scholarly journals Discrete Symmetry Group Approach for Numerical Solution of the Heat Equation

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 359
Author(s):  
Khudija Bibi ◽  
Tooba Feroze
Keyword(s):  
Finite Difference ◽  
Heat Equation ◽  
Difference Scheme ◽  
Exact Solutions ◽  
Finite Difference Scheme ◽  
Symmetry Transformation ◽  
Discrete Symmetry ◽  
Group Approach ◽  
Comparison Of The Results ◽  
Crank Nicolson

In this article, an invariantized finite difference scheme to find the solution of the heat equation, is developed. The scheme is based on a discrete symmetry transformation. A comparison of the results obtained by the proposed scheme and the Crank Nicolson method is carried out with reference to the exact solutions. It is found that the proposed invariantized scheme for the heat equation improves the efficiency and accuracy of the existing Crank Nicolson method.

Numerical identification of temperature dependent thermal conductivity using least squares method

2019 ◽  
Vol 30 (6) ◽  
pp. 3083-3099
Author(s):  
Anna Ivanova ◽  
Stanislaw Migorski ◽  
Rafal Wyczolkowski ◽  
Dmitry Ivanov
Keyword(s):  
Thermal Conductivity ◽  
Finite Difference ◽  
Heat Equation ◽  
Difference Scheme ◽  
Least Squares ◽  
Finite Difference Scheme ◽  
Nonlinear Heat Equation ◽  
Temperature Dependent ◽  
Content Type ◽  
Temperature Dependent Thermal Conductivity

Purpose This paper aims to considered the problem of identification of temperature-dependent thermal conductivity in the nonstationary, nonlinear heat equation. To describe the heat transfer in the furnace charge occupied by a homogeneous porous material, the heat equation is formulated. The inverse problem consists in finding the heat conductivity parameter, which depends on the temperature, from the measurements of the temperature in fixed points of the material. Design/methodology/approach A numerical method based on the finite-difference scheme and the least squares approach for numerical solution of the direct and inverse problems has been recently developed. Findings The influence of different numerical scheme parameters on the accuracy of the identified conductivity coefficient is studied. The results of the experiment carried out on real measurements are presented. Their results confirm the ones obtained earlier by using other methods. Originality/value Novelty is in a new, easy way to identify thermal conductivity by known temperature measurements. This method is based on special finite-difference scheme, which gives a resolvable system of algebraic equations. The results sensitivity on changes in the method parameters was studies. The algorithms of identification in the case of a purely mathematical experiment and in the case of real measurements, their differences and the practical details are presented.

A STUDY OF THE SHORT WAVE COMPONENTS IN COMPUTATIONAL ACOUSTICS

1993 ◽  
Vol 01 (01) ◽  
pp. 1-30 ◽  
Author(s):  
CHRISTOPHER K. W. TAM ◽  
JAY C. WEBB ◽  
ZHONG DONG
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Difference Scheme ◽  
Exact Solutions ◽  
Euler Equations ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Long Waves ◽  
Short Waves

It is shown by using a Dispersion-Relation-Preserving [Formula: see text] finite difference scheme that it is feasible to perform direct numerical simulation of acoustic wave propagation problems. The finite difference equations of the [Formula: see text] scheme have essentially the same Fourier-Laplace transforms and hence dispersion relations as the original linearized Euler equations over a broad range of wavenumbers (here referred to as long waves). Thus it is guaranteed that the acoustic waves, the entropy and the vorticity waves computed by the [Formula: see text] scheme are good approximations of those of the exact solutions of Euler equations as long as the wavenumbers are in the long wave range. Computed waves with higher wavenumber, or the short waves, generally have totally different propagation characteristics. There are no counterparts of such waves in the exact solutions. The short waves of a computation scheme are, therefore, contaminants of the numerical solutions. The characteristics of these short waves are analyzed here by group velocity consideration and standard dispersive wave theory. Numerical results of direct simulations of these waves are reported. These waves can be generated by discontinuous initial conditions. To purge the short waves so as to improve the quality of the numerical solution, it is suggested that artificial selective damping terms be added to the finite difference scheme. It is shown how the coefficients of such damping terms may be chosen so that damping is confined primarily to the high wavenumber range. This is important for then only the short waves are damped leaving the long waves basically unaffected. The effectiveness of the artificial selective damping terms is demonstrated by direct numerical simulations involving acoustic wave pulses with discontinuous wave fronts.

scholarly journals An Explicit Wavelet-Based Finite Difference Scheme for Solving One-Dimensional Heat Equation

Teknoin ◽  
2005 ◽  
Vol 10 (1) ◽  
Author(s):  
Mahmmod Aziz Muhammed ◽  
Adhi Susanto ◽  
F. Soesianto F. Soesianto ◽  
Soetrisno Soetrisno
Keyword(s):  
Finite Difference ◽  
Heat Equation ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
One Dimensional
Sign in / Sign up

Export Citation Format

Share Document