Generalized Trapezoidal Formulas for the Symmetric Heat Equation in Polar Coordinates

This paper has two objectives. We first describe one-step time integration schemes for the symmetric heat equation in polar coordinates: u t = v ( u rr +( a / r ) u r ) based on the generalized trapezoidal formulas (GTF( f ) of Chawla et al. [2]. This includes the case of cylindrical symmetry for a...

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Bibliographic Details
Published in:International journal of computer mathematics 2002, Vol.79 (6), p.729-745
Main Authors: Chawla, M.M., Al-Zanaidi, M.A., Evans, D.J.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Generalized Finite Hankel Transforms
Generalized Trapezoidal Formulas
Mathematical analysis
Mathematics
Ordinary differential equations
Polar Coordinates
Sciences and techniques of general use
Symmetric Heat Equation
Unconditional Stability
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Summary:This paper has two objectives. We first describe one-step time integration schemes for the symmetric heat equation in polar coordinates: u t = v ( u rr +( a / r ) u r ) based on the generalized trapezoidal formulas (GTF( f ) of Chawla et al. [2]. This includes the case of cylindrical symmetry for a =1 and of spherical symmetry for a =2. The obtained GTF( f ) time integration schemes are second order in time and unconditionally stable. We then introduce generalized finite Hankel transforms to obtain an analytical solution of the heat equation for all a S 1, with Dirichlet and Neumann type boundary conditions. Numerical experiments are provided to compare the accuracy and stability of the obtained GTF( f ) time integration schemes with the schemes based on the backward Euler, the classical arithmetic-mean trapezoidal formula and a third order time integration scheme.
ISSN:0020-7160
1029-0265
DOI:10.1080/00207160211292