A fast direct solver for the advection-diffusion equation using low-rank approximation of the Green's function
This work provides a faster solver for advection-diffusion equations, which is relevant for computational scientists solving stiff hyperbolic problems at large scales.
The authors present a fast direct solver for the advection-diffusion equation that achieves O(N) complexity via low-rank approximation of the Green's function, and demonstrate it is roughly an order of magnitude faster than Matlab's backslash operator.
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.