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 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 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 Simulation of differentiated thermal processing of railway rails by compressed air

2021 ◽  
Vol 63 (11-12) ◽  
pp. 907-914
Author(s):  
V. D. Sarychev ◽  
S. G. Molotkov ◽  
V. E. Kormyshev ◽  
S. A. Nevskii ◽  
E. V. Polevoi
Keyword(s):  
Experimental Data ◽  
Thermal Conductivity ◽  
Mathematical Model ◽  
Heat Conduction ◽  
Thermal Processing ◽  
Heat Conduction Equation ◽  
Rail Steel ◽  
Rail Head ◽  
The Third ◽  
Forced Cooling

Mathematical modeling of differentiated thermal processing of railway rails with air has been carried out. At the first stage, onedimensional heat conduction problem with boundary conditions of the third kind was solved analytically and numerically. The obtained temperature distributions at the surface of the rail head and at a depth of 20 mm from the rolling surface were compared with experimental data. As a result, value of the coefficients of heat transfer and thermal conductivity of rail steel was determined. At the second stage, mathematical model of temperature distribution in a rail template was created in conditions of forced cooling and subsequent cooling under natural convection. The proposed mathematical model is based on the Navier-Stokes and convective thermal conductivity equations for the quenching medium and thermal conductivity equation for rail steel. On the rail – air boundary, condition of heat flow continuity was set. In conditions of spontaneous cooling, change in temperature field was simulated by heat conduction equation with conditions of the third kind. Analytical solution of one-dimensional heat conduction equation has shown that calculated temperature values differ from the experimental data by 10 %. When cooling duration is more than 30 s, change of pace of temperature versus time curves occurs, which is associated with change in cooling mechanisms. Results of numerical analysis confirm this assumption. Analysis of the two-dimensional model of rail cooling by the finite element method has shown that at the initial stage of cooling, surface temperature of the rail head decreases sharply both along the central axis and along the fillet. When cooling duration is over 100 s, temperature stabilizes to 307 K. In the central zones of the rail head, cooling process is slower than in the surface ones. After forced cooling is stopped, heating of the surface layers is observed, due to change in heat flow direction from the central zones to the surface of the rail head, and then cooling occurs at speeds significantly lower than at the first stage. The obtained results can be used to correct differential hardening modes.

Sign in / Sign up

Export Citation Format

Share Document