Linear Heat Conduction Equation Research Articles

Development of a sixth-order two-dimensional convection–diffusion scheme via Cole–Hopf transformation

In this paper, we develop a two-dimensional finite-difference scheme for solving the time-dependent convection–diffusion equation. The numerical method exploits Cole–Hopf equation to transform the nonlinear scalar transport equation into the linear heat conduction equation. Within the semi-discretization context, the time derivative term in the transformed parabolic equation is approximated by a second-order accurate time-stepping scheme, resulting in an inhomogeneous Helmholtz equation. We apply the alternating direction implicit scheme of Polezhaev to solve the Helmholtz equation. As the key to success in the present simulation, we develop a Helmholtz scheme with sixth-order spatial accuracy. As is standard practice, we validated the code against test problems which were amenable to exact solutions. Results show excellent agreement for the one-dimensional test problems and good agreement with the analytical solution for the two-dimensional problem.

A novel approach to Burgers' equation

AbstractA new method is presented for the numerical solution of Burgers' equation. In this method, the Green's function of the linear heat conduction equation is employed to convert Burgers' equation to the corresponding integration equation system with respect to the time variable. Then the solution of this integration system is obtained by a numerical method. The computed results show that this method can deal with a wide range of Reynolds number and in the moderate Reynolds number case, even with the comparatively larger time step, good accuracy can be achieved. Furthermore, this method does not require spatial discretization. Copyright © 2001 John Wiley & Sons, Ltd.

Stability of Charlie's Method on Linear Heat Conduction Equation

Kaedah tak tersirat sangat menarik dalam mendapatkan penyelesaian beza terhingga bagi persamaan terbitan separa kerana sifatnya yang ringkas. Akan tetapi batasan kestabilan yang teruk yang dialami oleh kaedah ini mele\-mah\-kan ciri ini. Suatu kaedah yang mempunyai sifat kestabilan yang lebih baik ialah kaedah Charlie. Rantau kestabilan bagi kaedah ini untuk persamaan haba matra satu dibincangkan di sini. Katakunci: kaedah tak tersirat; kaedah Charlie; peramal-pembetul; kestabilan. Explicit schemes are attractive for obtaining finite difference solutions to partial differential equations because of their simplicity. However this feature is undermined by the severe restriction on stability that the schemes suffer. One method that appears to have better stability properties is Charlie's method. The stability region of this method applied to a one-dimensional heat conduction equation is discussed in this article. Keywords: Explicit schemes; Charlie's method; predictor-corrector; stability.

The effect of a local increase in impurity concentration in one-dimensional hydrodynamics

The system of non-linear equations of one-dimensional convective diffusion of a passive impurity in the case of a flow of fluid described by the model Burgers equation is solved using Cole-Hopf and Darboux transformations. For equal kinetic coefficients of the fluid its solution is reduced to solving linear heat-conduction equations. For integer Prandtl numbers, differing from unity, this reduction is successful for flows of uniformly moving shock waves, smoothed by the viscosity effect. Its possibility is closely related to the factorizability of second-order differential operators in this case, their decomposability into the product of first-order operators, and the additional internal symmetry (supersymmetry) of the problem. The interaction of shock waves and the impurity solitons they transfer has an absolutely inelastic character. A local increase in the impurity concentration is a result of the merging of impurity solitons. It is pointed out that, for certain equations of state, the reduction of the solutions of the non-linear convective diffusion equations to the solution of linear heat-conduction equations is also possible for an active impurity (when the kinetic coefficients of the fluid are equal).

The non-Fourier effect on the fin performance under periodic thermal conditions

This paper investigates the effect of thermal relaxation time (the non-Fourier effect) on the thermal performance of a convective fin under periodic thermal conditions. The environment heat transfer coefficient is assumed to be spatially varying. The periodic oscillation of the base temperature is considered. An efficient numerical scheme involving the hybrid application of the Laplace transform and control volume methods in conjunction with hyperbolic shape functions is used to solve the linear hyperbolic heat conduction equation. The thermal performance of the fin predicted by using the non-Fourier heat conduction model is compared with that predicted by using the Fourier model. Results show that the effects of the thermal relaxation time on the fin performance are significant for a short time after the initial transient. For the steady state, the non-Fourier effect is still great if the frequency of the temperature oscillation is high.

Unconditionally stable FEM for transient linear heat conduction analysis

AbstractThe paper presents a new method for the numerical solution of transient linear heat conduction problems. In the proposed method, the transient linear heat conduction equation is first integrated with respect to time over subsequent intervals. Then the resulting set of elliptic equations is discretized in space according to a finite‐element procedure. The method is unconditionally stable. At the end of the paper, the effectiveness of the proposed method is assessed through some numerical examples.

A multilayer heat conduction solution for magneto-optical disk recording

The Green’s function temperature expressions formulated by McGahan and Cole [J. Appl. Phys. 72, 1362 (1992)] are modified into a form suitable for solving the heat conduction problem encountered in magneto-optical (MO) disk recording situations. The temperature distribution within MO multilayer media heated by a pulsed scanning Gaussian laser beam is calculated by using Fourier-transformed Green’s functions. The linear heat conduction equation is solved exactly not in real space but in frequency space. The temperature in real space is efficiently recovered by the inverse fast Fourier transform; numerical integrations are unnecessary. Optical absorption in MO media is calculated exactly. Realistic and piecewise-linear models of the laser pulse’s time dependence are incorporated directly into the formalism. Elliptically shaped laser-beam cross sections are also easily included. At the same time, the extended method still preserves the conceptual simplicity and computational efficiency of the original theory. This paper describes the extended method, discusses some numerical issues arising from the modifications, and presents comparisons with previously published finite-difference calculations.

Optimal moving grids for time-dependent partial differential equations

Various adaptive moving grid techniques for the numerical solution of time-dependent partial differential equations have been proposed. The precise criterion for grid motion varies, but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented. We investigate moving grids on which the solutions of the linear heat conduction and viscous Burgers' equation in one space dimension are optimally approximated. Precisely, we report the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of PDE solutions in the least-squares norm.

SOLUTION CONTOUR-BASED ADAPTIVE REGRIDDIIMG SCHEME FOR TRANSIENT DIFFUSION PROBLEMS

An adaptive regridding scheme within the framework of irregular triangles and finite differences is presented. The new grid's nodal points are selected on solution contours, Boundary adaptation is accomplished by using a pseudo-triangle gradient level indicator. The technique is demonstrated by adoptively solving the linear two-dimensional transient heat conduction equation on both rectangular and circular domains. Results indicate that the adaptive regridding scheme works as accurately as a fine grid, especially in regions where a steep change of solution occurs, but at a significantly lower computational cost. Apart from promising to be a general method, the scheme is simple in that it does not require the evaluation of weight functions, the solution of a set of differential equations, or the construction of a solution monitor surface to obtain new nodes.

A heat conduction equation with three relaxation times. Particular solutions

A one-dimensional, linear heat conduction equation involving three relaxation times is derived for a rigid thermal conductor. Form preserving and plane wave solutions are investigated. Attention is focused on the influence on wave propagation of a dissipation term involving the time derivative of temperature in the expression for entropy. Some numerical results are presented.

Generating exact solutions of the two‐dimensional Burgers' equations

Burgers’ equation is well suited to modelling fluid flows as it incorporates directly the interaction between the non-linear convection processes and the diffusive viscous processes. In one dimension the Cole-Hopf procedure transforms Burgers’ equation into the linear heat conduction equation. As a result many exact solutions of Burgers’ equation are available in the literature. Thus Burgers’ equation has often been used as a model equation for comparing the accuracy of different computational algorithms. This aspect of Burgers’ equation is reviewed by Fletcher.’ The two-dimensional Burgers’ equations

Identifiability in the large of a linear heat-conduction equation subject to cauchy boundary conditions

The feasibility of simultaneously determining constant values of the specific heat, thermal conductivity, and heat-transfer coefficient from observations of a unique temperature field is studied.

Uniqueness of the solution of the inverse problem for the coefficients of the linear heat-conduction equation

We consider the problems of unambiguous determination of coefficients of the linear heat-conduction equation from measurements on a temperature field.

A Simple Method For Predicting Cap And Base Rock Heat Losses In' Thermal Reservoir Simulators

Abstract A fast simple method is given for calculating heat losses to cap and base rock in thermal reservoir simulators. The method can handle backflow of heat and cyclic temperature variations. Test comparisons against analytic results are given. The method gives remarkably accurate results on the test problems, considering its simplicity. Introduction In thermal reservoir simulations, it is necessary to calculate the heat loss from the reservoir to the surrounding environment, usually to the cap and base rock. A finite-difference calculation can become expensive, especially on computer storage, as the temperature profile can extend large distances into the overburden and underburden, requiring large numbers of grid blocks for its description. However, high accuracy in the solution of the heat loss calculations is not usually required in reservoir simulators owing to the uncertainty in cap and base rock thermal parameters. This paper describes an efficient semi-analytical method for the case where the one-dimensional linear heat conduction equation is adequate for the heat loss calculations. Semi-analytical methods have been used previously (1,2,3), and a study of the results given in these papers leads us to the following tentative conclusions:Conduction within the cap rock rapidly wipes out any sharp differences in temperature. Thus, the temperature in the cap rock varies smoothly, even for rather erratic changes at the interface.Longitudinal heat conduction in the cap rock can usually be neglected, because the Peclet number is normally high for reservoir floods(2).The most important physical features of the heat loss process are good descriptions of the boundary conditions at the interface and conservation of cap rock energy. The tail of the temperature distribution contains little energy and is relatively unimportant. These points suggest that the temperature profile into the cap or base rock can be adequately approximated by any reasonably flexible function containing a few parameters. If such a function could be made to fit the thermal diffusivity equation with appropriate boundary conditions, and the calculation of the parameters could be done in a simple fast fashion, the method would become extremely attractive further justification for such an approach is that high accuracy is not required, as the coefficient of thermal conduction of the cap and base rock is rarely known precisely. The Method We choose as the fitting function for the temperature profile into the cap or base rock: (Equation Available In Full Paper) θ is the temperature at the interface between the reservoir and the cap or base rock (the zero level has been defined at the initial interface temperature); p and q are the fitting parameters yet to be determined; d may be interpreted as the diffusion length and should be of order √kt where t is the time measured from the instant at which the interface temperature first begins to change. The deepest penetrating term in (1) is z2e−z/d, which has its maximum at z = 2d. Hence we choose for the diffusion length:

Methods for determining thermophysical characteristics of solid materials based on solutions of linear heat-conduction equations of order higher than the second

On the basis of solutions of the linearized heat-conduction equations, formulas are derived for the highly accurate determination of the thermophysical characteristics of solid materials.

Symmetric thermoelastic stress in cylinders by the lanczos-chebyshev method

The Lanczos-Chebyshev method is used to reduce the linear heat conduction equation to a set of ordinary differential equations. The eigenvalues and eigenvectors defining the axial decay of temperature components in a solid cylinder are obtained. The thermal stresses are found by applying the same reduction technique to the linear displacement equations, considering thermal and self-equilibrating stress fields separately. Numerical examples are used to illustrate the speed and accuracy of the method in comparison with semi-analytical Bessel function type solutions and to show the end effects in cylinders.

A comparison of the finite element and finite difference methods for the analysis of steady two dimensional heat conduction problems

A comparison between the finite difference method and the finite element method for solving the linear Two-dimensional heat conduction equation is presented. In all areas except computer core storage, the finite element method is demonstrated to be superior to the finite difference method for this type of problem.

Determination of the temperature dependence of thermophysical characteristics of solid materials

A method is proposed to determine the thermophysical characteristics of solid materials on the basis of solving third- and fourth-order linear heat-conduction equations. It is shown that linear heat-conduction equations of order greater than two possess a high degree of accuracy of the solution.

Analysis of inverse heat-conduction problems by means of similarity solutions

Similarity solutions are given for a linear heat-conduction equation for a wide class of boundary conditions. The solutions are used for analyzing inverse problems that are applied to the determination of external heating conditions.

An exact shock wave solution to Burgers' equation for parametric excitation of the boundary

It is well known that Burgers' equation for plane waves can be transformed into the linear heat conduction equation yielding a general solution for any driving waveform on the boundary. Of especial current interest is the extent to which the difference-frequency signal can be “pumped” when the primary waves are “saturated”: i.e., weak shock waves occur. A direct harmonic analysis of the general solution for parametric excitation at the boundary would yield information on the conversion efficiency even in the presence of weak shock waves. By past experience, it is difficult to maintain good numerical accuracy for acoustic Reynolds numbers above 50. It is at the higher Reynolds numbers that “saturation” effects become more pronounced and so an alternative method for calculating the conversion efficiency is desired. The saddle point method of integration is utilized to develop from the general plane wave solution of Burgers an exact weak shock solution for boundary excitation by either two signals or by one amplitude modulated signal. Approximate solutions are also given for spherical waves which are applicable to the Fraunhofer zone. Criteria for predicting the range at which weak shock waves appear are also given.