The Numerical Method of Lines facilitates the instruction of unsteady heat conduction in simple solid bodies with convective surfaces

2020 ◽  
pp. 030641902091042
Author(s):  
Antonio Campo
Keyword(s):  
Heat Conduction ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Method Of Lines ◽  
Adjoint System ◽  
One Dimensional ◽  
First Order ◽  
The Method Of Lines ◽  
Unsteady Heat Conduction

The present study on engineering education addresses the Method of Lines and its variant the Numerical Method of Lines as a reliable avenue for the numerical analysis of one-dimensional unsteady heat conduction in walls, cylinders, and spheres involving surface convection interaction with a nearby fluid. The Method of Lines transforms the one-dimensional unsteady heat conduction equation in the spatial and time variables x, t into an adjoint system of first-order ordinary differential equations in the time variable t. Subsequently, the adjoint system of first-order ordinary differential equations is channeled through the Numerical Method of Lines and the powerful fourth-order Runge–Kutta algorithm. The numerical solution of the adjoint system of first-order ordinary differential equations can be carried out by heat transfer students employing appropriate routines embedded in the computer codes Maple, Mathematica, Matlab, and Polymath. For comparison, the baseline solutions used are the exact, analytical temperature distributions that are available in the heat conduction literature.

A Numerical Study of Nonaxisymmetric Stokesian Flow in a Circular Tube

1981 ◽  
Vol 48 (3) ◽  
pp. 459-464
Author(s):  
J. Strigberger ◽  
A. Plotkin
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Circular Tube ◽  
Numerical Study ◽  
Mathematical Problem ◽  
Method Of Lines ◽  
Tube Axis ◽  
First Order ◽  
The Method Of Lines ◽  
Fixed Radius

A numerical study of the nonaxisymmetric Stokesian flow of a Newtonian fluid in a rigid circular tube of fixed radius has been performed. The analysis presented here is an integral part of the problem of modeling the flow of blood near the ostia of the intercostal arteries of rabbits in order to study a possible factor in the initiation of atherosclerosis. The method of lines is used to reduce the mathematical problem to one of solving a system of first-order ordinary differential equations along lines parallel to the tube axis. Solutions are obtained analytically using matrix eigenvalue techniques for the first two Fourier components of the flow and the accuracy of the numerical method is verified by suitable comparison with the results of independent computations.

The Transversal Method of Lines (TMOL) Applied to Unsteady Conduction in Large Plates, Long Cylinders and Spheres With Prescribed Surface Heat Flux

2011 ◽  
Author(s):  
Antonio Campo ◽  
Ramin Soujoudi ◽  
Adelina Davis
Keyword(s):  
Heat Flux ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Surface Heat Flux ◽  
Time Derivative ◽  
Method Of Lines ◽  
Surface Heat ◽  
First Order ◽  
Temperature Distributions ◽  
Cylinders And Spheres

The Transversal Method Of Lines (TMOL) or Rothe method is a general technique for solving parabolic partial differential equations that uses a two-point backward finite-difference formulation for the time derivative and differential spatial derivatives. This hybrid approach leads to transformed ordinary differential equations where the spatial coordinate is the independent variable and the time appears as an embedded parameter. The transformed ordinary differential equations may have constant or variable coefficients depending on the coordinate system and are first-order accurate. In this work, TMOL is applied to the 1-D heat equation for large plates, long cylinders and spheres with constant thermophysical properties, uniform initial temperature and prescribed surface heat flux. The analytic solutions of the adjoint heat equations are performed with the symbolic Maple software. It is demonstrated that the approximate semi-analytic TMOL temperature distributions for the three simple bodies are much better than first-order accurate. This signifies that TMOL temperature distributions are not only valid for short times, but they are valid for the entire heating period involving short, moderate and long times.

scholarly journals Solving Nonlinear Thermal Problems of Friction by Using Method of Lines

2015 ◽  
Vol 9 (1) ◽  
pp. 33-37
Author(s):  
Ewa Och
Keyword(s):  
Heat Conduction ◽  
Differential Equations ◽  
Heat Conduction Problem ◽  
Method Of Lines ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Partial Linearization ◽  
The Method Of Lines ◽  
Third Stage ◽  
Independent Variable

Abstract One-dimensional heat conduction problem of friction for two bodies (half spaces) made of thermosensitive materials was considered. Solution to the nonlinear boundary-value heat conduction problem was obtained in three stages. At the first stage a partial linearization of the problem was performed by using Kirchhoff transform. Next, the obtained boundary-values problem by using the method of lines was brought to a system of nonlinear ordinary differential equations, relatively to Kirchhoff’s function values in the nodes of the grid on the spatial variable, where time is an independent variable. At the third stage, by using the Adams's method from DIFSUB package, a numerical solution was found to the above-mentioned differential equations. A comparative analysis was conducted (Och, 2014) using the results obtained with the proposed method and the method of successive approximations.

scholarly journals A New Method for Numerical Integration of Higher-Order Ordinary Differential Equations Without Losing the Periodic Responses

2021 ◽  
Vol 7 ◽  
Author(s):  
John T. Katsikadelis
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Variable Coefficients ◽  
Space Representation ◽  
Parabolic Differential Equations ◽  
Linear Ordinary Differential Equations ◽  
First Order ◽  
Symmetrical Matrices ◽  
The Stability

A new numerical method is presented for the solution of initial value problems described by systems of N linear ordinary differential equations (ODEs). Using the state-space representation, a differential equation of order n > 1 is transformed into a system of L = n×N first-order equations, thus the numerical method developed recently by Katsikadelis for first-order parabolic differential equations can be applied. The stability condition of the numerical scheme is derived and is investigated using several well-corroborated examples, which demonstrate also its convergence and accuracy. The method is simply implemented. It is accurate and has no numerical damping. The stability does not require symmetrical and positive definite coefficient matrices. This advantage is important because the scheme can find the solution of differential equations resulting from methods in which the space discretization does not result in symmetrical matrices, for example, the boundary element method. It captures the periodic behavior of the solution, where many of the standard numerical methods may fail or are highly inaccurate. The present method also solves equations having variable coefficients as well as non-linear ones. It performs well when motions of long duration are considered, and it can be employed for the integration of stiff differential equations as well as equations exhibiting softening where widely used methods may not be effective. The presented examples demonstrate the efficiency and accuracy of the method.

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