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Truncation errors and computational cost are obstacles that still hinder large-scale applications of the Computational Fluid Dynamics method. The discontinuous Galerkin method is one of the high-order schemes utilized extensively in recent years, which is locally conservative, stable, and high-order accurate. Besides that, it can handle complex geometries and irregular meshes with hanging nodes. In this document, the nondimensional compressible Euler equations and Reynolds- Averaged Navier-Stokes equations are discretized by discontinuous Galerkin methods with a two-equations turbulence model on both structured and unstructured meshes. The traditional equation of state for an ideal gas model is substituted by a multispecies thermodynamics model in order to complete the governing equations. An approximate Riemann solver is used for computing the convective flux, and the diffusive flux is approximated with some internal penalty based schemes. The temporal discretization of the partial differential equations is either performed explicitly with the aid of Rung-Kutta methods or with semi-implicit methods. Inspired by the artificial viscosity diffusion based limiter for shock-capturing method, which has been extensively studied, a novel and robust technique based on the introduction of mass diffusion to the species governing equations to guarantee that the species mass fractions remain positive has been thoroughly investigated. This contact-surface-capturing method is conservative and a high order of accuracy can be maintained for the discontinuous Galerkin method. For each time step of the algorithm, any trouble cell is first caught by the contact-surface discontinuity detector. Then some amount of mass diffusions are added to the governing equations to change the gas mixtures and arrive at an equilibrium point satisfying some conditions. The species properties are reasonable without any oscillations. Computations are performed for many steady and unsteady flow problems. For general non-mixing fluid flows, the classical air-helium shock bubble interaction problem is the central test case for the high-order discontinuous Galerkin method with a mass diffusion based limiter chosen. The computed results are compared with experimental, exact, and empirical data to validate the fluid dynamic solver.

京都大学 / 新制・課程博士 / 博士(農学) / 甲第23527号 / 農博第2474号 / 新制||農||1087(附属図書館) / 学位論文||R3||N5358(農学部図書室) / 京都大学大学院農学研究科地域環境科学専攻 / (主査)教授 藤原 正幸, 教授 中村 公人, 准教授 宇波 耕一 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DGAM

A dynamical system originally invented by Boltzmann has had recent developments. The system consists of a particle in a gravitational potential with an added centrifugal force, which is subject to reflection against a wall that separates the system from the gravitational center. The recent developments are with regards to the integrability of the system in the special case of vanishing centrifugal term. The purpose of this essay is to explicate these developments. / Ett dynamiskt system, ursprungligen uppfunnet av Boltzmann, har nyligen sett utvecklingar. Systemet består av en partikel i en gravitationspotential med en tillagd centrifugalkraft, som reflekterar vid kontakt med en vägg som skiljer partikeln och gravitationscentrumet. De nya utvecklingarna är inom systemets integrabilitet i det specialfall att centrifugalkraften är borttagen. Syftet med denna uppsats är att explicera dessa framtaganden.

My drawings contemplate the unfixed nature of my experience. I draw from a state of uncertainty about the relationship between self and space, between a moment of experience and the one that follows it. My process involves intuitive mark-making in which instances of perception are indeterminate and discontinuous. I draw from the experience of unhinged moments, from silence and stillness, and from the indefinable, inarticulable, interstitial moments of perception between those that can be concretely described. The immediacy of drawing, the direct engagement with the mark on the surface, is central to my work. Intuitive mark-making is a way of engaging as directly as possible with the indeterminate nature of my experience. As the drawn marks allude to a once-fleeting present, the layers of marks interact to remind me of the non-linear nature of time and the unfixed nature of experience. The making of the mark punctuates a fragment of experience, dividing it into before and after the mark. The esoteric nature of drawing, the variety of marks engaging on and within the surface, the ethereal traces, the engagement with the space implied by the panel, the discontinuities revealed between adjacent drawings on panel, and the implied experience compressed in the seams between the panels address multiple and unresolveable ways of experiencing a moment in a space.

Coercivity is a characteristic property of the bilinear term in a weak form of a partial differential equation in both infinite space and the corresponding finite space utilized by a numerical scheme. This concept implies \textit{stability} and \textit{well-posedness} of the weak form in both the exact solution and the numerical solution. In fact, the loss of this property especially in finite dimension cases leads to instability of the numerical scheme. This phenomenon occurs in three major families of problems consisting of advection-diffusion equation with dominant advection term, elastic analysis of very thin beams, and associated plasticity and non-associated plasticity problems. There are two main paths to overcome the loss of coercivity, first manipulating and stabilizing a weak form to ensure that the discrete weak form is coercive, second using an automatically stable method to estimate the solution space such as the Discontinuous Petrov Galerkin (DPG) method in which the optimal test space is attained during the design of the method in such a way that the scheme keeps the coercivity inherently. In this dissertation, A stable numerical method for the aforementioned problems is proposed. A stabilized finite element method for the problem of migration risk problem which belongs to the family of the advection-diffusion problems is designed and thoroughly analyzed. Moreover, DPG method is exploited for a wide range of valuing option problems under the black-Scholes model including vanilla options, American options, Asian options, double knock barrier options where they all belong to family of advection-diffusion problem, and elastic analysis of Timoshenko beam theory. Besides, The problem of American option pricing, migration risk, and plasticity problems can be categorized as a free boundary value problem which has their extra complexity, and optimization theory and variational inequality are the main tools to study these families of the problems. Thus, an overview of the classic definition of variational inequalities and different tools and methods to study analytically and numerically this family of problems is provided and a novel adjoint sensitivity analysis of variational inequalities is proposed.

No / Nazarbayev University Seed Grant, entitled “Environmental assessment of sediment pollution impact on hydropower plants”.

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