A fast direct solver for the advection-diffusion equation using low-rank approximation of the Green's function

We present a fast direct solution method for the advection-diffusion equation in one and two dimensions with non-periodic boundaries. Computational cost is reduced to $\mathcal O(N)$ by making a low-rank approximation of the Green's function without sacrificing accuracy. Implicit treatment of the diffusion term reduces stiffness in advection-dominated problems. Results show that the solver is roughly an order of magnitude faster than a reference method, namely the Matlab backslash operator. This work motivates the use of hierarchical low-rank approximations for solution of stiff hyperbolic problems at very large scale, including those arising from high-order accurate spatial discretisations.