Programm--Universiẗat Tubingen.

Thesis (doctoral)--Faculté des sciences de Paris, 1915. / Includes bibliographical references.

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A problem encountered in the boundary element method is the difficulty caused by corners and/or discontinuous boundary conditions. An existing code using standard linear continuous elements is modified to overcome such problems using the multiple node method with an auxiliary boundary collocation approach. Another code is implemented applying the gradient approach as an alternative to handle such problems. Laplace problems posed on variety of domain shapes have been introduced to test the programs. For Poisson problems the programs have been developed using a transformation to a Laplace problem. This method cannot be applied to solve Poissontype equations. The dual reciprocity boundary element method (DRM) which is a generalised way to avoid domain integrals is introduced to solve such equations. The gradient approach to handle corner problems is co-opted in the program using DRM. The program is modified to solve non-linear problems using an iterative method. Newton's method is applied in the program to enhance the accuracy of the results and reduce the number of iterations. The program is further developed to solve coupled Poisson-type equations and such a formulation is considered for the biharmonic problems. A coupled pair of non-linear equations describing the ohmic heating problem is also investigated. Where appropriate results are compared with those from reference solutions or exact solutions. v

Moment methods from the kinetic theory of gases exist as an alternative to the Navier-Stokes model. Models in this family are described by first-order hyperbolic PDEs with local relaxation. They provide a natural treatment for non-equilibrium effects and expand the regime for which the model is physically applicable past the Navier-Stokes level (when the continuum assumption breaks down). Discontinuous-Galerkin (DG) methods are very well suited for distributed parallel solution of first-order PDEs. This is because the optimal locality of the method minimizes needed communication between computational processes. One highly efficient, coupled space-time DG method that achieves third-order accuracy in both space and time while using only linear elements is the discontinuous-Galerkin Hancock (DGH) scheme, which was specifically designed for the efficient solution of PDEs resulting from moment closures. Third-order accuracy is obtained through the use of a technique originally proposed by Hancock. The combination of moment methods with the DGH discretization leads to a very efficient numerical treatment for viscous compressible gas flows that is accurate both in and out of local thermodynamic equilibrium. This thesis describe the first-ever implementation of this scheme for the solution of moment equations on large-scale distributed-memory computers. This implementation uses solution-directed automatic mesh refinement to increase accuracy while reducing cost. A linear hyperbolic-relaxation equation is used to verify the order of accuracy of the scheme. Next a supersonic compressible Euler case is used to demonstrate the mesh refinement as well as the scheme’s ability to capture sharp discontinuities. Third, a moment-closure is then used to compute a viscous mixing layer. This serves to demonstrate the ability of the first-order PDEs and the DG scheme to efficiently compute viscous solutions. A moment-closure is used to compute the solution for Stokes flow past a circular cylinder. This case reinforces the hyperbolic PDEs’ ability to accurately predict viscous phenomena. As this case is very low speed, it also demonstrates the numerical technique’s ability to accurately solve problems that are ill-conditioned due to the extremely low Mach number. Finally, the parallel efficiency of the scheme is evaluated on Canada’s largest supercomputer. It may be surprising to some that viscous flow behaviour can be accurately predicted by first-order PDEs. However, the applicability of hyperbolic moment methods to both continuum and non-equilibrium gas flows is now well established. Such a first-order treatment brings many physical and computational advantages to gas flow prediction.

<p> This thesis is concerned with the cooperative growth problem of the discontinuous precipitation reaction. Previous theories are examined, and the kinetic details of a model, which assumes the existence of a metastable monotectoid reaction and the adherence & local equilibrium, are derived. As with other attempt to describe the parameters of duplex growth situations we cannot find a unique relation between the rate of growth and the lamellar spacing. The various optimal or variational procedures used to remove this degree of freedom are considered. The Cu-In system is subject to quantitative experimental study, and the extent of interference of the concurrent general precipitation reaction is determined. Finally the kinetic data and auxiliary information are used to test the various theories. </p> / Thesis / Doctor of Philosophy (PhD)

As entropy is also an important quantity in physics, we relate our results to physical processes by demonstrating that the discontinuity of entropy can occur in a Markov chain. We also discuss the possible implications of this phenomenon in thermodynamics and cosmology. / For two probability distributions with finite alphabets, some bounds on the difference of their entropies as a function of their alphabet sizes and variational distance are obtained. These bounds, which are tighter than some existing results, show that entropy estimation by finite alphabet approximation may not work as we expected. On the other hand, we show that there always exists a finite alphabet approximation that works for entropy estimation provided that the alphabet used is sufficiently large. Some necessary and sufficient conditions under which the entropy of a sequence of probability distributions converges are given in terms of four new information divergence measures, where the square root of two of them are metrics. / In information theory, weak typicality and strong typicality are essential tools for proving coding theorems. Strong typicality, which is more powerful than weak typicality, can be applied to finite alphabet only, while weak typicality can be applied to both finite and countably infinite alphabets. We introduce a unified typicality for finite or countably infinite alphabet which is stronger than both weak typicality and strong typicality. With this unified typicality, the asymptotic equipartition property and the structural properties of strong typicality are preserved. / The Shannon information measures are well known to be continuous functions of the probability distribution for finite alphabet. In this thesis, however, we show that these measures are discontinuous with respect to almost all commonly used "distance" measures when the alphabet is countably infinite. Such "distance" measures include the Kullback-Leibler divergence and the variational distance. Specifically, we show that all the Shannon information measures are in fact discontinuous at all probability distributions. / Ho Siu Wai. / "August 2006." / Adviser: Wai Ho Raymond Yeung. / Source: Dissertation Abstracts International, Volume: 68-03, Section: B, page: 1824. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (p. 121-123). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

A Discontinuous Galerkin Time Domain method (DGTD), using a fourth order Runge-Kutta time-stepping of Maxwell's equations, was applied to the simulation of the optical properties of dielectric particles in two-dimensional (2-D) geometry. As examples of the numerical implementation of this method, the single-scattering properties of 2D circular and hexagonal particles are presented. In the case of circular particles, the scattering phase matrix was computed using the DGTD method and compared with the exact solution. For hexagonal particles, the DGTD method was used to compute single-scattering properties of randomly oriented 2-D hexagonal ice crystals, and results were compared with those calculated using a geometric optics method. Both shortwave (visible) and longwave (infrared) cases are considered, with particle size parameters 50 and 100. Ice in shortwave and longwave cases is absorptive and non-absorptive, respectively. The comparisons between DG solutions and the exact solutions in computing the optical properties of circular ice crystals reveal the applicability of the DG method to calculations of both absorptive and non-absorptive particles. In the hexagonal case scattering results are also presented as a function of both incident and scattering angles, revealing structure apparently not reported before. Using the geometric optics method we are able to interpret this structure in terms of contributions from varying numbers of internal reflections within the crystal.

In this paper a continuous Galerkin multiscale method (CGMM) and a discontinuous Galerkin multiscale method (DGMM) are proposed, both based on the variational multiscale method for solving partial differential equations numerically. The solution is decoupled into a coarse and a fine scale contribution, where the fine-scale contribution is computed on patches with localized right hand side. Numerical experiments are presented where exponential decay of the error is observed when increasing the size of the patches for both CGMM and DGMM. DGMM gives much better accuracy when the same size of the patches are used.