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

TL;DR

This paper introduces a fast, low-rank approximation-based direct solver for the advection-diffusion equation that significantly reduces computational cost while maintaining accuracy, especially useful for large-scale hyperbolic problems.

Contribution

The paper develops a novel low-rank approximation method for the Green's function, enabling an $ ext{O}(N)$ direct solver for advection-diffusion equations in 1D and 2D.

Findings

Solver is roughly ten times faster than Matlab backslash.

Low-rank approximation maintains accuracy in large-scale problems.

Method effectively handles stiff hyperbolic problems with high-order discretizations.

Abstract

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.