Iterative methods of solution of difference schemes for the heat-conduction equation with nonlinear boundary conditions

1989 ◽  
Vol 44 (1) ◽  
pp. 18-26
Author(s):  
M. M. Karchevskii ◽  
E. M. Fedotov
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Iterative Methods ◽  
Heat Conduction Equation ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Difference Schemes ◽  
Conduction Equation

Difference Schemes with Nonlocal Boundary Conditions

2001 ◽  
Vol 1 (1) ◽  
pp. 62-71 ◽  
Author(s):  
Alexei V. Goolin ◽  
Nikolai I. Ionkin ◽  
Valentina A. Morozova
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Conduction Equation ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Difference Schemes ◽  
Conduction Equation ◽  
The Stability ◽  
The One ◽  
Necessary And Sufficient

AbstractThe paper deals with the stability, with respect to initial data, of difference schemes that approximate the heat-conduction equation with constant coefficients and nonlocal boundary conditions. Some difference schemes are considered for the one-dimensional heat-conduction equation, the energy norm is constructed, and the necessary and sufficient stability conditions in this norm are established for explicit and weighted difference schemes.

Stability Criterion of Difference Schemes for the Heat Conduction Equation with Nonlocal Boundary Conditions

2006 ◽  
Vol 6 (1) ◽  
pp. 31-55 ◽  
Author(s):  
A. Gulin ◽  
N. Ionkin ◽  
V. Morozova
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Stability Theory ◽  
Heat Conduction Equation ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Difference Schemes ◽  
Energy Norm ◽  
Conduction Equation ◽  
L2 Norm

Abstract The elements of the stability theory of nonselfadoint difference schemes for nonstationary problems of mathematical physics are discussed. Difference schemes for the heat conduction equation with nonlocal boundary conditions are considered in detail from the viewpoint of the general stability theory of two-layer operator-difference schemes. The necessary and sufficient stability conditions in the sense of the initial data in special energy norm have been found. The equivalence of the energy norm to the grid L2-norm has been proved. A priori estimates expressing the difference schemes stability in the sense of the right-hand side have been constructed.

Assessment of Models for Extracting Thermal Conductivity From Frequency Domain Thermoreflectance Experiments

2020 ◽  
Author(s):  
Siddharth Saurav ◽  
Sandip Mazumder
Keyword(s):  
Thermal Conductivity ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Frequency Domain ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Hyperbolic Heat Conduction ◽  
Phase Lag ◽  
Computational Domain ◽  
Fourier Heat Conduction

Abstract The Fourier heat conduction and the hyperbolic heat conduction equations were solved numerically to simulate a frequency-domain thermoreflectance (FDTR) experimental setup. Numerical solutions enable use of realistic boundary conditions, such as convective cooling from the various surfaces of the substrate and transducer. The equations were solved in time domain and the phase lag between the temperature at the center of the transducer and the modulated pump laser signal were computed for a modulation frequency range of 200 kHz to 200 MHz. It was found that the numerical predictions fit the experimentally measured phase lag better than analytical frequency-domain solutions of the Fourier heat equation based on Hankel transforms. The effects of boundary conditions were investigated and it was found that if the substrate (computational domain) is sufficiently large, the far-field boundary conditions have no effect on the computed phase lag. The interface conductance between the transducer and the substrate was also treated as a parameter, and was found to have some effect on the predicted thermal conductivity, but only in certain regimes. The hyperbolic heat conduction equation yielded identical results as the Fourier heat conduction equation for the particular case studied. The thermal conductivity value (best fit) for the silicon substrate considered in this study was found to be 108 W/m/K, which is slightly different from previously reported values for the same experimental data.

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