scholarly journals Construction and investigation of new numerical algorithms for the heat equation : Part 2

2020 ◽  
Vol 10 (4) ◽  
pp. 339-348
Author(s):  
Mahmoud Saleh ◽  
Ádám Nagy ◽  
Endre Kovács
Keyword(s):  
Heat Conduction ◽  
Numerical Methods ◽  
Heat Conduction Equation ◽  
Space Dimension ◽  
Numerical Test ◽  
Numerical Algorithms ◽  
Heat Conducting ◽  
Heat Sources ◽  
Test Results ◽  
New Algorithms

This paper is the second part of a paper-series in which we create and examine new numerical methods for solving the heat conduction equation. Now we present numerical test results of the new algorithms which have been constructed using the known, but non-conventional UPFD and odd-even hopscotch methods in Part 1. Here all studied systems have one space dimension and the physical properties of the heat conducting media are uniform. We also examine different possibilities of treating heat sources.

scholarly journals Construction and investigation of new numerical algorithms for the heat equation : Part 3

2020 ◽  
Vol 10 (4) ◽  
pp. 349-360
Author(s):  
Mahmoud Saleh ◽  
Ádám Nagy ◽  
Endre Kovács
Keyword(s):  
Heat Conduction ◽  
Numerical Methods ◽  
Heat Conduction Equation ◽  
Space Dimension ◽  
Numerical Test ◽  
Numerical Algorithms ◽  
Test Results ◽  
The Third ◽  
Three Space ◽  
New Algorithms

This paper is the third part of a paper-series in which we create and examine new numerical methods for solving the heat conduction equation. Now we present additional numerical test results of the new algorithms which were constructed using the known, but non-conventional UPFD and odd-even hopscotch methods in Part 1. In Part 2 these methods were tested in one space dimension, while in this part of the series, we present numerical case studies for two and three space dimensions, as well as for inhomogeneous media.

scholarly journals Fractional Heat Conduction Models and Thermal Diffusivity Determination

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Monika Žecová ◽  
Ján Terpák
Keyword(s):  
Heat Conduction ◽  
Thermal Diffusivity ◽  
Numerical Methods ◽  
Fractional Order ◽  
Experimental Verification ◽  
Heat Conduction Equation ◽  
Historical Overview ◽  
One Dimensional ◽  
The One ◽  
Fractional Heat Conduction

The contribution deals with the fractional heat conduction models and their use for determining thermal diffusivity. A brief historical overview of the authors who have dealt with the heat conduction equation is described in the introduction of the paper. The one-dimensional heat conduction models with using integer- and fractional-order derivatives are listed. Analytical and numerical methods of solution of the heat conduction models with using integer- and fractional-order derivatives are described. Individual methods have been implemented in MATLAB and the examples of simulations are listed. The proposal and experimental verification of the methods for determining thermal diffusivity using half-order derivative of temperature by time are listed at the conclusion of the paper.

scholarly journals Probabilistic numerics and uncertainty in computations

2015 ◽  
Vol 471 (2179) ◽  
pp. 20150142 ◽  
Author(s):  
Philipp Hennig ◽  
Michael A. Osborne ◽  
Mark Girolami
Keyword(s):  
Numerical Methods ◽  
Numerical Algorithms ◽  
Climate Science ◽  
Complex Data ◽  
Open Problems ◽  
Error Sources ◽  
Contemporary Science ◽  
Empirical Performance ◽  
And Control ◽  
New Algorithms

We deliver a call to arms for probabilistic numerical methods : algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation with limited time or hardware, are important for much contemporary science and industry. Within applications such as climate science and astrophysics, the need to make decisions on the basis of computations with large and complex data have led to a renewed focus on the management of numerical uncertainty. We describe how several seminal classic numerical methods can be interpreted naturally as probabilistic inference. We then show that the probabilistic view suggests new algorithms that can flexibly be adapted to suit application specifics, while delivering improved empirical performance. We provide concrete illustrations of the benefits of probabilistic numeric algorithms on real scientific problems from astrometry and astronomical imaging, while highlighting open problems with these new algorithms. Finally, we describe how probabilistic numerical methods provide a coherent framework for identifying the uncertainty in calculations performed with a combination of numerical algorithms (e.g. both numerical optimizers and differential equation solvers), potentially allowing the diagnosis (and control) of error sources in computations.

scholarly journals NONSTATIONARY HEAT CONDUCTION IN MULTILAYER GLAZING SUBJECTED TO DISTRIBUTED HEAT SOURCES

2020 ◽  
Vol 10 (2) ◽  
pp. 28-31
Author(s):  
Natalia Smetankina ◽  
Oleksii Postnyi
Keyword(s):  
Heat Conduction ◽  
Laplace Transformation ◽  
Heat Conduction Equation ◽  
Constant Thickness ◽  
Heat Sources ◽  
Nonstationary Heat Conduction ◽  
Multilayer Plate ◽  
Expansion Theorem ◽  
Thermal Fields ◽  
Transformation Series

A method for calculation of nonstationary thermal fields in a multilayer glazing of vehicles under the effect of impulse film heat sources is offered. The glazing is considered as a rectangular multilayer plate made up of isotropic layers with constant thickness. Film heat sources are arranged on layers' interfaces. The heat conduction equation is solved using the Laplace transformation, series expansion and the second expansion theorem. The method offered can be used for designing a safe multilayer glazing under operational and emergency thermal and force loading in vehicles.

Error Estimates for the Finite Difference Solution of the Heat Conduction Equation: Consideration of Boundary Conditions and Heat Sources

2013 ◽  
Vol 336 ◽  
pp. 195-207
Author(s):  
Mohammad Mahdi Davoudi ◽  
Andreas Öchsner
Keyword(s):  
Heat Conduction ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Heat Conduction Equation ◽  
Exact Result ◽  
Numerical Approach ◽  
Conduction Equation ◽  
Heat Sources ◽  
Numerical Implementation ◽  
Dependent Sources

This contribution investigates the numerical solution of the steady-state heat conduction equation. The finite difference method is applied to simple formulations of heat sources where still analytical solutions can be derived. Thus, the results of the numerical approach can be related to the exact solutions and conclusions on the accuracy obtained. In addition, the numerical implementation of different forms of boundary conditions, i.e. temperature and flux condition, is compared to the exact solution. It is found that the numerical implementation of coordinate dependent sources gives the exact result while temperature dependent sources are only approximately represented. Furthermore, the implementation of the mentioned boundary conditions gives the same results as the analytical reference solution.

Peridynamics for Heat Conduction

2020 ◽  
Author(s):  
Yozo Mikata
Keyword(s):  
Heat Conduction ◽  
Constitutive Equation ◽  
Heat Equation ◽  
Governing Equation ◽  
Heat Conduction Equation ◽  
Transient Heat Conduction ◽  
Anisotropic Materials ◽  
Time Dependent ◽  
Heat Sources ◽  
Exact Analytical Solutions

Abstract Peridynamics for transient heat conduction problems in general anisotropic materials is developed. In order to develop a new peridynamic governing equation for heat conduction problems, the microconductivity (or microdiffusivity), which contains equivalent information as the constitutive equation for classical heat conduction, is determined by directly requiring the resulting peridynamic equation to converge to a classical heat conduction equation for anisotropic materials as the generalized material horizon approaches 0. Therefore, the convergence proof is built into the theory from the perspective of the governing equation. For the application of the newly obtained peridynamic governing equation, a time-dependent 3D peridynamic heat equation is analytically solved with two types of heat sources, and the results are discussed. These are believed to be the first exact analytical solutions for peridynamic heat conduction.

Meshless Local B-Spline Collocation Method for Two-Dimensional Heat Conduction Problems With Nonhomogenous and Time-Dependent Heat Sources

2017 ◽  
Vol 139 (7) ◽  
Author(s):  
Mas Irfan P. Hidayat ◽  
Bambang Ariwahjoedi ◽  
Setyamartana Parman ◽  
T. V. V. L. N. Rao
Keyword(s):  
Heat Conduction ◽  
Numerical Methods ◽  
Time Integration ◽  
Spline Approximation ◽  
Method Comparison ◽  
Field Variable ◽  
Time Dependent ◽  
Heat Sources ◽  
Spline Collocation ◽  
B Spline

This paper presents a new approach of meshless local B-spline based finite difference (FD) method for transient 2D heat conduction problems with nonhomogenous and time-dependent heat sources. In this method, any governing equations are discretized by B-spline approximation which is implemented as a generalized FD technique using local B-spline collocation scheme. The key aspect of the method is that any derivative is stated as neighboring nodal values based on B-spline interpolants. The set of neighboring nodes is allowed to be randomly distributed. This allows enhanced flexibility to be obtained in the simulation. The method is truly meshless as no mesh connectivity is required for field variable approximation or integration. Galerkin implicit scheme is employed for time integration. Several transient 2D heat conduction problems with nonuniform heat sources in arbitrary complex geometries are examined to show the efficacy of the method. Comparison of the simulation results with solutions from other numerical methods in the literature is given. Good agreement with reference numerical methods is obtained. The method is shown to be simple and accurate for the time-dependent problems.

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