Дисертації з теми "First-order hyperbolic partial differential equations"
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1
Cheema, Tasleem Akhter. "Higher-order finite-difference methods for partial differential equations." Thesis, Brunel University, 1997. http://bura.brunel.ac.uk/handle/2438/7131.
Анотація:
This thesis develops two families of numerical methods, based upon rational approximations having distinct real poles, for solving first- and second-order parabolic/ hyperbolic partial differential equations. These methods are thirdand fourth-order accurate in space and time, and do not require the use of complex arithmetic. In these methods first- and second-order spatial derivatives are approximated by finite-difference approximations which produce systems of ordinary differential equations expressible in vector-matrix forms. Solutions of these systems satisfy recurrence relations which lead to the development of parallel algorithms suitable for computer architectures consisting of three or four processors. Finally, the methods are tested on advection, advection-diffusion and wave equations with constant coefficients.
2
Strogies, Nikolai. "Optimization of nonsmooth first order hyperbolic systems." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2016. http://dx.doi.org/10.18452/17633.
Анотація:
Wir betrachten Optimalsteuerungsprobleme, die von partiellen Differentialgleichungen beziehungsweise Variationsungleichungen mit Differentialoperatoren erster Ordnung abhängen. Wir führen die Reformulierung eines Tagebauplanungsproblems, das auf stetigen Funktionen beruht, ein. Das Resultat ist ein Optimalsteuerungsproblem für Viskositätslösungen einer Eikonalgleichung. Die Existenz von Lösungen dieses und bestimmter Hilfsprobleme, die von semilinearen PDG‘s mit künstlicher Viskosität abhängen, wird bewiesen, Stationaritätsbedingungen hergeleitet und ein schwaches Konsistenzresultat für stationäre Punkte präsentiert. Des Weiteren betrachten wir Optimalsteuerungsprobleme, die von stationären Variationsungleichungen erster Art mit linearen Differentialoperatoren erster Ordnung abhängen. Wir diskutieren Lösbarkeit und Stationaritätskonzepte für diese Probleme. Für letzteres vergleichen wir Ergebnisse, die entweder durch die Anwendung von Penalisierungs- und Regularisierungsansätzen direkt auf Ebene von Differentialoperatoren erster Ordnung oder als Grenzwertprozess von Stationaritätssystemen für viskositätsregularisierte Optimalsteuerungsprobleme unter passenden Annahmen erhalten werden. Um die Konsistenz von ursprünglichem und regularisierten Problemen zu sichern, wird ein bekanntes Ergebnis für Lösungen von VU’s mit degeneriertem Differentialoperator erweitert. In beiden Fällen ist die erhaltene Stationarität schwächer als W-stationarität. Die theoretischen Ergebnisse werden anhand numerischer Beispiele verifiziert. Wir erweitern diese Ergebnisse auf Optimalsteuerungsprobleme bezüglich zeitabhängiger VU’s mit Differentialoperatoren erster Ordnung. Hierfür wird die Existenz von Lösungen bewiesen und erneut ein Stationaritätssystem mit Hilfe verschwindender Viskositäten unter bestimmten Beschränktheitsannahmen hergeleitet. Die erhaltenen Ergebnisse werden anhand von numerischen Beispielen verifiziert.We consider problems of optimal control subject to partial differential equations and variational inequality problems with first order differential operators. We introduce a reformulation of an open pit mine planning problem that is based on continuous functions. The resulting formulation is a problem of optimal control subject to viscosity solutions of a partial differential equation of Eikonal Type. The existence of solutions to this problem and auxiliary problems of optimal control subject to regularized, semilinear PDE’s with artificial viscosity is proven. For the latter a first order optimality condition is established and a mild consistency result for the stationary points is proven. Further we study certain problems of optimal control subject to time-independent variational inequalities of the first kind with linear first order differential operators. We discuss solvability and stationarity concepts for such problems. In the latter case, we compare the results obtained by either utilizing penalization-regularization strategies directly on the first order level or considering the limit of systems for viscosity-regularized problems under suitable assumptions. To guarantee the consistency of the original and viscosity-regularized problems of optimal control, we extend known results for solutions to variational inequalities with degenerated differential operators. In both cases, the resulting stationarity concepts are weaker than W-stationarity. We validate the theoretical findings by numerical experiments for several examples. Finally, we extend the results from the time-independent to the case of problems of optimal control subject to VI’s with linear first order differential operators that are time-dependent. After establishing the existence of solutions to the problem of optimal control, a stationarity system is derived by a vanishing viscosity approach under certain boundedness assumptions and the theoretical findings are validated by numerical experiments.
3
Postell, Floyd Vince. "High order finite difference methods." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/28876.
4
Smith, James. "Global time estimates for solutions to higher order strictly hyperbolic partial differential equations." Thesis, Imperial College London, 2006. http://hdl.handle.net/10044/1/1267.
Анотація:
In this thesis we consider the Cauchy problem for general higher order constant coefficient strictly hyperbolic PDEs with lower order terms and show how the behaviour of the characteristic roots determine the rate of decay in the associated Lp-Lq estimates. In particular, we show under what conditions the solution behaves like that of the standard wave equation, the wave equation with dissipation or the Klein-Gordon equation. We explain the various factors involved, such as the presence of multiple roots, the size of the sets of multiplicity and the order with which characteristics meet the real axis, yield different rates of decay. As an example, we show how the results obtained can be applied to the Fokker-Planck equation. In the second part, we derive Lp-Lq estimates for wave equations with a bounded time dependent coefficient. A classification of the oscillating behaviour of the coefficient is given and related to the estimate which can be obtained.
5
Jurás, Martin. "Geometric Aspects of Second-Order Scalar Hyperbolic Partial Differential Equations in the Plane." DigitalCommons@USU, 1997. https://digitalcommons.usu.edu/etd/7139.
Анотація:
The purpose of this dissertation is to address various geometric aspects of second-order scalar hyperbolic partial differential equations in two independent variables and one dependent variable
F(x, y, u, u_x, u_y, u_xx, u_xy, u_yy )= 0 (1)
We find a characterization of hyperbolic Darboux integrable equations at level k (1) in terms of the vanishing of the generalized Laplace invariants and provide an invariant characterization of various cases in the Goursat general classification of hyperbolic Darboux integrable equations (1). In particular we give a contact invariant characterization of equations integrable by the methods of general and intermediate integrals. New relative invariants that control the existence of the first integrals of the characteristic Pfaffian systems are found and used to obtain an invariant characterization for the class of -Gordon equations. A notion of a hyperbolic Darboux system is introduced and we show by examples that the classical Laplace transformation is just a special case of a diffeomorphism of hyperbolic Darboux systems. We also construct new examples of homomorphisms between certain hyperbolic systems. We characterize Monge-Ampere equations and explicitly exhibit two invariants whose vanishing is a necessary and sufficient condition for the equation to be of the Monge-Ampere type. The solution to the inverse problem of the calculus of variations for hyperbolic equations (1) in terms of the generalized Laplace invariants is presented. We also obtain some partial results on symplectic conservation laws. We characterize, up to contact equivalence, some classical equations using the generalized Laplace invariants. These results contain characterizations of the wave, Liouville, Klein-Gordon, and certain types of Euler-Poisson equations.
6
Pefferly, Robert J. "Finite difference approximations of second order quasi-linear elliptic and hyperbolic stochastic partial differential equations." Thesis, University of Edinburgh, 2001. http://hdl.handle.net/1842/11244.
Анотація:
This thesis covers topics such as finite difference schemes, mean-square convergence, modelling, and numerical approximations of second order quasi-linear stochastic partial differential equations (SPDE) driven by white noise in less than three space dimensions. The motivation for discussing and expanding these topics lies in their implications in such physical phenomena as signal and information flow, gravitational and electromagnetic fields, large scale weather systems, and macro-computer networks. Chapter 2 delves into the hyperbolic SPDE in one space and one time dimension. This is an important equation to such fields as signal processing, communications, and information theory where singularities propagate throughout space as a function of time. Chapter 3 discusses some concepts and implications of elliptic SPDE's driven by additive noise. These systems are key for understanding steady state phenomena. Chapter 4 presents some numerical work regarding elliptic SPDE's driven by multiplicative and general noise. These SPDE's are open topics in the theoretical literature, hence numerical work provides significant insight into the nature of the process. Chapter 5 presents some numerical work regarding quasi-geostrophic geophysical fluid dynamics involving stochastic noise and demonstrates how these systems can be represented as a combination of elliptic and hyperbolic components.
7
Luo, BiYong. "Shooting method-based algorithms for solving control problems associated with second-order hyperbolic partial differential equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/NQ66358.pdf.
8
Haque, Md Z. "An adaptive finite element method for systems of second-order hyperbolic partial differential equations in one space dimension." Ann Arbor, Mich. : ProQuest, 2008. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3316356.
Анотація:
Thesis (Ph.D. in Computational and Applied Mathematics)--S.M.U.Title from PDF title page (viewed Mar. 16, 2009). Source: Dissertation Abstracts International, Volume: 69-08, Section: B Adviser: Peter K. Moore. Includes bibliographical references.
9
Sroczinski, Matthias [Verfasser]. "Global existence and asymptotic decay for quasilinear second-order symmetric hyperbolic systems of partial differential equations occurring in the relativistic dynamics of dissipative fluids / Matthias Sroczinski." Konstanz : KOPS Universität Konstanz, 2019. http://d-nb.info/1184795460/34.
10
Yang, Lixiang. "Modeling Waves in Linear and Nonlinear Solids by First-Order Hyperbolic Differential Equations." The Ohio State University, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=osu1303846979.
11
Park, Elinor Jane. "Regularizations of first order partial differential equations by generators of semigroups." Thesis, Swansea University, 2005. https://cronfa.swan.ac.uk/Record/cronfa42982.
Анотація:
This thesis investigates the limiting behaviour of solutions to certain partial and pseudo differential equations. Included is a study of the notion of generalised solutions and particular examples, with emphasis on hyperbolic conservation laws. A probabilistic interpretation of some results is also presented.
12
Aziz, Waleed. "Analytic and algebraic aspects of integrability for first order partial differential equations." Thesis, University of Plymouth, 2013. http://hdl.handle.net/10026.1/1468.
Анотація:
This work is devoted to investigating the algebraic and analytic integrability of first order polynomial partial differential equations via an understanding of the well-developed area of local and global integrability of polynomial vector fields. In the view of characteristics method, the search of first integrals of the first order partial differential equations P(x,y,z)∂z(x,y) ∂x +Q(x,y,z)∂z(x,y) ∂y = R(x,y,z), (1) is equivalent to the search of first integrals of the system of the ordinary differential equations dx/dt= P(x,y,z), dy/dt= Q(x,y,z), dz/dt= R(x,y,z). (2) The trajectories of (2) will be found by representing these trajectories as the intersection of level surfaces of first integrals of (1). We would like to investigate the integrability of the partial differential equation (1) around a singularity. This is a case where understanding of ordinary differential equations will help understanding of partial differential equations. Clearly, first integrals of the partial differential equation (1), are first integrals of the ordinary differential equations (2). So, if (2) has two first integrals φ1(x,y,z) =C1and φ2(x,y,z) =C2, where C1and C2 are constants, then the general solution of (1) is F(φ1,φ2) = 0, where F is an arbitrary function of φ1and φ2. We choose for our investigation a system with quadratic nonlinearities and such that the axes planes are invariant for the characteristics: this gives three dimensional Lotka– Volterra systems x' =dx/dt= P = x(λ +ax+by+cz), y' =dy/dt= Q = y(µ +dx+ey+ fz), z' =dz/dt= R = z(ν +gx+hy+kz), where λ,µ,ν 6= 0. v Several problems have been investigated in this work such as the study of local integrability and linearizability of three dimensional Lotka–Volterra equations with (λ:µ:ν)–resonance. More precisely, we give a complete set of necessary and sufficient conditions for both integrability and linearizability for three dimensional Lotka-Volterra systems for (1:−1:1), (2:−1:1) and (1:−2:1)–resonance. To prove their sufficiency, we mainly use the method of Darboux with the existence of inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable. Also, more general three dimensional system have been investigated and necessary and sufficient conditions are obtained. In another approach, we also consider the applicability of an entirely different method which based on the monodromy method to prove the sufficiency of integrability of these systems. These investigations, in fact, mean that we generalized the classical centre-focus problem in two dimensional vector fields to three dimensional vector fields. In three dimensions, the possible mechanisms underling integrability are more difficult and computationally much harder. We also give a generalization of Singer’s theorem about the existence of Liouvillian first integrals in codimension 1 foliations in Cnas well as to three dimensional vector fields. Finally, we characterize the centres of the quasi-homogeneous planar polynomial differential systems of degree three. We show that at most one limit cycle can bifurcate from the periodic orbits of a centre of a cubic homogeneous polynomial system using the averaging theory of first order.
13
Stanistreet, Timothy Francis. "Numerical methods for first order partial differential equations describing steady-state forming processes." Thesis, Imperial College London, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.398232.
14
Jonasson, Jens. "Systems of Linear First Order Partial Differential Equations Admitting a Bilinear Multiplication of Solutions." Doctoral thesis, Linköping : Department of Mathematics, Linköpings universitet, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-9949.
15
Bezerra, Rufino Ferreira Paiva Eduardo. "Wind Velocity Estimation for Wind Farms." Electronic Thesis or Diss., Université Paris sciences et lettres, 2023. http://www.theses.fr/2023UPSLM046.
Анотація:
Cette thèse propose des algorithmes pour estimer la vitesse et la direction du vent pour des éoliennes et des parcs éoliens.Tout d'abord, nous proposons des méthodes basées sur les données pour estimer la vitesse effective du rotor (REWS) sans nécessiter la connaissance de certains paramètres physiques de l'éolienne, qui pourraient être inconnus de l'opérateur. Nous fournissons deux méthodes basées sur les données, l'une basée sur la régression par processus gaussien et l'autre combinant la régression par processus gaussien avec un observateur grand gain.Ensuite, en nous basant sur cette estimation locale de la REWS, au niveau d'une éolinenne, nous abordons la question de l'estimation du vent en écoulement libre au niveau du parc éolien.Nous commençons par nous concentrer sur l'estimation de la vitesse du vent, pour une direction du vent connue. Pour un parc éolien de géométrie simple, nous démontrons qu'une mesure locale de la vitesse perturbée par la présence des éoliennes peut être utilisée pour estimer la vitesse du vent en écoulement libre. Nous fondons notre méthodologie d'estimation sur une modélisation simplifiée de l'effet de sillage qui consiste en des équations aux dérivées partielles hyperboliques du premier ordre en cascade, et dont la vitesse de transport est la vitesse du vent en écoulement libre. Nous proposons d'utiliser une solution analytique de ces équations, impliquant des retards de transport, pour effectuer une estimation de la mesure locale et mettre à jour l'estimation de la vitesse du vent en écoulement libre. Nous démontrons formellement la convergence de cette estimation et illustrons numériquement l'efficacité de cette méthode.Enfin, nous passons à une configuration plus générale où à la fois la vitesse et la direction du vent en écoulement libre sont inconnues. Nous proposons d'utiliser une modélisation bidimensionelle du sillage et de nous appuyer sur une méthode basée sur l'optimisation. Le problème d'identification que nous formulons se révèle être particulièrement difficile en raison de l'apparition de retards de transport, mais nous montrons comment contourner cette difficulté en considérant une valeur moyenne de l'historique de la vitesse du vent en écoulement libre. Des résultats de simulation obtenus avec le simulateur FAST.Farm illustrent l'intérêt de la méthode proposéeThis thesis designs algorithms to estimate the wind speed and direction for wind turbines and wind farms.First, we propose data-based methods to estimate the Rotor Effective Wind Speed (REWS) for a single turbine without prior knowledge of certain physical parameters of the turbine that might be unknown to an operator.We provide two data-based methods, based respectively on Gaussian Process Regression (GPR) and on an combination of GPR with high-gain observers.Second, grounding on this REWS estimation at the local level of one turbine, we address the question of estimating the free-flow wind at the level of a wind farm.We start by focusing on wind speed estimation, for a given known wind direction. For a wind farm with a simple geometry, we prove that a local speed measurement disturbed by the presence of the turbines can be used to estimate the free-flow wind speed. We ground our estimation methodology on a simplified wake model, which consists of first-order hyperbolic partial differential equations, the transport speed of which is the free-flow wind speed. We propose to use an analytical solution of these equations, involving transport delays, to perform an estimate of the local measurement and to update the free-flow wind speed estimate. We formally prove the convergence of this estimate and numerically illustrate the efficiency of this method.Finally, we move to a more general setup where both the free-flow wind speed and direction are unknown. We propose to use a two-dimensional wake model and to rely on an optimization-based method. This identification problem reveals to be particularly challenging due to the appearance of transport delays, but we illustrate how to circumvent this issue by considering an average value of the free flow wind speed history. Simulation results obtained with the simulator FAST.Farm illustrate the interest of the proposed method
16
Studener, Stephan [Verfasser]. "Embedded Control and Parameter Estimation Algorithms for Transport Process Systems : modeled by first-order Partial Differential Equations / Stephan Studener." Aachen : Shaker, 2011. http://d-nb.info/1069049832/34.
17
Gorgone, Matteo. "Symmetries, Equivalence and Decoupling of First Order PDE's." Doctoral thesis, Università di Catania, 2017. http://hdl.handle.net/10761/3901.
Анотація:
The present Ph.D. Thesis is concerned with first order PDE's and to the structural conditions allowing for their transformation into an equivalent, and somehow simpler, form.
Most of the results are framed in the context of the classical theory of the Lie symmetries of differential equations, and on the analysis of some invariant quantities.
The thesis is organized in 5 main sections. The first two Chapters present the basic elements of the Lie theory and some introductory facts about first order PDE's, with special emphasis on quasilinear ones.
Chapter 3 is devoted to investigate equivalence transformations, i.e., point transformations suitable to deal with classes of differential equations involving arbitrary elements.
The general framework of equivalence transformations is then applied to a class of systems of first order PDE's, consisting of a linear conservation law and four general balance laws involving some arbitrary continuously differentiable functions, in order to identify the elements of the class that can be mapped to a system of autonomous conservation laws.
Chapter 4 is concerned with the transformation of nonlinear first order systems of differential equations to a simpler form. At first, the reduction to an equivalent first order autonomous and homogeneous quasilinear form is considered. A theorem providing necessary conditions is given, and the reduction to quasilinear form is performed by constructing the canonical variables associated to the Lie point symmetries admitted by the nonlinear system.
Then, a general nonlinear system of first order PDE's involving the derivatives of the unknown variables in polynomial form is considered, and a theorem giving necessary and sufficient conditions in order to map it to an autonomous system polynomially homogeneous in the derivatives is established. Several classes of first order Monge-Ampere systems, either with constant coefficients or with coefficients depending on the field variables, provided that the coefficients entering their equations satisfy some constraints, are reduced to quasilinear (or linear) form.
Chapter 5 faces the decoupling problem of general quasilinear first order systems. Starting from the direct decoupling problem of hyperbolic quasilinear first order systems in two independent variables and two or three dependent variables, we observe that the decoupling conditions can be written in terms of the eigenvalues and eigenvectors of the coefficient matrix.
This allows to obtain a completely general result. At first, general autonomous and homogeneous quasilinear first order systems (either hyperbolic or not) are discussed, and the necessary and sufficient conditions for the decoupling in two or more subsystems proved. Then, the analysis is extended to the case of nonhomogeneous and/or nonautonomous systems. The conditions, as one expects, involve just the properties of the eigenvalues and the eigenvectors (together with the generalized eigenvectors, if needed) of the coefficient matrix; in particular, the conditions for the full decoupling of a hyperbolic system in non-interacting subsystems have a physical interpretation since require the vanishing both of the change of characteristic speeds of a subsystem across a wave of the other subsystems, and of the interaction coefficients between waves of different subsystems.
Moreover, when the required decoupling conditions are satisfied, we have also the differential constraints whose integration provides the variable transformation leading to the (partially or fully) decoupled system. All the results are extended to the decoupling of nonhomogeneous and/or nonautonomous quasilinear first order systems.
18
Schnücke, Gero [Verfasser], Christian [Gutachter] Klingenberg, and Manfred [Gutachter] Dobrowolski. "Arbitrary Lagrangian-Eulerian Discontinous Galerkin methods for nonlinear time-dependent first order partial differential equations / Gero Schnücke ; Gutachter: Christian Klingenberg, Manfred Dobrowolski." Würzburg : Universität Würzburg, 2016. http://d-nb.info/1117477290/34.
19
Ježková, Jitka. "Modelování dopravního toku." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2015. http://www.nusl.cz/ntk/nusl-232180.
Анотація:
Tato diplomová práce prezentuje problematiku dopravního toku a jeho modelování. Zabývá se především několika LWR modely, které následně rozebírá a hledá řešení pro počáteční úlohy. Ukazuje se, že ne pro všechny počáteční úlohy lze řešení definovat na celém prostoru, ale jen v určitém okolí počáteční křivky. Proto je dále odvozena metoda výpočtu velikosti tohoto okolí a to nejen zcela obecně, ale i pro dané modely. Teoretický rozbor LWR modelů a řešení počátečních úloh jsou demonstrovány několika příklady, které zřetelně ukazují, jak se dopravní tok simulovaný danými modely chová.
20
Shu, Yupeng. "Numerical Solutions of Generalized Burgers' Equations for Some Incompressible Non-Newtonian Fluids." ScholarWorks@UNO, 2015. http://scholarworks.uno.edu/td/2051.
Анотація:
The author presents some generalized Burgers' equations for incompressible and isothermal flow of viscous non-Newtonian fluids based on the Cross model, the Carreau model, and the Power-Law model and some simple assumptions on the flows. The author numerically solves the traveling wave equations for the Cross model, the Carreau model, the Power-Law model by using industrial data. The author proves existence and uniqueness of solutions to the traveling wave equations of each of the three models. The author also provides numerical estimates of the shock thickness as well as maximum strain $\varepsilon_{11}$ for each of the fluids.
21
Chiocchetti, Simone. "High order numerical methods for a unified theory of fluid and solid mechanics." Doctoral thesis, Università degli studi di Trento, 2022. http://hdl.handle.net/11572/346999.
Анотація:
This dissertation is a contribution to the development of a unified model of
continuum mechanics, describing both fluids and elastic solids as a general
continua, with a simple material parameter choice being the distinction
between inviscid or viscous fluid, or elastic solids or visco-elasto-plastic
media. Additional physical effects such as surface tension, rate-dependent
material failure and fatigue can be, and have been, included in the same
formalism.
The model extends a hyperelastic formulation of solid mechanics in
Eulerian coordinates to fluid flows by means of stiff algebraic relaxation
source terms. The governing equations are then solved by means of high
order ADER Discontinuous Galerkin and Finite Volume schemes on fixed
Cartesian meshes and on moving unstructured polygonal meshes with
adaptive connectivity, the latter constructed and moved by means of a in-
house Fortran library for the generation of high quality Delaunay and Voronoi
meshes.
Further, the thesis introduces a new family of exponential-type and semi-
analytical time-integration methods for the stiff source terms governing
friction and pressure relaxation in Baer-Nunziato compressible multiphase
flows, as well as for relaxation in the unified model of continuum mechanics,
associated with viscosity and plasticity, and heat conduction effects.
Theoretical consideration about the model are also given, from the
solution of weak hyperbolicity issues affecting some special cases of the
governing equations, to the computation of accurate eigenvalue estimates, to
the discussion of the geometrical structure of the equations and involution
constraints of curl type, then enforced both via a GLM curl cleaning method,
and by means of special involution-preserving discrete differential operators,
implemented in a semi-implicit framework.
Concerning applications to real-world problems, this thesis includes
simulation ranging from low-Mach viscous two-phase flow, to shockwaves in
compressible viscous flow on unstructured moving grids, to diffuse interface
crack formation in solids.
22
Chien, Chih-Cheng, and 簡志成. "The viscosity solution of first-order partial differential equations." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/50047238279156587483.
Анотація:
碩士東海大學
數學系
96
The study of this thesis is to discuss the definition and property of the viscosity solution of first-order partial differential equations. To claim the value function in viscosity sense and guides the Hamilton -Jacobi-Bellman equation which is satisfied with . Then we prove that is the viscosity solution of this equation. Moreover, we also prove the uniqueness, comparison and stability of viscosity solutions.
23
Schnücke, Gero. "Arbitrary Lagrangian-Eulerian Discontinous Galerkin methods for nonlinear time-dependent first order partial differential equations." Doctoral thesis, 2016. https://nbn-resolving.org/urn:nbn:de:bvb:20-opus-139579.
Анотація:
The present thesis considers the development and analysis of arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods with time-dependent approximation spaces for conservation laws and the Hamilton-Jacobi equations. Fundamentals about conservation laws, Hamilton-Jacobi equations and discontinuous Galerkin methods are presented. In particular, issues in the development of discontinuous Galerkin (DG) methods for the Hamilton-Jacobi equations are discussed. The development of the ALE-DG methods based on the assumption that the distribution of the grid points is explicitly given for an upcoming time level. This assumption allows to construct a time-dependent local affine linear mapping to a reference cell and a time-dependent finite element test function space. In addition, a version of Reynolds’ transport theorem can be proven. For the fully-discrete ALE-DG method for nonlinear scalar conservation laws the geometric conservation law and a local maximum principle are proven. Furthermore, conditions for slope limiters are stated. These conditions ensure the total variation stability of the method. In addition, entropy stability is discussed. For the corresponding semi-discrete ALE-DG method, error estimates are proven. If a piecewise $\mathcal{P}^{k}$ polynomial approximation space is used on the reference cell, the sub-optimal $\left(k+\frac{1}{2}\right)$ convergence for monotone fuxes and the optimal $(k+1)$ convergence for an upwind flux are proven in the $\mathrm{L}^{2}$-norm. The capability of the method is shown by numerical examples for nonlinear conservation laws. Likewise, for the semi-discrete ALE-DG method for nonlinear Hamilton-Jacobi equations, error estimates are proven. In the one dimensional case the optimal $\left(k+1\right)$ convergence and in the two dimensional case the sub-optimal $\left(k+\frac{1}{2}\right)$ convergence are proven in the $\mathrm{L}^{2}$-norm, if a piecewise $\mathcal{P}^{k}$ polynomial approximation space is used on the reference cell. For the fullydiscrete method, the geometric conservation is proven and for the piecewise constant forward Euler step the convergence of the method to the unique physical relevant solution is discussedDie vorliegende Arbeit beschäftigt sich mit der Entwicklung und Analyse von arbitrar Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) Methoden mit zeitabhängigen Testfunktionen Räumen für Erhaltungs- und Hamilton-Jacobi Gleichungen. Grundlagen über Erhaltungsgleichungen, Hamilton-Jacobi Gleichungen und discontinuous Galerkin Methoden werden präsentiert. Insbesondere werden Probleme bei der Entwicklung von discontinuous Galerkin Methoden für die Hamilton-Jacobi Gleichungen untersucht. Die Entwicklung der ALE-DG Methode basiert auf der Annahme, dass die Verteilung der Gitterpunkte zu einem kommenden Zeitpunkt explizit gegeben ist. Diese Annahme ermöglicht die Konstruktion einer zeitabhängigen lokal affin-linearen Abbildung auf eine Referenzzelle und eines zeitabhängigen Testfunktionen Raums. Zusätzlich kann eine Version des Reynolds’schen Transportsatzes gezeigt werden. Für die vollständig diskretisierte ALE-DG Methode für nichtlineare Erhaltungsgleichungen werden der geometrischen Erhaltungssatz und ein lokales Maximumprinzip bewiesen. Des Weiteren werden Bedingungen für Limiter angegeben. Diese Bedingungen sichern die Stabilität der Methode im Sinne der totalen Variation. Zusätzlich wird die Entropie-Stabilität der Methode diskutiert. Für die zugehörige semi-diskretisierte ALE-DG Methode werden Fehlerabschätzungen gezeigt. Wenn auf der Referenzzelle ein Testfunktionen Raum, der stückweise Polynome vom Grad $k$ enthält verwendet wird, kann für einen monotonen Fluss die suboptimale Konvergenzordnung $\left(k+\frac{1}{2}\right)$ und für einen upwind Fluss die optimale Konvergenzordnung $\left(k+1\right)$ in der $\mathrm{L}^{2}$-Norm gezeigt werden. Die Leistungsfähigkeit der Methode wird anhand numerischer Beispiele für nichtlineare Erhaltungsgleichungen untersucht. Ebenso werden für die semi-diskretisierte ALE-DG Methode für nichtlineare Hamilton-Jacobi Gleichungen Fehlerabschätzungen gezeigt. Wenn auf der Referenzzelle ein Testfunktionen Raum, der stückweise Polynome vom Grad k enthält verwendet wird, kann im eindimensionalen Fall die optimale Konvergenzordnung $\left(k+1\right)$ und im zweidimensionalen Fall die suboptimale Konvergenzordnung $\left(k+\frac{1}{2}\right)$ in der $\mathrm{L}^{2}$-Norm gezeigt werden. Für die vollständig diskretisierte ALE-DG Methode werden der geometrischen Erhaltungssatz bewiesen und für die stückweise konstante explizite Euler Diskretisierung wird die Konvergenz gegen die eindeutige physikalisch relevante Lösung diskutiert
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Lamothe, Vincent. "Analyse de groupe d’un modèle de la plasticité idéale planaire et sur les solutions en termes d’invariants de Riemann pour les systèmes quasilinéaires du premier ordre." Thèse, 2013. http://hdl.handle.net/1866/10343.
Анотація:
Les objets d’étude de cette thèse sont les systèmes d’équations quasilinéaires du premier ordre. Dans une première partie, on fait une analyse du point de vue du groupe de Lie classique des symétries ponctuelles d’un modèle de la plasticité idéale. Les écoulements planaires dans les cas stationnaire et non-stationnaire sont étudiés. Deux nouveaux champs de vecteurs ont été obtenus, complétant ainsi l’algèbre de Lie du cas stationnaire dont les sous-algèbres sont classifiées en classes de conjugaison sous l’action du groupe. Dans le cas non-stationnaire, une classification des algèbres de Lie admissibles selon la force choisie est effectuée. Pour chaque type de force, les champs de vecteurs sont présentés. L’algèbre ayant la dimension la plus élevée possible a été obtenues en considérant les forces monogéniques et elle a été classifiée en classes de conjugaison. La méthode de réduction par symétrie est appliquée pour obtenir
des solutions explicites et implicites de plusieurs types parmi lesquelles certaines s’expriment en termes d’une ou deux fonctions arbitraires d’une variable et d’autres en termes de fonctions elliptiques de Jacobi. Plusieurs solutions sont interprétées physiquement pour en déduire la forme de filières d’extrusion réalisables. Dans la seconde partie, on s’intéresse aux solutions s’exprimant en fonction d’invariants de Riemann pour les systèmes quasilinéaires du premier ordre. La méthode des caractéristiques généralisées ainsi qu’une méthode basée sur les symétries conditionnelles pour les invariants de Riemann sont étendues pour être applicables à des systèmes dans leurs régions elliptiques. Leur applicabilité est démontrée par des exemples de la plasticité idéale non-stationnaire pour un flot irrotationnel ainsi que les équations de la mécanique des fluides. Une nouvelle approche basée sur l’introduction de matrices de rotation satisfaisant certaines conditions algébriques est développée.
Elle est applicable directement à des systèmes non-homogènes et non-autonomes
sans avoir besoin de transformations préalables. Son efficacité est illustrée par des exemples comprenant un système qui régit l’interaction non-linéaire d’ondes et de particules. La solution générale est construite de façon explicite.The objects under consideration in this thesis are systems of first-order quasilinear equations. In the first part of the thesis, a study is made of an ideal plasticity model from the point of view of the classical Lie point symmetry group. Planar flows are investigated in both the stationary and non-stationary cases. Two new vector fields are obtained. They complete the Lie algebra of the stationary case, and the subalgebras are classified into conjugacy classes under the action of the group. In the non-stationary case, a classification of the Lie algebras admissible under the chosen force is performed. For each type of force, the vector fields are presented. For monogenic forces, the algebra is of the highest possible dimension. Its classification into conjugacy classes is made. The symmetry reduction method is used to obtain explicit and implicit solutions of several types. Some of them can be expressed in terms of one or two arbitrary functions of one variable. Others can be expressed in terms of Jacobi elliptic functions. Many solutions are interpreted physically in order to determine the shape of realistic extrusion dies. In the second part of the thesis, we examine solutions expressed in terms of Riemann invariants for first-order quasilinear systems. The generalized method of characteristics, along with a method based on conditional symmetries for Riemann invariants are extended so as to be applicable to systems in their elliptic regions. The applicability of the methods is illustrated by examples such as non-stationary ideal plasticity for an irrotational flow as well as fluid mechanics equations. A new approach is developed, based on the introduction of rotation matrices which satisfy certain algebraic conditions. It is directly applicable to non-homogeneous and non-autonomous systems. Its efficiency is illustrated by examples which include a system governing the non-linear superposition of waves and particles. The general solution is constructed in explicit form.