Generalized Trapezoidal Formulas for the Symmetric Heat Equation in Polar Coordinates

Abstract

This paper has two objectives. We first describe one-step time integration schemes for the symmetric heat equation in polar coordinates: u t ¼ nðu rr þ ða=rÞu r Þ based on the generalized trapezoidal formulas (GTF(a)) of Chawla et al. [2]. This includes the case of cylindrical symmetry for a ¼ 1 and of spherical symmetry for a ¼ 2. The obtained GTF(a) 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 1, with Dirichlet and Neumann type boundary conditions. Numerical experiments are provided to compare the accuracy and stability of the obtained GTF(a) time integration schemes with the schemes based on the backward Euler, the classical arithmetic-mean trapezoidal formula and a third order time integration scheme.