In this work we apply the method of hydrodynamic reductions to study the integrability of the class of second order quasilinear equations.

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It is well known that a classical solution of the initial value problem for a scalar conservation law may fail to exist on the whole domain of definition of the problem. For this reason, suitable generalized solutions of such problems, known as weak solutions, have been considered and studied extensively. However, weak solutions are not unique. In order to obtain a unique solution that is physically relevant, the vanishing viscosity method, amongst others, has been employed to single out a unique solution known as the entropy solution. In this thesis we present an alternative approach to the study of the entropy solution of conservation laws. The main novelty of our approach is that the theory of entropy solution of conservation law is presented in an operator theoretic setting. In this regard, the Order Completion Method for nonlinear PDEs, in the context of convergence vector spaces, is modified to obtain an operator equation which generalizes the initial value problem. This equation admits at most one solution, which may be represented as a Hausdorff continuous function. As a particular case, we apply our method to obtain the entropy solution of the Burger's equation. Copyright / Dissertation (MSc)--University of Pretoria, 2011. / Mathematics and Applied Mathematics / Unrestricted

Identifying antibiotic resistance in blood infections requires separating bacteria from whole blood. A hollow spinning disk rapidly removes suspended red blood cells by leveraging hydrodynamic differences between bacteria and whole blood components in a centrifugal field. Once the red cells are removed, the supernatant plasma which contains bacteria is collected for downstream antibiotic testing. This work improves upon previous work by modifying the disk design to maximize fractional plasma recovery and minimize fractional red cell recovery. V-shaped channels induce plasma flow and increase fractional plasma recovery. Additionally, diluting a blood sample spiked with bacteria prior to spinning it increased the fractional bacteria recovery. A numerical model for red cell sedimentation shows that red cells are removed from solution more rapidly as the blood is diluted. Diluting blood is beneficial but may create too much biological waste. The benefits of diluting are formulated as an optimization problem subject to the end user’s needs.

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<p dir="ltr">This thesis introduces the evolving neural network method for solving scalar hyperbolic conservation laws. This method uses neural networks to compute solutions with an optimal moving mesh that evolves with the solution over time. The motivation for this method was to produce solutions with high accuracy near shocks while reducing the overall computational cost. The evolving neural network method first approximates initial data with a neural network producing a continuous piecewise linear approximation. Then, the neural network representation is evolved in time according to a combination of characteristics and a finite volume-type method.</p><p dir="ltr">It is shown numerically and theoretically that the evolving neural network method out performs traditional fixed-mesh methods with respect to computational cost. Numerical results for benchmark test problems including Burgers’ equation and the Buckley-Leverett equation demonstrate that this method can accurately capture shocks and rarefaction waves with a minimal number of mesh points.</p>

In this dissertation we present an analysis for the discontinuous Galerkin discretization error of multi-dimensional first-order linear symmetric and symmetrizable hyperbolic systems of conservation laws. We explicitly write the leading term of the local DG error, which is spanned by Legendre polynomials of degree p and p+1 when p-th degree polynomial spaces are used for the solution. For special hyperbolic systems, where the coefficient matrices are nonsingular, we show that the leading term of the error is spanned by (p+1)-th degree Radau polynomials. We apply these asymptotic results to observe that projections of the error are pointwise O(h^p+2)-superconvergent in some cases and establish superconvergence results for some integrals of the error. We develop an efficient implicit residual-based a posteriori error estimation scheme by solving local finite element problems to compute estimates of the leading term of the discretization error. For smooth solutions we obtain error estimates that converge to the true error under mesh refinement. We first show these results for linear symmetric systems that satisfy certain assumptions, then for general linear symmetric systems. We further generalize these results to linear symmetrizable systems by considering an equivalent symmetric formulation, which requires us to make small modifications in the error estimation procedure. We also investigate the behavior of the discretization error when the Lax-Friedrichs numerical flux is used, and we construct asymptotically exact a posteriori error estimates. While no superconvergence results can be obtained for this flux, the error estimation results can be recovered in most cases. These error estimates are used to drive h- and p-adaptive algorithms and assess the numerical accuracy of the solution. We present computational results for different fluxes and several linear and nonlinear hyperbolic systems in one, two and three dimensions to validate our theory. Examples include the wave equation, Maxwell's equations, and the acoustic equation. / Ph. D.

Neste trabalho, discutem-se aspectos das transformações conformes na eletrodinâmica clássica com ênfase na invariância e nas leis de conservação. Inicialmente, abordaram-se aspectos gerais das transformações conformes e fez-se um resumo histórico da evolução dessas transformações. Procurou-se fazer uma apresentação didática, revisando-se a formulação Lagrangiana e o Teorema de Noether para campos aplicado à eletrodinâmica. Estudaram-se as transformações conformes no espaço plano, onde se mostrou que para dimensões maiores ou iguais a três o número de transformações é finito. A partir das equações de Maxwell em coordenadas curvilíneas, chegou-se à condição para que essas equações mantivessem sua forma cartesiana. Com essa condição, mostrou-se que a eletrodinâmica clássica é invariante para o grupo de transformações conformes. Foram discutidas as leis de conservação associadas à invariância conforme da eletrodinâmica clássica a partir do teorema de Noether. Das simetrias por translações no espaço-tempo, obtiveram-se as leis de conservação do momento linear e da energia. Das simetrias associadas às rotações, obtiveram-se seis quantidades conservadas: três delas ligadas à conservação do momento angular e, com relação às três restantes, observou-se, a partir de analogias com a mecânica, que estavam associadas ao movimento do centro de energia do campo. Para a interpretação da grandeza conservada por simetria de escala, verificou-se, também a partir de uma analogia mecânica, que essa simetria somente é verificada para partículas não massivas ou para partículas massivas a altas energias. Finalmente, para as transformações conformes especiais, verificou-se que as leis de conservação resultantes são consequências das leis anteriores de conservação para o campo eletromagnético, e neste caso, essa simetria também somente se manifesta para partículas de massa nula ou para altas energias. / In this work, aspects of conformal transformations in classical electrodynamics are discussed with emphasis on the invariance and conservation laws. Initially, a general view of conformal transformations was shown and a summary of the historical evolution of those transformations was presented. The work was approached didactically, and Noethers theorem based on the electrodynamics Lagrangian formulation was revised. The conformal transformations were studied in plane spaces and it was shown that, for dimensions greater than or equal to three, the number of transformations is finite. Starting from Maxwells equations in curvilinear coordinates, a condition for maintaining those equations in Cartesian form was established. With that condition, it was shown that the classical electrodynamics laws are invariant for the group of conformal transformations. The conservation laws associated with the conformal invariance of classical electrodynamics were discussed, based on Noethers theorem. From the space-time translation symmetry, the laws of conservation of linear momentum and of energy were obtained. From rotational symmetry, six conserved quantities were obtained: three of them associated with angular momentum and the remaining three, observed, starting from analogies with mechanics, were associated with the movement of the center of energy of the field. For the interpretation of the quantity conserved by scale symmetry, it was verified, also from a mechanical analogy, that that symmetry is only valid for null mass particles or for high energies. Finally, for the special conformal transformations, it was verified that the resultant laws of conservation are consequences of the previous laws, and in that case, symmetry is also valid only for particles of null mass or for high energies.

L’analyse de sensibilité (AS) concerne la quantification des changements dans la solution d’un système d’équations aux dérivées partielles (EDP) dus aux varia- tions des paramètres d’entrée du modèle. Les techniques standard d’AS pour les EDP, comme la méthode d’équation de sensibilité continue, requirent de dériver la variable d’état. Cependant, dans le cas d’équations hyperboliques l’état peut présenter des dis- continuités, qui donc génèrent des Dirac dans la sensibilité. Le but de ce travail est de modifier les équations de sensibilité pour obtenir un syst‘eme valable même dans le cas discontinu et obtenir des sensibilités qui ne présentent pas de Dirac. Ceci est motivé par plusieurs raisons : d’abord, un Dirac ne peut pas être saisi numériquement, ce qui pourvoit une solution incorrecte de la sensibilité au voisinage de la discontinuité ; deuxièmement, les pics dans la solution numérique des équations de sensibilité non cor- rigées rendent ces sensibilités inutilisables pour certaines applications. Par conséquent, nous ajoutons un terme de correction aux équations de sensibilité. Nous faisons cela pour une hiérarchie de modèles de complexité croissante : de l’équation de Burgers non visqueuse au système d’Euler quasi-1D. Nous montrons l’influence de ce terme de correction sur un problème d’optimisation et sur un de quantification d’incertitude. / Sensitivity analysis (SA) concerns the quantification of changes in Partial Differential Equations (PDEs) solution due to perturbations in the model input. Stan- dard SA techniques for PDEs, such as the continuous sensitivity equation method, rely on the differentiation of the state variable. However, if the governing equations are hyperbolic PDEs, the state can exhibit discontinuities yielding Dirac delta functions in the sensitivity. We aim at modifying the sensitivity equations to obtain a solution without delta functions. This is motivated by several reasons: firstly, a Dirac delta function cannot be seized numerically, leading to an incorrect solution for the sensi- tivity in the neighbourhood of the state discontinuity; secondly, the spikes appearing in the numerical solution of the original sensitivity equations make such sensitivities unusable for some applications. Therefore, we add a correction term to the sensitivity equations. We do this for a hierarchy of models of increasing complexity: starting from the inviscid Burgers’ equation, to the quasi 1D Euler system. We show the influence of such correction term on an optimization algorithm and on an uncertainty quantification problem.