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 =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.

Keywords:

• Symmetric Heat Equation
• Polar Coordinates
• Generalized Trapezoidal Formulas
• Unconditional Stability
• Generalized Finite Hankel Transforms