A Direct Solver for the Advection-diffusion Equation Using Green’s Functions and Low-rank Approximation

A new direct solution method for the advection-diffusion equation is presented. By employing a semi-implicit time discretisation, the equation is rewritten as a heat equation with source terms. The solution is obtained by discretely approximating the integral convolution of the associated Green’s function with advective source terms. The heat equation has an exponentially decaying Green’s function, allowing for a reduction of effort via low-rank matrix approximation. Simple low-rank approximations of the Green’s function matrix are investigated as a precursor to using the Fast Multipole Method in higher dimensions. Results show that fast, stable and accurate computations can be achieved by this method. Low-rank approximation saves computational time at the expense of some accuracy. The new method is a template for developing fast, scalable preconditioners for advection-dominated problems including the unsteady Navier-Stokes equations.