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

Jonathan R. Bull

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arXiv · 1612.09273

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

Jonathan R. Bull

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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.

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Jonathan R. Bull. 2016-12-29. A fast direct solver for the advection-diffusion equation using low-rank approximation of the Green's function. https://arxiv.org/abs/1612.09273

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