Journal articles: '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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Tullis, Stephen, and Cameron Galipeau. "Fully Coupled Modeling of Athlete Force Application and Power Transfer in Rowing Ergometry." Proceedings 49, no. 1 (June 15, 2020): 108. http://dx.doi.org/10.3390/proceedings2020049108.

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A fully coupled model of an athlete’s muscular force output combined with a load resistance is developed and investigated in context of ergometer rowing. The athlete force is based on a simple Hill equation hyperbolic-in-speed, and parabolic-in-length model. Coupling this force function with the dynamics of the ergometer load and inertia and athlete’s own body mass inertia produces a trajectory of the resultant motion in force-speed-length space. The coupled equations were solved using a first order time-marching procedure, and iteratively calculated starting conditions based on ergometer spin-down during the recovery period between strokes. The results agree well with experimental measurements available from Kleshnev particularly given the relatively simple, and untuned, athlete force model used. Changing the load resistance changed the trajectory of the stroke, with qualitative agreement with the expected outcomes.

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Bergounioux, Maïtine, Xavier Bonnefond, Thomas Haberkorn, and Yannick Privat. "An optimal control problem in photoacoustic tomography." Mathematical Models and Methods in Applied Sciences 24, no. 12 (August 15, 2014): 2525–48. http://dx.doi.org/10.1142/s0218202514500286.

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This paper is devoted to the introduction and study of a photoacoustic tomography model, an imaging technique based on the reconstruction of an internal photoacoustic source distribution from measurements acquired by scanning ultrasound detectors over a surface that encloses the body containing the source under study. In a nutshell, the inverse problem consists in determining absorption and diffusion coefficients in a system coupling a hyperbolic equation (acoustic pressure wave) with a parabolic equation (diffusion of the fluence rate), from boundary measurements of the photoacoustic pressure. Since such kinds of inverse problems are known to be generically ill-posed, we propose here an optimal control approach, introducing a penalized functional with a regularizing term in order to deal with such difficulties. The coefficients we want to recover stand for the control variable. We provide a mathematical analysis of this problem, showing that this approach makes sense. We finally write necessary first-order optimality conditions and give preliminary numerical results.

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Kadioglu, Samet Y. "An Essentially Nonoscillatory Spectral Deferred Correction Method for Hyperbolic Problems." International Journal of Computational Methods 13, no. 03 (May 31, 2016): 1650017. http://dx.doi.org/10.1142/s0219876216500171.

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We present a computational method based on the spectral deferred corrections (SDC) time integration technique and the essentially nonoscillatory (ENO) finite volume method for hyperbolic problems. The SDC technique is used to advance the solutions in time with high-order of accuracy. The ENO method is used to define high-order cell edge quantities that are then used to evaluate numerical fluxes. The coupling of the SDC method with a high-order finite volume method (piece-wise parabolic method (PPM)) is first carried out by Layton et al. [J. Comput. Phys. 194(2) (2004) 697]. Issues about this approach have been addressed and some improvements have been added to it in Kadioglu et al. [J. Comput. Math. 1(4) (2012) 303]. Here, we investigate the implications when the PPM method is replaced with the well-known ENO method. We note that the SDC-PPM method is fourth-order accurate in time and space. Therefore, we kept the order of accuracy of the ENO procedure as fourth-order in order to be able to make a consistent comparison between the two approaches (SDC-ENO versus SDC-PPM). We have tested the new SDC-ENO technique by solving smooth and nonsmooth hyperbolic problems. Our numerical results indicate that the fourth-order of accuracy in both space and time has been achieved for smooth problems. On the other hand, the new method performs very well when it is applied to nonlinear problems that involve discontinuous solutions. In other words, we have obtained highly resolved discontinuous solutions with essentially no-oscillations at or around the discontinuities.

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Friedly, J. C. "Transient Response of a Coupled Conduction and Convection Heat Transfer Problem." Journal of Heat Transfer 107, no. 1 (February 1, 1985): 57–62. http://dx.doi.org/10.1115/1.3247402.

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Systems of dynamic models involving the coupling of both conduction and convection offer significant theoretical challenges because of the interaction between parabolic and hyperbolic types of responses. Recent results of state space theory for coupled partial differential equation models are applied to conjugate heat transfer problems in an attempt to understand this interaction. Definition of a matrix of Green’s functions for such problems permits the transient responses to be resolved directly in terms of the operators’ spectral properties when they can be obtained. Application of the theory to a simple conjugate heat transfer problem is worked out in detail. The model consists of the transient energy storage or retrieval in a stationary, single dimensioned matrix through which an energy transport fluid flows. Even though the partial differential operator is nonself-adjoint, it is shown how its spectral properties can be obtained and used in the general solution. Computations are presented on the effect of parameters on the spectral properties and the nature of the solution. Comparison is made with several readily solvable limiting cases of the equations.

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SCHINNERL, M., M. KALTENBACHER, U. LANGER, R. LERCH, and J. SCHÖBERL. "A Survey in Mathematics for Industry An efficient method for the numerical simulation of magneto-mechanical sensors and actuators." European Journal of Applied Mathematics 18, no. 2 (April 2007): 233–71. http://dx.doi.org/10.1017/s0956792507006882.

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The dynamic behaviour of magneto-mechanical sensors and actuators can be completely described by Maxwell's and Navier-Lamé's partial differential equations (PDEs) with appropriate coupling terms reflecting the interactions of these fields and with the corresponding initial, boundary and interface conditions. Neglecting the displacement currents, which can be done for the classes of problems considered in this paper, and introducing the vector potential for the magnetic field, we arrive at a system of degenerate parabolic PDEs for the vector potential coupled with the hyperbolic PDEs for the displacements.Usually the computational domain, the finite element discretization, the time integration, and the solver are different for the magnetic and mechanical parts. For instance, the vector potential is approximated by edge elements whereas the finite element discretization of the displacements is based on nodal elements on different meshes. The most time consuming modules in the solution procedure are the solvers for both, the magnetical and the mechanical finite element equations arising at each step of the time integration procedure. We use geometrical multigrid solvers which are different for both parts. These multigrid solvers enable us to solve quite efficiently not only academic test problems, but also transient 3D technical magneto-mechanical systems of high complexity such as solenoid valves and electro-magnetic-acoustic transducers. The results of the computer simulation are in very good agreement with the experimental data.

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Ahmed, N. U. "Mathematical problems in modeling artificial heart." Mathematical Problems in Engineering 1, no. 3 (1995): 245–54. http://dx.doi.org/10.1155/s1024123x95000159.

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In this paper we discuss some problems arising in mathematical modeling of artificial hearts. The hydrodynamics of blood flow in an artificial heart chamber is governed by the Navier-Stokes equation, coupled with an equation of hyperbolic type subject to moving boundary conditions. The flow is induced by the motion of a diaphragm (membrane) inside the heart chamber attached to a part of the boundary and driven by a compressor (pusher plate). On one side of the diaphragm is the blood and on the other side is the compressor fluid. For a complete mathematical model it is necessary to write the equation of motion of the diaphragm and all the dynamic couplings that exist between its position, velocity and the blood flow in the heart chamber. This gives rise to a system of coupled nonlinear partial differential equations; the Navier-Stokes equation being of parabolic type and the equation for the membrane being of hyperbolic type. The system is completed by introducing all the necessary static and dynamic boundary conditions. The ultimate objective is to control the flow pattern so as to minimize hemolysis (damage to red blood cells) by optimal choice of geometry, and by optimal control of the membrane for a given geometry. The other clinical problems, such as compatibility of the material used in the construction of the heart chamber, and the membrane, are not considered in this paper. Also the dynamics of the valve is not considered here, though it is also an important element in the overall design of an artificial heart. We hope to model the valve dynamics in later paper.

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Ebenfeld, Stefan. "Non-linear initial boundary value problemsof hyperbolic-parabolic type. A general investigationof admissible couplings between systems of higher order. Part 1: linear theory." Mathematical Methods in the Applied Sciences 25, no. 3 (February 2002): 179–212. http://dx.doi.org/10.1002/mma.283.

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Ebenfeld, Stefan. "Non-linear initial boundary value problems of hyperbolic-parabolic type. A general investigation of admissible couplings between systems of higher order. Part 3: applications." Mathematical Methods in the Applied Sciences 25, no. 3 (February 2002): 241–62. http://dx.doi.org/10.1002/mma.285.

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Ebenfeld, Stefan. "Non-linear initial boundary value problems of hyperbolic-parabolic type. A general investigation of admissible couplings between systems of higher order. Part 2: abstract quasilinear theory." Mathematical Methods in the Applied Sciences 25, no. 3 (February 2002): 213–40. http://dx.doi.org/10.1002/mma.284.

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Lévi, Laurent. "Obstacle problems for a coupling of quasilinear hyperbolic-parabolic equations." Interfaces and Free Boundaries, 2007, 331–54. http://dx.doi.org/10.4171/ifb/167.

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Fan, Long, Cheng-Jie Liu, and Lizhi Ruan. "Local well-posedness of solutions to the boundary layer equations for compressible two-fluid flow." Electronic Research Archive, 2021, 0. http://dx.doi.org/10.3934/era.2021070.

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<p style='text-indent:20px;'>In this paper, we consider the two-dimensional (2D) two-fluid boundary layer system, which is a hyperbolic-degenerate parabolic-elliptic coupling system derived from the compressible isentropic two-fluid flow equations with nonslip boundary condition for the velocity. The local existence and uniqueness is established in weighted Sobolev spaces under the monotonicity assumption on tangential velocity along normal direction based on a nonlinear energy method by employing a nonlinear cancelation technic introduced in [R. Alexandre, Y.-G. Wang, C.-J. Xu and T. Yang, J. Amer. Math. Soc., 28 (2015), 745-784; N. Masmoudi and T.K. Wong, Comm. Pure Appl. Math., 68(2015), 1683-1741] and developed in [C.-J. Liu, F. Xie and T. Yang, Comm. Pure Appl. Math., 72(2019), 63-121].</p>

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Soszyńska, Martyna, and Thomas Richter. "Adaptive time-step control for a monolithic multirate scheme coupling the heat and wave equation." BIT Numerical Mathematics, April 1, 2021. http://dx.doi.org/10.1007/s10543-021-00854-3.

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AbstractWe study the dynamics of a parabolic and a hyperbolic equation coupled on a common interface. We develop time-stepping schemes that can use different time-step sizes for each of the subproblems. The problem is formulated in a strongly coupled (monolithic) space-time framework. Coupling two different step sizes monolithically gives rise to large algebraic systems of equations. There, multiple states of the subproblems must be solved at once. For efficiently solving these algebraic systems, we inherit ideas from the partitioned regime. Therefore we present two decoupling methods, namely a partitioned relaxation scheme and a shooting method. Furthermore, we develop an a posteriori error estimator serving as a mean for an adaptive time-stepping procedure. The goal is to optimally balance the time-step sizes of the two subproblems. The error estimator is based on the dual weighted residual method and relies on the space-time Galerkin formulation of the coupled problem. As an example, we take a linear set-up with the heat equation coupled to the wave equation. We formulate the problem in a monolithic manner using the space-time framework. In numerical test cases, we demonstrate the efficiency of the solution process and we also validate the accuracy of the a posteriori error estimator and its use for controlling the time-step sizes.

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Journal articles on the topic 'Parabolic-hyperbolic coupling'

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