An angular spectral method for solution of the heat equation in spheroidal geometries

Abstract

A spectral numerical method is presented for solving the heat equation in oblate or prolate spheroids. Cartesian coordinates are scaled to transform the spheroidal geometry into a spherical geometry. The diffusion term in the transformed equation is anisotropic, being enhanced in the polar directions. The transformed equation is discretised in angle using a truncated spherical harmonic expansion of the temperature in transformed spherical polar coordinates. The anisotropic diffusion term is reduced to block tridiagonal form using recurrence relations for spherical harmonics. The radial coordinate is discretised using finite differences in scaled radius but other radial schemes are possible. Without heat sources and with a homogeneous Dirichlet boundary condition the problem reduces to an eigenproblem for the decay rate. The results are compared to the separated variables solution, which employs oblate or prolate spheroidal wave functions. The method directly extends to other scalar problems in spheroidal geometries, which have the highest derivatives in Laplacian form, including passive advection-diffusion of a scalar. The method may be extended, with difficulty, to problems with vector diffusion or ellipsoidal geometries.