A series method for the eigenvalues of the advection diffusion equation

Abstract

In steady hill-slope seepage problems, the advection diffusion equation can be conformally transformed to a semi-regular solution domain using (?,?) coordinates. Uniform flow in the (?,?) domain reduces the advection diffusion equation to a simpler version with constant coefficients. The solutions depend on finding the eigenvalues (or natural frequencies ) of an (elliptic) Helmholtz equation. In the absence of natural frequencies, this equation can be solved for nonzero boundary conditions using analytic series methods. In this paper, we present a pseudo-spectral approach to solve for the series coefficients. At the natural frequencies, the determinant of the coefficient matrix becomes zero, thus marking the natural frequencies. We present some preliminary results and identify natural frequencies for a set of test problems.