Hyperbolic Heat Equation Research Articles

A survey on well-balanced and asymptotic preserving schemes for hyperbolic models for chemotaxis

Chemotaxis is a biological phenomenon widely studied these last years, at the biological level but also at the mathematical level. Various models have been proposed, from microscopic to macroscopic scales. In this article, we consider in particular two hyperbolic models for the density of organisms, a semi-linear system based on the hyperbolic heat equation (or dissipative waves equation) and a quasi-linear system based on incompressible Euler equation. These models possess relatively stiff solutions and well-balanced and asymptotic-preserving schemes are necessary to approximate them accurately. The aim of this article is to present various techniques of well-balanced and asymptotic-preserving schemes for the two hyperbolic models for chemotaxis.

Existence and approximation for Navier–Stokes system with Tresca’s friction at the boundary and heat transfer governed by Cattaneo’s law

We consider an unsteady non-isothermal incompressible fluid flow. We model heat conduction with Cattaneo’s law instead of the commonly used Fourier’s law, in order to overcome the physical paradox of infinite propagation speed. We assume that the fluid viscosity depends on the temperature, while the thermal capacity depends on the velocity field. The problem is thus described by a Navier–Stokes system coupled with the hyperbolic heat equation. Furthermore, we consider non-standard boundary conditions with Tresca’s friction law on a part of the boundary. By using a time-splitting technique, we construct a sequence of decoupled approximate problems and we prove the convergence of the corresponding approximate solutions, leading to an existence theorem for the coupled fluid flow/heat transfer problem. Finally, we present some numerical results.

MATHEMATICAL MODELING OF THERMAL PROCESSES AT SURFACE TREATMENT OF METAL PRODUCTS WITH HIGHLY CONCENTRATED ENERGY FLOWS

The modeling tasks of thermal effect of power impulse action on a surface of the plate of VК10 (КS) alloy were considered and solved. As modeling tasks for homogeneous equations of parabolic and hyperbolic heat conductions, a wave equation in a cylindrical solid of final sizes with boundary conditions of III kind were chosen. The action of power impulse from an exterior radiant was modeled by sudden appearance of initial high temperature, which spreads on a plate body under the laws expressed by various heat conduction equations, on one of the surface ends of a cylinder. Approaches of temperature fields were received in the form of a series segment of functions from eigenvalues of tasks, gradients of fields were defined. Simultaneous presence in the equation of heat conductivity of private derivatives on time of the first and second usages (the hyperbolic equation), statement of the task for it with boundary conditions of III kind and the entry condition at a cylinder end surface provides two ways (modes) of the problem’s decision, both of diffusion type. For the value of the relaxation time of the heat flux of 10–11 s, the complete cooling of the cylindrical sample (tungsten carbide) in the first mode is minutes, in the second – 10–10 s. It can be concluded that the modes for solving the problem for the hyperbolic heat equation do not correspond to the actual pattern of heat propagation. However, the linear combination of these modes as a solution of the problem preserves the possibility of obtaining a diffusion dynamics adequate to the actual process. Gradients of the temperature field in the solutions of the problems for the parabolic heat conduction equation and the wave equation are in the same order of values. The temperature field of the moving thermal wave for several of its first reflections in experimental samples should be taken into account when evaluating phase transformations and temperature stresses. The results of the theoretical analysis are compared with changes in the microstructure of the near-surface layer of a plate of alloy VK10 (KS), subjected to electric explosive loading by plasma of a titanium foil.

Regularity of solutions of the anisotropic hyperbolic heat equation with nonregular heat sources and homogeneous boundary conditions

The research of this author was partially supported by Generalitat Valenciana, Conselleria d'Educacio, Cultura i Esport, Spain, Grant PROMETEO/2013/058.

Intensive wave power and steel quenching 3-D model for cylindrical sample. Time direct and reverse formulations and solutions

In this paper we develop mathematical models for three dimensional hyperbolic heat equations (wave equation or telegraph equation) with inner source power and construct their analytical solutions for the determination of the initial heat flux for cylindrical sample. As additional conditions the temperature and heat flux at the end time are given. In some cases we give expression of wave energy. In some cases we give expression of wave energy. Some solutions of time inverse problems are obtained in the form of first kind Fredholm integral equation, but others has been obtained in closed analytical form as series. We viewed both direct and inverse problems at the time. For the time inverse problem we use inversion in the time argument.

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

We studied physical problems related to heat transport and the corresponding differential equations, which describe a wider range of physical processes. The operational method was employed to construct particular solutions for them. Inverse differential operators and operational exponent as well as operational definitions and operational rules for generalized orthogonal polynomials were used together with integral transforms and special functions. Examples of an electric charge in a constant electric field passing under a potential barrier and of heat diffusion were compared and explored in two dimensions. Non-Fourier heat propagation models were studied and compared with each other and with Fourier heat transfer. Exact analytical solutions for the hyperbolic heat equation and for its extensions were explored. The exact analytical solution for the Guyer-Krumhansl type heat equation was derived. Using the latter, the heat surge propagation and relaxation was studied for the Guyer-Krumhansl heat transport model, for the Cattaneo and for the Fourier models. The comparison between them was drawn. Space-time propagation of a power–exponential function and of a periodic signal, obeying the Fourier law, the hyperbolic heat equation and its extended Guyer-Krumhansl form were studied by the operational technique. The role of various terms in the equations was explored and their influence on the solutions demonstrated. The accordance of the solutions with maximum principle is discussed. The application of our theoretical study for heat propagation in thin films is considered. The examples of the relaxation of the initial laser flash, the wide heat spot, and the harmonic function are considered and solved analytically.

Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes

We prove the uniform AP convergence on unstructured meshes in 2D of a generalization, of the Gosse-Toscani 1D scheme for the hyperbolic heat equation. This scheme is also a nodal extension in 2D of the Jin-Levermore scheme described in [18] for the 1D case. In 2D, the proof is performed using a new diffusion scheme.

Analytical solutions for heat diffusion beyond Fourier law

We obtain solutions for differential equations, describing a broad range of physical problems by the operational method with recourse to inverse differential operators, integral transforms and operational exponent. Generalized families of orthogonal polynomials and special functions are also employed with recourse to their operational definitions. The evolutional type problems for heat transfer in various heat conduction models are studied. Exact analytical solutions for Guyer–Krumhansl hyperbolic heat equation are obtained and compared with those of Fourier and Cattaneo equations. Modelling heat pulse propagation from a laser source is performed in the framework of Fourier, Cattaneo and Guyer–Krumhansl heat transfer models. Compliance of obtained solutions with the maximum principle is studied.

Modeling ballistic effects in frequency-dependent transient thermal transport using diffusion equations

Understanding ballistic phonon transport effects in transient thermoreflectance experiments and explaining the observed deviations from classical theory remains a challenge. Diffusion equations are simple and computationally efficient but are widely believed to break down when the characteristic length scale is similar or less than the phonon mean-free-path. Building on our prior work, we demonstrate how well-known diffusion equations, namely, the hyperbolic heat equation and the Cattaneo equation, can be used to model ballistic phonon effects in frequency-dependent periodic steady-state thermal transport. Our analytical solutions are found to compare excellently to rigorous numerical results of the phonon Boltzmann transport equation. The correct physical boundary conditions can be different from those traditionally used and are paramount for accurately capturing ballistic effects. To illustrate the technique, we consider a simple model problem using two different, commonly used heating conditions. We demonstrate how this framework can easily handle detailed material properties, by considering the case of bulk silicon using a full phonon dispersion and mean-free-path distribution. This physically transparent approach provides clear insights into the nonequilibrium physics of quasi-ballistic phonon transport and its impact on thermal transport properties.

Cattaneo–Vernotte equation versus Fourier thermoelastic hyperbolic heat equation

The aim of this short note is to point out the inconsistency of the Cattaneo–Vernotte law. This law is assumed to be valid at a scale L of a few mean free paths of phonons lfp. Thus a separation of scales is absent and the Cattaneo–Vernotte law looses its status of equivalent macroscopic law. On an other hand, the classical thermoelastic system, which is well supported by experimental investigations, is valid for L ≫ lfp where a separation of scales is present. In this domain of validity, an acoustic wave is adiabatic. It results in an apparent hyperbolic heat equation which remains valid for L > 10 lfp. This equation shows a similar structure to the Cattaneo–Vernotte equation, however with a different relaxation time.

Exact solution of Guyer–Krumhansl type heat equation by operational method

We construct particular solutions for some heat transport differential equations, in particular, for extended forms of hyperbolic heat equation and of Guyer–Krumhansl (GK) equation. The operational approach, integral transforms, generalized orthogonal polynomials and special functions are used. Examples of heat propagation in non-Fourier models are studied and compared with each other. Analytical solutions for some three-dimensional heat transport equations are obtained. The exact analytical solutions for GK type heat equation with linear term are derived. The description of an instant heat surge propagation and of power-exponential pulse is given in heat transport models of Fourier, Cattaneo and Guyer–Krumhansl. Space–time propagation of a periodic function, obeying telegraph and GK equations with linear terms is studied by the operational technique. The exact bounded analytical solutions are obtained. The role of various terms in the equations is illustrated and their influence on the solutions is elucidated. The application for ballistic heat flow study with account for Knudsen number is provided.

Identification of a time-dependent source term in nonlinear hyperbolic or parabolic heat equation

A problem of identification of the heat transfer sources in nonlinear media is considered with account for the time of heat flux relaxation. In order to solve the identification problem we use the approach based on the theory of inverse infinite-dimensional open dynamical systems (DS) or, in other words, input-state-output systems. For solving the identification problems we used finite-difference methods. The results of numerical simulation of the problems of internal heat sources identification in nonlinear hyperbolic or parabolic equation are presented.

Numerical simulation on hyperbolic diffusion equations using modified cubic B-spline differential quadrature methods

In this article, a modified cubic B-spline differential quadrature method (MCB-DQM) is proposed to solve a hyperbolic diffusion problem in which flow motion is affected by both convection and diffusion. One dimensional hyperbolic non-homogeneous heat, wave and telegraph equations are also considered along with two dimensional hyperbolic diffusion problem. The method reduces the hyperbolic problem into a system of nonlinear ordinary differential equations. The system is then solved by the optimal four stage three order strong stability-preserving time stepping Runge–Kutta (SSP-RK43) scheme. The reliability and efficiency of the method has been tested on seven examples. The stability of the method is also discussed and found to be unconditionally stable.

A Counterexample about the Regularity of the Solutions of the Hyperbolic Heat Equation

We give an example of a fundamental solution of the Neumann problem for the hyperbolic heat equation in a disk D which is not in L2(D ×]0,T[) whatever be T > 0.

A simple Boltzmann transport equation for ballistic to diffusive transient heat transport

Developing simplified, but accurate, theoretical approaches to treat heat transport on all length and time scales is needed to further enable scientific insight and technology innovation. Using a simplified form of the Boltzmann transport equation (BTE), originally developed for electron transport, we demonstrate how ballistic phonon effects and finite-velocity propagation are easily and naturally captured. We show how this approach compares well to the phonon BTE, and readily handles a full phonon dispersion and energy-dependent mean-free-path. This study of transient heat transport shows (i) how fundamental temperature jumps at the contacts depend simply on the ballistic thermal resistance, (ii) that phonon transport at early times approach the ballistic limit in samples of any length, and (iii) perceived reductions in heat conduction, when ballistic effects are present, originate from reductions in temperature gradient. Importantly, this framework can be recast exactly as the Cattaneo and hyperbolic heat equations, and we discuss how the key to capturing ballistic heat effects is to use the correct physical boundary conditions.

Existence for non-isothermal fluid flows with Tresca's friction and Cattaneo's heat law

We consider a two dimensional non-isothermal incompressible fluid flow involving heat transfer and friction. The problem is described by the non-stationary Navier–Stokes system and by the hyperbolic heat equation derived from Cattaneo's heat flux law, which allows to overcome the paradox of infinite propagation speed. We assume a strong coupling due to temperature dependent viscosity and velocity dependent heat capacity. Moreover we consider Tresca's friction boundary conditions for the flow. We prove an existence result for this problem by using a fixed point technique.

Fourier, hyperbolic and relativistic heat transfer equations: a comparative analytical study

Parabolic heat equation based on Fourier's theory (FHE), and hyperbolic heat equation (HHE), has been used to mathematically model the temperature distributions of biological tissue during thermal ablation. However, both equations have certain theoretical limitations. The FHE assumes an infinite thermal energy propagation speed, whereas the HHE might possibly be in breach of the second law of thermodynamics. The relativistic heat equation (RHE) is a hyperbolic-like equation, whose theoretical model is based on the theory of relativity and which was designed to overcome these theoretical impediments. In this study, the three heat equations for modelling of thermal ablation of biological tissues (FHE, HHE and RHE) were solved analytically and the temperature distributions compared. We found that RHE temperature values were always lower than those of the FHE, while the HHE values were higher than the FHE, except for the early stages of heating and at points away from the electrode. Although both HHE and RHE are mathematically hyperbolic, peaks were only found in the HHE temperature profiles. The three solutions converged for infinite time or infinite distance from the electrode. The percentage differences between the FHE and the other equations were larger for higher values of thermal relaxation time in HHE.

Modeling of ultrashort pulsed laser irradiation in the cornea based on parabolic and hyperbolic heat equations using electrical analogy

Hyperbolic and parabolic heat equations are formulated to study a nonperfused homogeneous transparent cornea irradiated by high power and ultrashort pulsed laser in the Laser Thermo Keratoplasty (LTK) surgery. Energy absorption inside the cornea is modeled using the Beer–Lambert law that is incorporated as an exponentially decaying heat source. The hyperbolic and parabolic bioheat models of the tissue were solved by exploiting the mathematical analogy between thermal and electrical systems, by using robust circuit simulation program called Hspice to get the solutions of simultaneous RLC and RC transmission line networks. This method can be used to rapidly calculate the temperature in laser-irradiated tissue at time and space domain. It is found that internal energy gained from the irradiated field results in a rapid rise of temperature in the cornea surface during the early heating period, while the hyperbolic wave model predicts a higher temperature rise than the classical heat diffusion model. In addition, this paper investigates and examines the effect of some critical parameters such as relaxation time, convection coefficient, radiation, tear evaporation and variable thermal conductivity of cornea. Accordingly, it is found that a better accordance between hyperbolic and parabolic models will be achieved by time.

Asymptotic Preserving Schemes on Distorted Meshes for Friedrichs Systems with Stiff Relaxation: Application to Angular Models in Linear Transport

In this paper we propose an asymptotic preserving scheme for a family of Friedrichs systems on unstructured meshes based on a decomposition between the hyperbolic heat equation and a linear hyperbolic which not involved in the diffusive regime. For the hyperbolic heat equation we use asymptotic preserving schemes recently designed in [7]---[20]. To discretize the second part we use classical Rusanov or upwind schemes. To finish we apply this method for the discretization of the $$P_N$$ P N and $$S_N$$ S N models which are widely used in transport codes.

Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions

We consider a non-isothermal modified Cahn--Hilliard equation which was previously analyzed by M. Grasselli et al. Such an equation is characterized by an inertial term and a viscous term and it is coupled with a hyperbolic heat equation. The resulting system was studied in the case of no-flux boundary conditions. Here we analyze the case in which the order parameter is subject to a dynamic boundary condition. This assumption requires a more refined strategy to extend the previous results to the present case. More precisely, we first prove the well-posedness for solutions with bounded energy as well as for weak solutions. Then we establish the existence of a global attractor. Finally, we prove the convergence of any given weak solution to a single equilibrium by using a suitable Lojasiewicz--Simon inequality.