An implicit-iterative solution of the heat conduction equation with a radiation boundary condition

1977 ◽  
Vol 11 (10) ◽  
pp. 1605-1619 ◽  
Author(s):  
S. D. Williams ◽  
Donald M. Curry
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Iterative Solution ◽  
Conduction Equation ◽  
Radiation Boundary Condition ◽  
Radiation Boundary

scholarly journals Determining An Unknown Boundary Condition by An Iteration Method

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 409
Author(s):  
Dejian Huang ◽  
Yanqing Li ◽  
Donghe Pei
Keyword(s):  
Boundary Condition ◽  
Exact Solution ◽  
Heat Conduction ◽  
Iteration Method ◽  
Heat Conduction Equation ◽  
Heat Conduction Problem ◽  
Variational Iteration Method ◽  
Conduction Equation ◽  
Unknown Boundary ◽  
Initial Condition

This paper investigates the boundary value in the heat conduction problem by a variational iteration method. Applying the iteration method, a sequence of convergent functions is constructed, the limit approximates the exact solution of the heat conduction equation in a few iterations using only the initial condition. This method does not require discretization of the variables. Numerical results show that this method is quite simple and straightforward for models that are currently under research.

Reconstruction of the Robin boundary condition and order of derivative in time fractional heat conduction equation

2018 ◽  
Vol 13 (1) ◽  
pp. 5 ◽  
Author(s):  
Rafał Brociek ◽  
Damian Słota
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Fractional Order ◽  
Heat Conduction Equation ◽  
Robin Boundary Condition ◽  
Conduction Equation ◽  
Additional Information ◽  
Bee Colony ◽  
Robin Boundary ◽  
Fractional Heat Conduction

This paper describes an algorithm for reconstruction the boundary condition and order of derivative for the heat conduction equation of fractional order. This fractional order derivative was applied to time variable and was defined as the Caputo derivative. The heat transfer coefficient, occurring in the boundary condition of the third kind, was reconstructed. Additional information for the considered inverse problem is given by the temperature measurements at selected points of the domain. The direct problem was solved by using the implicit finite difference method. To minimize functional defining the error of approximate solution an Artificial Bee Colony (ABC) algorithm and Nelder-Mead method were used. In order to stabilize the procedure the Tikhonov regularization was applied. The paper presents examples to illustrate the accuracy and stability of the presented algorithm.

Iterative solution of 3D acoustic wave equation with perfectly matched layer boundary condition and multigrid preconditioner

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. T133-T140 ◽  
Author(s):  
Guangdong Pan ◽  
Aria Abubakar
Keyword(s):  
Boundary Condition ◽  
Acoustic Wave ◽  
Frequency Domain ◽  
Iterative Solution ◽  
Perfectly Matched Layer ◽  
Computational Time ◽  
Radiation Boundary Condition ◽  
Low Frequencies ◽  
Speed Up ◽  
Radiation Boundary

We tested a biconjugate gradient stabilized (BiCGSTAB) solver using a multigrid-based preconditioner for solving the acoustic wave (Helmholtz) equation in the frequency domain. The perfectly matched layer (PML) method was used as the radiation boundary condition (RBC). The equation was discretized using either a second- or fourth-order finite-difference (FD) scheme. The convergence of an iterative solver depended strongly on the RBC used because the spectrum of the discretized equation also depends on it. We used a geometric multigrid approach to construct a preconditioner for our FD frequency-domain (FDFD) forward solver equipped with the PML boundary condition. For efficiency, this preconditioner was only constructed using a second-order FD scheme with negligible attenuation inside the PML domain. The preconditioner was used for accelerating the convergence rate of the FDFD forward solver for cases when the discretization grids were oversampled (i.e., when the number of discretization points per minimum wavelength was greater than 10). The number of multigrid levels was also chosen adaptively depending on the number of discretization grids. We found that the multigrid preconditioner can speed up the total computational time of the BiCGSTAB solver for oversampled cases or at low frequencies. We also observed that the BiCGSTAB solver using an accurate PML boundary condition converged for realistic SEG benchmark models at high frequencies.

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