Recently there have been numerous advances in the development of numerical algorithms to solve conservation laws. Even though the analytical theory (existence-uniqueness) is complete in the case of scalar conservation laws, there are many numerically robust methods for which the question of convergence and error estimates are still open. Usually high order schemes are constructed to be Total Variation Diminishing (TVD) which only guarantees convergence of such schemes to a weak solution. The standard approach in proving convergence to the entropy solution is to try to establish cell entropy inequalities. However, this typically requires additional non-homogeneous limitations on the numerical method, which reduces the modified scheme to first order when the mesh is refined. There are only a few results on the convergence which do not impose such limitations and all of them assume some smoothness on the initial data in addition to L^infinity bound. The Nessyahu-Tadmor (NT) scheme is a typical example of a high order scheme. It is a simple yet robust second order non-oscillatory scheme, which relies on a non-linear piecewise linear reconstruction. A standard reconstruction choice is based on the so-called minmod limiter which gives a maximum principle for the scheme. Unfortunately, this limiter reduces the reconstruction to first order at local extrema. Numerical evidence suggests that this limitation is not necessary. By using MAPR-like limiters, one can allow local nonlinear reconstructions which do not reduce to first order at local extrema. However, use of such limiters requires a new approach when trying to prove a maximum principle for the scheme. It is also well known that the NT scheme does not satisfy the so-called strict cell entropy inequalities, which is the main difficulty in proving convergence to the entropy solution. In this work, the NT scheme with MAPR-like limiters is considered. A maximum principle result for a conservation law with any Lipschitz flux and also with any k-monotone flux is proven. Using this result it is also proven that in the case of strictly convex flux, the NT scheme with a properly selected MAPR-like limiter satisfies an one-sided Lipschitz stability estimate. As a result, convergence to the unique entropy solution when the initial data satisfies the so-called one-sided Lipschitz condition is obtained. Finally, compensated compactness arguments are employed to prove that for any bounded initial data, the NT scheme based on a MAPR-like limiter converges strongly on compact sets to the unique entropy solution of the conservation law with a strictly convex flux.

[by] Norman J. Padelford. / Includes bibliographical references.

The objective of this dissertation is to develop robust and accurate numerical methods for solving the compressible, non-linear Euler equations of gas dynamics in one and two space dimensions. In theory, solutions of the Euler equations can display various characteristics including shock waves, rarefaction waves and contact discontinuities. To capture these features correctly, highly accurate numerical schemes are designed. In this thesis, two different projects have been studied to show the accuracy and utility of these numerical schemes. Firstly, the compressible, non-linear Euler equations of gas dynamics in one space dimension are considered. Since the non-linear partial differential equations (PDEs) can develop discontinuities (shock waves), the numerical code is designed to obtain stable numerical solutions of the Euler equations in the presence of shocks. Discontinuous solutions are defined in a weak sense, which means that there are many different solutions of the initial value problems of PDEs. To choose the physically relevant solution among the others, the entropy condition was applied to the problem. This condition is then used to derive a bound on the solution in order to satisfy L2-stability. Also, it provides information on how to add an adequate amount of diffusion to smooth the numerical shock waves. Furthermore, numerical solutions are obtained using far-field and no penetration (wall) boundary conditions. Grid interfaces were also included in these numerical computations. Secondly, the two dimensional compressible, non-linear Euler equations are considered. These equations are used to obtain numerical solutions for compressible ow in a shock tube with a 90° circular bend for two channels of different curvatures. The cell centered finite volume numerical scheme is employed to achieve these numerical solutions. The accuracy of this numerical scheme is tested using two different methods. In the first method, manufactured solutions are used to the test the convergence rate of the code. Then, Sod's shock tube test case is implemented into the numerical code to show the correctness of the code in both ow directions. The numerical method is then used to obtain numerical solutions which are compared with experimental data available in the literature. It is found that the numerical solutions are in a good agreement with these experimental results.

<p>There are many real models in which unbounded solution to conservation law system occur. Most often we have some kind of delta function in the solution as a result of the accumulation of mass or some other variable. There is no general method of approaching<br />such problems with nonlinearities. This dissertation provides solutions to conservation law systems that contain division by a dependent variable, which is a problematic part when working with measures. For example, a basic model of chromatography and similar chemical processes has a division with a variable that is unbounded in some cases. The denition of the split delta shock and the general method of using it in such systems is given. Finally, the solution for the singular chromatography model is given.<br />&nbsp;</p> / <p>Postoji mnogo realnih modela u kojima se javljaju neoranicena resenja zakona odrzanja. Najcesce imamo neku vrstu delta funkcije u resenju kao posledicu nagomilavanja mase ili neke druge velicine. Ne postoji opsti metod prilaza takvim problemima sa nelinearnostima. U ovoj disertaciji su data resenja problema zakona odrzanja koja sadrze delenje zavisnom promenljivom, sto je problematican deo kod rada sa merama. Na primer, osnovni model hromatograje i slicnih hemijskih procesa ima delenje promenljivom koja je neogranicena u nekim slucajevima. Data je denicija inverza delenjog delta udarnog talasa i opsti metod primene u takvim sistemima. Na kraju je dato resenje kod modela singularne hromatograje.<br />&nbsp;</p>

Fluid flows occurring in moving and/or deforming environments are influenced by the transient nature of their containment. In Computational Fluid Dynamics (CFD), simulating such flow fields requires effort to maintain the geometric integrity of the transient flow domain. Convective fluxes in such domains are evaluated with respect to the motion of the boundaries of the control volume. These simulations demand conservation of space in addition to the conservation of mass, momentum and energy as the solution continues in time. The Space Conservation Law in its continuous form can be inferred by using the rules of fundamental calculus. However, implementing it in a discrete form poses substantial challenges. During mesh motion, the surfaces enclosing the control volumes sweep through three-dimensional space. As per the Space Conservation Law, the change in the control volume has to match the sum of the swept volumes of all its faces exactly. The Space Conservation Law must be satisfied accurately and consistently in order to avoid the occurrence of non-physical masses and to prevent the violation of the continuity equation. In this work we have attempted to address the consistency issues surrounding the implementation of the Space Conservation Law in OpenFOAM. The existing method for calculation of swept volumes falls short in terms of consistency. Moreover, its capabilities are limited when it comes to complex three-dimensional mesh motions. The existing method of calculation treats swept volumes as net fluxes emanating from cell faces. We have implemented an alternate algorithm in which the swept volumes are treated as intermittent virtual cells whose volumes can be calculated in a unique and consistent manner. We will conclude by validating our approach for mesh motions of varying degrees of complexity.

abstract: The main part of this work establishes existence, uniqueness and regularity properties of measure-valued solutions of a nonlinear hyperbolic conservation law with non-local velocities. Major challenges stem from in- and out-fluxes containing nonzero pure-point parts which cause discontinuities of the velocities. This part is preceded, and motivated, by an extended study which proves that an associated optimal control problem has no optimal $L^1$-solutions that are supported on short time intervals. The hyperbolic conservation law considered here is a well-established model for a highly re-entrant semiconductor manufacturing system. Prior work established well-posedness for $L^1$-controls and states, and existence of optimal solutions for $L^2$-controls, states, and control objectives. The results on measure-valued solutions presented here reduce to the existing literature in the case of initial state and in-flux being absolutely continuous measures. The surprising well-posedness (in the face of measures containing nonzero pure-point part and discontinuous velocities) is directly related to characteristic features of the model that capture the highly re-entrant nature of the semiconductor manufacturing system. More specifically, the optimal control problem is to minimize an $L^1$-functional that measures the mismatch between actual and desired accumulated out-flux. The focus is on the transition between equilibria with eventually zero backlog. In the case of a step up to a larger equilibrium, the in-flux not only needs to increase to match the higher desired out-flux, but also needs to increase the mass in the factory and to make up for the backlog caused by an inverse response of the system. The optimality results obtained confirm the heuristic inference that the optimal solution should be an impulsive in-flux, but this is no longer in the space of $L^1$-controls. The need for impulsive controls motivates the change of the setting from $L^1$-controls and states to controls and states that are Borel measures. The key strategy is to temporarily abandon the Eulerian point of view and first construct Lagrangian solutions. The final section proposes a notion of weak measure-valued solutions and proves existence and uniqueness of such. In the case of the in-flux containing nonzero pure-point part, the weak solution cannot depend continuously on the time with respect to any norm. However, using semi-norms that are related to the flat norm, a weaker form of continuity of solutions with respect to time is proven. It is conjectured that also a similar weak continuous dependence on initial data holds with respect to a variant of the flat norm. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2019

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第19108号 / 情博第554号 / 新制||情||98(附属図書館) / 32059 / 京都大学大学院情報学研究科知能情報学専攻 / (主査)教授 阿久津 達也, 教授 西田 豊明, 教授 山本 章博 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

No description available.

No description available.

A project was completed that 1) developed a list of potential primary law enforcement objectives; 2) analyzed the arrest component of agency and agent objectives; and 3) developed a computer model that produced an agent arrest score. A methodology to select law enforcement objectives using a hierarchy was developed. The objective hierarchy and example objectives are shown. A crime wildlife related list was developed. Game wardens and wildlife biologists with the Virginia Department of Game and Inland Fisheries, foresters with the Virginia Department of Forestry, and members of 2 private interest groups were surveyed to determine the relative importance of the crimes. Respondents ranked violations on a scale from 1 ("not very wrong") to 9 (livery wrong") relative to a given standard violation. The survey contained 3 sections: (A) specific violation list, (8) species list, and (C) list of violation categories. Differences among groups, consistency in responses, and relative consensus about importance of violations were analyzed. / Master of Science