jet for a powerlaw
uid with
n 6= 1 are discussed. In conjunction with the attempt at obtaining conservation
laws for the twodimensional wall jet we present tenable reasons for its failure,
and a feasible way forward.
Similarity solutions for the twodimensional free jet have been derived for
both the velocity components as well as for the stream function. The associated
Lie point symmetry approach is also presented for the stream function. A
parametric solution has been obtained for shear thinning
uid free jets for
0 < n < 1 and shear thickening
uid free jets for n > 1. It is observed that for
values of n > 1 in the range 1=2 < n < 1, the velocity pro le extends over a
nite range.
For the twodimensional liquid jet, along with a similarity solution the
complete Lie point symmetries have been obtained. By associating the Lie
point symmetry with the elementary conserved vector an invariant solution
is found. A parametric solution for the twodimensional liquid jet is derived
for 1=2 < n < 1. The solution does not exist for n = 1=2 and the range 0 < n < 1=2 requires further investigation.

7 
A local extrapolation method for hyperbolic conservation laws: the ENO and GoodmanLeVeque underlying schemes and sufficient conditions for TVD propertyAdongo, Donald Omedo January 1900 (has links)
Doctor of Philosophy / Department of Mathematics / Marianne Korten / Charles N. Moore / We start with linear single variable conservation laws and examine the conditions under
which a local extrapolation method (LEM) with upwinding underlying scheme is total
variation diminishing TVD. The results are then extended to nonlinear conservation laws.
For this later case, we restrict ourselves to convex flux functions f, whose derivatives are
positive, that is, f A0 and f A0. We next show that the GoodmanLeVeque flux satisfies
the conditions for the LEM to be applied to it. We make heavy use of the CFL conditions,
the geometric properties of convex functions apart from the martingle type properties of
functions which are increasing, continuous, and differentiable.

8 
Analytical solutions and conservation laws of models describing heat transfer through extended surfacesNdlovu, Partner Luyanda 29 July 2013 (has links)
A dissertation submitted to the Faculty of Science,
University of the Witwatersrand, in fulfillment of the
requirements for the degree of Master of Science.
March 28, 2013 / The search for solutions to the important differential equations arising in extended
surface heat transfer continues unabated. Extended surfaces, in the
form of longitudinal fins are considered. First we consider the steady state
problem and then the transient heat transfer models. Here, thermal conductivity
and heat transfer coefficient are assumed to be functions of temperature.
Thermal conductivity is considered to be given by the power law in one case
and by the linear function of temperature in the other; whereas heat transfer
coefficient is only given by the power law. Explicit analytical expressions for
the temperature profile, fin efficiency and heat flux for steady state problems
are derived using the onedimensional Differential Transform Method (1D DTM).
The obtained results from 1D DTM are compared with the exact solutions
to verify the accuracy of the proposed method. The results reveal that the 1D
DTM can achieve suitable results in predicting the solutions of these problems.
The effects of some physical parameters such as the thermogeometric
fin parameter and thermal conductivity gradient, on temperature distribution
are illustrated and explained. Also, we apply the twodimensional Differential
Transform Method (2D DTM) to models describing transient heat transfer in
longitudinal fins. Furthermore, conservation laws for transient heat conduction
equations are derived using the direct method and the multiplier method, and
finally we find Lie point symmetries associated with the conserved vectors.

9 
Symmetries, conservation laws and reductions of Schrodinger systems of equationsMasemola, Phetogo 12 June 2014 (has links)
One of the more recently established methods of analysis of di erentials involves the
invariance properties of the equations and the relationship of this with the underlying
conservation laws which may be physical. In a variational system, conservation laws
are constructed using a well known formula via Noether's theorem. This has been
extended to non variational systems too. This association between symmetries and
conservation laws has initiated the double reduction of di erential equations, both
ordinary and, more recently, partial. We apply these techniques to a number of well
known equations like the damped driven Schr odinger equation and a transformed
PT symmetric equation(with Schr odinger like properties), that arise in a number
of physical phenomena with a special emphasis on Schr odinger type equations and
equations that arise in Optics.

10 
Adjointbased optimization for optimal control problems governed by nonlinear hyperbolic conservation lawsYohana, Elimboto 05 September 2012 (has links)
Research considered investigates the optimal control problem which is constrained by a hyperbolic
conservation law (HCL). We carried out a comparative study of the solutions of the
optimal control problem subject to each one of the two di erent types of hyperbolic relaxation
systems [64, 92]. The objective was to employ the adjointbased optimization to minimize the
cost functional of a matching type between the optimal solution and the target solution. Numerical
tests were then carried out and promising results obtained. Finally, an extension was
made to the adjointbased optimization approach to apply secondorder schemes for the optimal
control problem in which also good numerical results were observed.

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