scholarly journals New explicit algorithm based on the asymmetric hopscotch structure to solve the heat conduction equation

2021 ◽  
Vol 11 (5) ◽  
pp. 233-244
Author(s):  
Issa Omle
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Stage Structure ◽  
Two Dimensional ◽  
Random Parameters ◽  
Time Step ◽  
Acceptable Accuracy ◽  
New Methods ◽  
Large Systems ◽  
Short Time

In this paper we will consider a new four-stage structure inspired by the well-known odd-even hopscotch method to construct new schemes for the numerical solution of the two-dimensional heat or diffusion equation. In this structure the first and the last time step are halved stage and therefore the time steps are shifted compared to each other for odd and even cells. We insert 10 concrete formulas into this structure to obtain 104 different combinations. First we test all of these in case of small systems with random parameters, and then examine the competitiveness of the best algorithms by testing them in case of large systems against popular solvers. We select the top 5 combinations, and demonstrate that these new methods are indeed effective if the goal is to produce results with acceptable accuracy in very short time.

scholarly journals New Stable, Explicit, Shifted-Hopscotch Algorithms for the Heat Equation

2021 ◽  
Vol 26 (3) ◽  
pp. 61
Author(s):  
Ádám Nagy ◽  
Mahmoud Saleh ◽  
Issa Omle ◽  
Humam Kareem ◽  
Endre Kovács
Keyword(s):  
Heat Equation ◽  
Numerical Algorithms ◽  
Nonlinear Problems ◽  
Half Time ◽  
Random Parameters ◽  
Time Step ◽  
Order Accuracy ◽  
New Methods ◽  
Large Systems ◽  
Short Time

Our goal was to find more effective numerical algorithms to solve the heat or diffusion equation. We created new five-stage algorithms by shifting the time of the odd cells in the well-known odd-even hopscotch algorithm by a half time step and applied different formulas in different stages. First, we tested 105 = 100,000 different algorithm combinations in case of small systems with random parameters, and then examined the competitiveness of the best algorithms by testing them in case of large systems against popular solvers. These tests helped us find the top five combinations, and showed that these new methods are, indeed, effective since quite accurate and reliable results were obtained in a very short time. After this, we verified these five methods by reproducing a recently found non-conventional analytical solution of the heat equation, then we demonstrated that the methods worked for nonlinear problems by solving Fisher’s equation. We analytically proved that the methods had second-order accuracy, and also showed that one of the five methods was positivity preserving and the others also had good stability properties.

Enhanced Explicit Scheme to Solve Transient Heat Conduction Problem

2007 ◽  
Vol 2 (1) ◽  
Author(s):  
Ganesh Hegde ◽  
Madhu Gattumane
Keyword(s):  
Heat Conduction ◽  
Correction Factor ◽  
Truncation Error ◽  
Transient Heat Conduction ◽  
Explicit Scheme ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Time Step ◽  
One Dimensional ◽  
Partial Derivatives

Improvement in accuracy without sacrificing stability and convergence of the solution to unsteady diffusion heat transfer problems by computational method of enhanced explicit scheme (EES), has been achieved and demonstrated, through transient one dimensional and two dimensional heat conduction. The truncation error induced in the explicit scheme using finite difference technique is eliminated by optimization of partial derivatives in the Taylor series expansion, by application of interface theory developed by the authors. This theory, in its simple terms gives the optimum values to the decision vectors in a redundant linear equation. The time derivatives and the spatial partial derivatives in the transient heat conduction, take the values depending on the time step chosen and grid size assumed. The time correction factor and the space correction factor defined by step sizes govern the accuracy, stability and convergence of EES. The comparison of the results of EES with analytical results, show decreased error as compared to the result of explicit scheme. The paper has an objective of reducing error in the explicit scheme by elimination of truncation error introduced by neglecting the higher order terms in the expansion of the governing function. As the pilot examples of the exercise, the implementation is aimed at solving one-dimensional and two-dimensional problems of transient heat conduction and compared with the results cited in the referred literature.

scholarly journals A Revised Tikhonov Regularization Method for a Cauchy Problem of Two-Dimensional Heat Conduction Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Songshu Liu ◽  
Lixin Feng
Keyword(s):  
Cauchy Problem ◽  
Heat Conduction ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Tikhonov Regularization Method ◽  
Two Dimensional ◽  
Convergence Estimate ◽  
Ill Posed

In this paper we investigate a Cauchy problem of two-dimensional (2D) heat conduction equation, which determines the internal surface temperature distribution from measured data at the fixed location. In general, this problem is ill-posed in the sense of Hadamard. We propose a revised Tikhonov regularization method to deal with this ill-posed problem and obtain the convergence estimate between the approximate solution and the exact one by choosing a suitable regularization parameter. A numerical example shows that the proposed method works well.

scholarly journals Solving of Two-Dimensional Unsteady Inverse Heat Conduction Problems Based on Boundary Element Method and Sequential Function Specification Method

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Shoubin Wang ◽  
Yuanzheng Deng ◽  
Xiaogang Sun
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Measuring Error ◽  
Two Dimensional ◽  
Future Time ◽  
Time Step ◽  
Measuring Point ◽  
Specification Method ◽  
Sequential Function Specification ◽  
Function Specification Method

The boundary element method (BEM) and sequential function specification method (SFSM) are used to research the inverse problem of boundary heat flux identification in the two-dimensional heat conduction system. The future time step in the SFSM is optimized by introducing the residual error principles to get the more accurate inversion results. For the forward problems, the BEM is used to calculate the required temperature value of discrete point; for the inverse problems, the impacts of different future time steps, measuring point position, and measuring error on the inversion results are discussed. Furthermore, the comparison is made for the optimal future time step obtained by introducing the residual error principle and the inherent future time step. The example analysis shows that the method proposed still has higher accuracy when the measuring error exists or the measuring point position is far away from the boundary heat flux.

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