Relevant bibliographies by topics / Parabolic-hyperbolic coupling

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Parabolic-hyperbolic coupling'

Author: Grafiati
Published: 4 June 2021
Last updated: 17 February 2022

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Journal articles on the topic "Parabolic-hyperbolic coupling"

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AGUILAR, GLORIA, LAURENT LÉVI, and MONIQUE MADAUNE-TORT. "COUPLING OF MULTIDIMENSIONAL PARABOLIC AND HYPERBOLIC EQUATIONS." Journal of Hyperbolic Differential Equations 03, no. 01 (March 2006): 53–80. http://dx.doi.org/10.1142/s0219891606000720.

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This paper deals with the mathematical analysis of a quasilinear parabolic-hyperbolic problem in a multidimensional bounded domain Ω. In a region Ωp a diffusion-advection-reaction type equation is set, while in the complementary Ωh ≡ Ω\Ωp, only advection-reaction terms are taken into account. To begin we provide a definition of a weak solution through an entropy inequality on the whole domain. Since the interface ∂Ωp ∩ ∂Ωh contains outward characteristics for the first-order operator in Ωh, the uniqueness proof starts by considering first the hyperbolic zone and then the parabolic one. The existence property uses the vanishing viscosity method and to pass to the limit on the hyperbolic zone, we refer to the notion of process solution.
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CHOQUET, CATHERINE. "PARABOLIC AND DEGENERATE PARABOLIC MODELS FOR PRESSURE-DRIVEN TRANSPORT PROBLEMS." Mathematical Models and Methods in Applied Sciences 20, no. 04 (April 2010): 543–66. http://dx.doi.org/10.1142/s0218202510004337.

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We consider two models of flow and transport in porous media, the first one for consolidational flow in compressible sedimentary basins, the second one for flow in partially saturated media. Despite the differences in these physical settings, they lead to quite similar mathematical models with a strong pressure coupling. The first model is a coupled system of pde's of parabolic type. The second one involves a coupled system of pdes of degenerate parabolic–hyperbolic type. We state an existence result of weak solutions for both models.
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Avalos, George. "The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics." Abstract and Applied Analysis 1, no. 2 (1996): 203–17. http://dx.doi.org/10.1155/s1085337596000103.

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We show here the uniform stabilization of a coupled system of hyperbolic and parabolic PDE's which describes a particular fluid/structure interaction system. This system has the wave equation, which is satisfied on the interior of a bounded domainΩ, coupled to a “parabolic–like” beam equation holding on∂Ω, and wherein the coupling is accomplished through velocity terms on the boundary. Our result is an analog of a recent result by Lasiecka and Triggiani which shows the exponential stability of the wave equation via Neumann feedback control, and like that work, depends upon a trace regularity estimate for solutions of hyperbolic equations.
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Han, Zhong-Jie, Gengsheng Wang, and Jing Wang. "Explicit decay rate for a degenerate hyperbolic-parabolic coupled system." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 116. http://dx.doi.org/10.1051/cocv/2020040.

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This paper studies the stability of a 1-dim system which comprises a wave equation and a degenerate heat equation in two connected bounded intervals. The coupling between these two different components occurs at the interface with certain transmission conditions. We find an explicit polynomial decay rate for solutions of this system. This rate depends on the degree of the degeneration for the diffusion coefficient near the interface. Besides, the well-posedness of this degenerate coupled system is proved by the semigroup theory.
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Gastaldi, Fabio, and Alfio Quarteroni. "On the coupling of hyperbolic and parabolic systems: analytical and numerical approach." Applied Numerical Mathematics 6, no. 1-2 (December 1989): 3–31. http://dx.doi.org/10.1016/0168-9274(89)90052-4.

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AMIRAT, Y., K. HAMDACHE, and A. ZIANI. "MATHEMATICAL ANALYSIS FOR COMPRESSIBLE MISCIBLE DISPLACEMENT MODELS IN POROUS MEDIA." Mathematical Models and Methods in Applied Sciences 06, no. 06 (September 1996): 729–47. http://dx.doi.org/10.1142/s0218202596000316.

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We discuss a three-dimensional displacement model of one miscible compressible fluid by another in a porous medium. The motion is modeled by a nonlinear system of parabolic type coupling the pressure and the concentration. We give an existence result of weak solutions for a model with diffusion and dispersion, using the Schauder fixed point theorem. We also study a model in the absence of diffusion and dispersion. The system becomes of parabolic-hyperbolic type, the existence of global weak solutions is then obtained through a compensated compactness argument.
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Avalos, George, Irena Lasiecka, and Roberto Triggiani. "Higher Regularity of a Coupled Parabolic-Hyperbolic Fluid-Structure Interactive System." gmj 15, no. 3 (September 2008): 403–37. http://dx.doi.org/10.1515/gmj.2008.403.

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Abstract This paper considers an established model of a parabolic-hyperbolic coupled system of two PDEs, which arises when an elastic structure is immersed in a fluid. Coupling occurs at the interface between the two media. Semigroup well-posedness on the space of finite energy for {𝑤, 𝑤𝑡, 𝑢} was established in [Contemp. Math. 440: 15–54, 2007]. Here, [𝑤, 𝑤𝑡] are the displacement and the velocity of the structure, while 𝑢 is the velocity of the fluid. The domain D(A) of the generator A does not carry any smoothing in the 𝑤-variable (its resolvent 𝑅(λ, A) is not compact on this component space). This raises the issue of higher regularity of solutions. This paper then shows that the mechanical displacement, fluid velocity, and pressure terms do enjoy a greater regularity if, in addition to the I.C. {𝑤0, 𝑤1, 𝑢0} ∈ D(A), one also has 𝑤0 in (𝐻2(Ω𝑠))𝑑.
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Aloui, Lassaad, and Amal Arama. "Diffusion phenomenon for indirectly damped hyperbolic systems coupled by velocities in exterior domains." Journal of Hyperbolic Differential Equations 17, no. 03 (September 2020): 475–500. http://dx.doi.org/10.1142/s0219891620500137.

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We consider a system of two coupled wave equations in an exterior domain, where only one equation is directly damped. We prove that the solutions are [Formula: see text]-approximated by special functions, classified into three patterns depending on the values of the damping and the coupling terms, as well as on the speeds of the waves. In particular, when the damping term is sufficiently large, the waves are asymptotically equal to solutions of parabolic-type equations as [Formula: see text]. This result generalizes the standard diffusion phenomenon for directly damped hyperbolic systems.
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Bulíček, Miroslav, Piotr Gwiazda, Endre Süli, and Agnieszka Świerczewska-Gwiazda. "Analysis of a viscosity model for concentrated polymers." Mathematical Models and Methods in Applied Sciences 26, no. 08 (June 7, 2016): 1599–648. http://dx.doi.org/10.1142/s0218202516500391.

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The paper is concerned with a class of mathematical models for polymeric fluids, which involves the coupling of the Navier–Stokes equations for a viscous, incompressible, constant-density fluid with a parabolic–hyperbolic integro-differential equation describing the evolution of the polymer distribution function in the solvent, and a parabolic integro-differential equation for the evolution of the monomer density function in the solvent. The viscosity coefficient, appearing in the balance of linear momentum equation in the Navier–Stokes system, includes dependence on the shear rate as well as on the weight-averaged polymer chain length. The system of partial differential equations under consideration captures the impact of polymerization and depolymerization effects on the viscosity of the fluid. We prove the existence of global-in-time, large-data weak solutions under fairly general hypotheses.
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Majumdar, Angshuman, Chintan Kumar Mandal, and Sankar Gangopadhyay. "Laser Diode to Single-Mode Circular Core Parabolic Index Fiber Coupling via Upside-Down Tapered Hyperbolic Microlens on the Tip of the Fiber: Prediction of Coupling Optics by ABCD Matrix Formalism." Journal of Optical Communications 40, no. 3 (July 26, 2019): 171–80. http://dx.doi.org/10.1515/joc-2017-0040.

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Abstract We employ ABCD matrix formalism in order to investigate the coupling optics involving laser diode to single-mode circular core parabolic index fiber excitation via upside-down tapered hyperbolic microlens on the fiber tip. Analytic expressions for coupling efficiencies both in absence and in presence of transverse and angular mismatches are formulated. The concerned investigations are made for two practical wavelengths namely 1.3 µm and 1.5 µm. The execution of the prescribed formulations involves little computation. It has been found that the wavelength 1.5 µm is more efficient in respect of coupling. It is also seen that the present coupling device at both the wavelengths shows more tolerance with respect to angular mismatch. As regards tolerance with respect to transverse mismatch, the result is poor at both the wavelengths used. Consequently, it is desirable that designers should not to exceed transverse mismatch beyond 1 μm.
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Dissertations / Theses on the topic "Parabolic-hyperbolic coupling"

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Santatriniaina, Nirina. "Thermomécanique des milieux continus : modèles théoriques et applications au comportement de l'hydrogel en ingénierie biomédicale." Thesis, Rennes 1, 2015. http://www.theses.fr/2015REN1S047/document.

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Dans la première partie on propose un outil mathématique pour traiter les conditions aux limites dynamiques d'un problème couplé d'EDP. La simulation avec des conditions aux limites dynamiques nécessite quelques fois une condition de "switch" en temps des conditions aux limites de Dirichlet en Neumann. La méthode numérique (St DN) a été validée avec des mesures expérimentales pour le cas de la contamination croisée en industrie micro-électronique. Cet outil sera utilisé par la suite pour simuler le phénomène de « self-heating » dans les polymères et les hydrogels sous sollicitations dynamiques. Dans la deuxième partie, on s'intéresse à la modélisation du phénomène de self-heating dans les polymères, les hydrogels et les tissus biologiques. D'abord, nous nous sommes focalisés sur la modélisation de la loi constitutive de l'hydrogel de type HEMA-EGDMA. Nous avons utilisé la théorie des invariants polynomiaux pour définir la loi constitutive du matériau. Ensuite, nous avons mis en place un modèle théorique en thermomécanique couplée d'un milieu continu classique pour analyser la production de chaleur dans ce matériau. Deux potentiels thermodynamiques ont été proposés et identifiés avec les mesures expérimentales. Une nouvelle forme d'équation du mouvement non-linéaire et couplée a été obtenue (un système d'équation aux dérivées partielles parabolique et hyperbolique non-linéaire couplé avec des conditions aux limites dynamiques). Dans la troisième partie, une méthode numérique des équations thermomécaniques (couplage parabolique-hyperbolique) pour les modèles a été utilisée. Cette étape nous a permis, entre autres, de résoudre ce système couplé. La méthode est basée sur la méthode des éléments finis. Divers résultats expérimentaux obtenus sur ce phénomène de self-heating sont présentés dans ce travail suivi d'une étude de corrélations des résultats théoriques et expérimentaux. Dans la dernière partie de ce travail, ces divers résultats sont repris et leurs conséquences sur la modélisation du comportement de l'hydrogel naturel utilisé dans le domaine biomédical sont discutées
In the first part, we propose a mathematical tool for treating the dynamic boundary conditions. The simulation within dynamic boundary condition requires sometimes ''switch'' condition in time of the Dirichlet to Neumann boundary condition (St DN). We propose a numerical method validated with experimental measurements for the case of cross-contamination in microelectronics industry. This tool will be used to compute self-heating in the polymers and hydrogels under dynamic loading. In the second part we focus on modeling the self-heating phenomenon in polymers, hydrogels and biological tissues. We develop constitutive law of the hydrogel type HEMA-EGDMA, focusing on the heat e.ects (dissipation) in this material. Then we set up a theoretical model of coupled thermo-mechanical classic continuum for a better understanding of the heat production in this media. We use polynomial invariants theory to define the constitutive law of the media. Two original thermodynamic potentials are proposed. Original non-linear and coupled governing equations were obtained and identified with the experimental measurements (non-linear parabolic-hyperbolic system with the dynamic boundary condition). In the third part, numerical methods were used to solve thermo-mechanical formalism for the model. This step deals with a numerical method of a coupled partial di.erential equation system of the self-heating (parabolic-hyperbolic coupling). Then, is step allows us, among other things, to propose an appropriate numerical methods to solve this system. The numerical method is based on the finite element methods. Numerous experimental results on the self-heating phenomenon are presented in this work together with correlations studies between the theoretical and experimental results. In the last part of the thesis, these various results will be presented and their impact on the modeling of the behavior of the natural hydrogel used in the biomedical field will be discussed
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Jordão, Daniela Sofia Domingues. "Coupling hyperbolic and parabolic equations." Master's thesis, 2016. http://hdl.handle.net/10316/48035.

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Dissertação de Mestrado em Matemática, área de Especialização em Análise Aplicada e Computação, apresentada à Faculdade de Ciências e Tecnologia da Universidade de Coimbra.
Nesta tese estudamos um sistema de equações diferenciais parciais constituído por uma equação hiperbólica e uma equação parabólica que surge, frequentemente, na descrição da libertação controlada de fármacos. Neste contexto, a evolução da concentração é definida por uma equação de difusão-convecção-reação em que a velocidade convectiva é induzida por um campo elétrico. Apresentamos um estudo qualitativo e quantitativo para o modelo contínuo e para o modelo discreto construído de forma conveniente. Realçamos que, para este último, estabelecemos resultados de convergência que mostram que os métodos numéricos propostos são supraconvergentes.
In this work we study a system of two PDEs: a hyperbolic and a parabolic equation. This system arise often in the mathematical modelling of the controlled drug release. In this scope, the time and space evolution of the concentration is described by a convective-diffusion-reaction equation, where the convective velocity is induced by an electric field. We present a qualitative and quantitative study for the continuous and the proposed discrete models. We remark that in the quantitative analysis we include supraconvergence results.
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Jordão, Daniela Sofia Domingues. "Coupling Hyperbolic and Parabolic IBVP: Applications to Drug Delivery." Doctoral thesis, 2020. http://hdl.handle.net/10316/94361.

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Tese no âmbito do Programa Interuniversitário de Doutoramento em Matemática, apresentada à Faculdade de Ciências e Tecnologia da Universidade de Coimbra
In this thesis, we study a system of partial differential equations defined by a hyperbolic equation - a wave equation, and two parabolic equations - a quasilinear diffusion-reaction equation and a convection-diffusion-reaction equation. In this system, the reaction term of the first parabolic equation depends on the solution of the wave equation, the convective velocity of the second parabolic equation depends on the solution of the wave equation and its gradient, and the diffusion coefficient of the convection-diffusion-reaction equation depends on the solutions of the other two equations. This system arises in the mathematical modeling of several multiphysics processes, as for instance in ultrasound enhanced drug delivery. In this case, the propagation of the acoustic pressure wave, which is described by the hyperbolic equation, induces an increase in the temperature of the target tissue, an increase of the convective drug transport, and the increase of the temperature induces an increase of the diffusion drug transport. Here we propose an algorithm to solve this coupled problem defined in a two-dimensional spatial domain. Our numerical method can be seen, simultaneously, as a fully discrete in space, piecewise linear finite element method, where special quadrature rules are considered, and as a finite difference method defined in nonuniform rectangular grids. We provide the theoretical convergence support where we show that the numerical approximations for the solution of the hyperbolic equation are second order convergent with respect to a discrete $H^1$- norm. This result allows us to conclude that the numerical approximations for the gradient do not deteriorate the quality of the numerical approximations for the solution of the last parabolic equation. For the numerical approximations for the two parabolic equations, we also establish second order convergence but with respect to a discrete $L^2$- norm. These convergence results are proved assuming lower regularity conditions than those usually imposed. In the scope of the finite difference methods, our results can be seen as supraconvergence results because the method uses nonuniform rectangular grids where the correspondent truncation errors are only first order convergent with respect to the norm $\| . \|_\infty$. As the method can be constructed considering piecewise linear finite element method, in the language of the finite element methods our results can be seen as superconvergence results. In fact, it is well known that piecewise linear finite element methods for elliptic equations lead to first order convergent approximations with respect to the usual $H^1$- norm. Numerical results illustrating the theoretical support are also included, highlighting the sharpness of the smoothness assumption on the solutions of the multiphysics problem. It is reported in the literature the use of ultrasound to increase the drug transport and its absorption within the target tissue in different contexts, as for instance in cancer treatment. A simple version of the mathematical problem studied in this work is considered to illustrate the effectiveness of the use of ultrasound to enhance the drug transport.
Nesta tese estudamos um sistema de equações diferenciais de derivadas parciais definido por uma equação hiperbólica – uma equação de onda, e duas equações parabólicas – uma equação de difusão-reação quase linear e uma equação de convecção-difusão-reação. Neste sistema, o termo reativo da primeira equação parabólica depende da solução da equação da onda, e a velocidade convectiva da segunda equação parabólica depende da solução da primeira equação e do seu gradiente. O coeficiente de difusão da última equação depende também das soluções das duas primeiras equações. O problema matemático que motivou esta dissertação surge no contexto de diversos problemas físicos, como por exemplo, no contexto da libertação controlada de fármacos estimulada por ultrassons. Neste caso, a propagação da onda de pressão acústica descrita pela equação hiperbólica, induz um aumento da temperatura no tecido alvo, um aumento no transporte do fármaco, e o aumento da temperatura induz um aumento do transporte difusivo do fármaco. Neste trabalho, propomos um método numérico para o sistema diferencial definido num domínio espacial de duas dimensões. O nosso método pode ser visto, simultaneamente, como um método de elementos finitos segmentado linear discreto no espaço, e como um método de diferenças finitas definido em malhas retangulares não uniformes. Para este método provamos a segunda ordem de convergência, relativamente a uma norma que pode ser vista como uma versão discreta da norma usual de $H^1$, para a discretização da equação hiperbólica. Este resultado permite concluir que a aproximação para o gradiente não deteriora a qualidade da aproximação para a concentração. Estabelecemos que as aproximações para a temperatura e para a concentração também são de segunda ordem, mas relativamente a uma norma que pode ser vista como uma discretização da norma usual de $L^2$. Os resultados de convergência são demonstrados utilizando condições de regularidade mais fracas do que as usadas usualmente. No contexto dos métodos de diferenças finitas, uma vez que consideramos malhas não uniformes onde os erros de truncatura associados são de primeira ordem relativamente à norma $\| . \|_\infty$, os nossos resultados podem ser vistos como resultados de supraconvergência. Visto que o método proposto pode ser visto como um método de elementos finitos segmentado linear, no contexto dos métodos de elementos finitos os nossos resultados podem ser vistos como resultados de superconvergência. De facto, é bem conhecido que os métodos de elementos finitos segmentados lineares para equações elípticas levam a aproximações convergentes de primeira ordem, relativamente à norma usual de $H^1$. Os resultados teóricos obtidos são ilustrados numericamente. A precisão das condições de regularidade impostas às soluções do sistema diferencial contínuo é também analisada numericamente. Podemos encontrar na literatura que o uso de ultrassons leva a um aumento do transporte do fármaco e da sua absorção pelo tecido alvo em diferentes contextos, como por exemplo em tratamentos de cancro. Uma versão simples do sistema estudado neste trabalho é considerada para ilustrar a eficiência do uso dos ultrassons como estímulo ao transporte de fármacos.
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Books on the topic "Parabolic-hyperbolic coupling"

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Gastaldi, Fabio. On the coupling of hyperbolic and parabolic systems: Analytical and numerical approach. Hampton, Va: ICASE, 1988.

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Alfio, Quarteroni, and Langley Research Center, eds. On the coupling of hyperbolic and parabolic systems: Analytical and numerical approach. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1989.

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Book chapters on the topic "Parabolic-hyperbolic coupling"

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Aguillon, Nina. "Numerical Simulations of a Fluid-Particle Coupling." In Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, 759–67. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-05591-6_76.

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Galdi, Giovanni Paolo, Mahdi Mohebbi, Rana Zakerzadeh, and Paolo Zunino. "Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems." In Fluid-Structure Interaction and Biomedical Applications, 197–256. Basel: Springer Basel, 2014. http://dx.doi.org/10.1007/978-3-0348-0822-4_3.

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Gastaldi, Fabio, and Alfio Quarteroni. "On the Coupling of Hyperbolic and Parabolic Systems: Analitical and Numerical Approach." In Proceedings of the Third German-Italian Symposium Applications of Mathematics in Industry and Technology, 123–65. Wiesbaden: Vieweg+Teubner Verlag, 1989. http://dx.doi.org/10.1007/978-3-322-96692-6_8.

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Rybak, Iryna. "Coupling Free Flow and Porous Medium Flow Systems Using Sharp Interface and Transition Region Concepts." In Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, 703–11. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-05591-6_70.

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Fuhrmann, Jürgen, Alexander Linke, and Christian Merdon. "Coupling of Fluid Flow and Solute Transport Using a Divergence-Free Reconstruction of the Crouzeix-Raviart Element." In Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, 587–95. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-05591-6_58.

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Brenner, Konstantin, Roland Masson, Laurent Trenty, and Yumeng Zhang. "Coupling of a Two Phase Gas Liquid Compositional 3D Darcy Flow with a 1D Compositional Free Gas Flow." In Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, 517–25. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-05591-6_51.

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Crestetto, Anaïs, Nicolas Crouseilles, and Mohammed Lemou. "Asymptotic-Preserving Scheme Based on a Finite Volume/Particle-In-Cell Coupling for Boltzmann-BGK-Like Equations in the Diffusion Scaling." In Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, 827–35. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-05591-6_83.

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Conference papers on the topic "Parabolic-hyperbolic coupling"

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Chouly, Franz, Miguel A. Fernández, Theodore E. Simos, George Psihoyios, and Ch Tsitouras. "An Enhanced Parareal Algorithm for Partitioned Parabolic-Hyperbolic Coupling." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS: International Conference on Numerical Analysis and Applied Mathematics 2009: Volume 1 and Volume 2. AIP, 2009. http://dx.doi.org/10.1063/1.3241387.

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Dutta, Ashim, Kyunghan Kim, Kunal Mitra, and Zhixiong Guo. "Experimental Measurements and Numerical Modeling Validation of Temperature Distribution in Tissue Medium During Short Pulse Laser Irradiation." In ASME 2007 International Mechanical Engineering Congress and Exposition. ASMEDC, 2007. http://dx.doi.org/10.1115/imece2007-41295.

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The objective of this paper is to analyze the temperature distributions and heat affected zone in skin tissue medium when irradiated with either a collimated or a focused laser beam from a short pulse laser source. Single-layer and three-layer tissue phantoms containing embedded inhomogeneities are used as a model of human skin tissue having subsurface tumor. Q-switched Nd:YAG laser is used in this study. Experimental measurements of axial and radial temperature distribution in the tissue phantom are compared with the numerical modeling results. For numerical modeling, the transient radiative transport equation is first solved using discrete ordinates method for obtaining the intensity distribution and radiative heat flux inside the tissue medium. Then the temperature distribution is obtained by coupling the bio-heat transfer equation with either hyperbolic non-Fourier or parabolic Fourier heat conduction model. The hyperbolic heat conduction equation is solved using MacCormack’s scheme with error terms correction. It is observed that experimentally measured temperature distribution is in good agreement with that predicted by hyperbolic heat conduction model. The experimental measurements also demonstrate that converging laser beam focused directly at the subsurface location can produce desired high temperature at that location as compared to that produced by collimated laser beam for the same laser parameters.
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Hagani, Fouad, M'hamed Boutaous, Ronnie Knikker, Shihe Xin, and Dennis Siginer. "Numerical Modeling of Non-Affine Viscoelastic Fluid Flow Including Viscous Dissipation Through a Square Cross-Section Duct: Heat Transfer Enhancement due to the Inertia and the Elastic Effects." In ASME 2020 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/imece2020-23558.

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Abstract Non-isothermal laminar flow of a viscoelastic fluid including viscous dissipation through a square cross–section duct is analyzed. Viscoelastic stresses are described by Giesekus modele orthe Phan-Thien–Tanner model and the solvent shear stress is given by the linear Newtonian constitutive relationship. The flow through the tube is governed by the conservation equations of energy, mass, momentum associated with to one non–affine rheological model mentioned above. The mixed type of the governing system of equations (elliptic–parabolic–hyperbolic) requires coupling between discretisation methods designed for elliptic–type equations and techniques adapted to transport equations. To allow appropriate spatial discretisation of the convection terms, the system is rewritten in a quasi-linear first-order and homogeneous form without the continuity and energy equations. With the rheological models of the Giesekus type, the conformation tensor is by definition symmetrical and positive-definite, with the PTT model the hyperbolicity condition is subject to restrictions related to the rheological parameters. Based on this hyperbolicity condition, the contribution of the hyperbolic part is approximated by applying the characteristic method to extract pure advection terms which are then discretized by high ordre schemes WENO and HOUC. The algorithm thus developed makes it possible, to avoid the problems of instabilities related to the high Weissenberg number without the use of any stabilization method. Finally, a Nusselt number analysis is given as a function of inertia, elasticity, viscous dissipation, for constant solvent viscosity ratio and constant material and rheological parameters.
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Bahmani, Bahador, and Amir R. Khoei. "Modeling Convective Heat Propagation in a Fractured Domain With X-FEM and Least Square Method." In ASME 2017 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/imece2017-71167.

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The main goal of the current study is developing an advanced and robust numerical tool for accurate capturing heat front propagation. In some applications such as impermeable medium, Heat transfer in the surrounding domain of fracture acts just as a conduction process but the heat transfer through the fractures appears as a convection process. From a mathematical point of view, a parabolic partial differential equation (PDE) should be solved in the surrounding domain whereas a hyperbolic PDE should be solved in the domain of fractures. In fact, they have completely different treatments and this is one of the complicated problems in this area. In this paper, the presence of fractures and discontinuities are considered with the aim of eXtended Finite Element Method (X-FEM). In the proposed numerical approach, the domain is decomposed into local and global scales. Global and local domains are solved by the X-FEM and Least Square Method (LSM) techniques, respectively. As a final result, it is determined that the treatment of coupling term between two scales is one of the most important factors for system performance. Increasing its effect can significantly improve the efficiency of the whole system.
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