A Fast Direct Solver for the Advection-Diffusion Equation Using Low-Rank Approximation of the Green’s Function

Abstract

We describe a new direct solution method for the advection-diffusion equation at high Reynolds number on simple bounded two-dimensional domains. The key step is to treat advection explicitly, leading to a new class of time integration schemes based on Green’s functions. As a proof of concept a first-order Euler scheme is presented. We compare the accuracy and computational cost of the new scheme to existing solution techniques. Low-rank approximation of the Green’s function is found to reduce cost without loss of accuracy. Stabilisation via numerical dissipation is required for high Reynolds number problems on coarse grids. Linear scaling of computational cost is achieved in 1D and 2D. This work is a building block for constructing fast direct solvers and preconditioners for the Navier-Stokes equations.