Abstract
A finite volume method for solving general time-homogeneous convection-diffusion equations with zero source term is presented. Computational efficiency of the method is improved by performing linear algebra in the tensor train format. To our knowledge this is the first time that the tensor train format and the finite volume method have been combined for this purpose.
Finite volume methods, tensors and tensor decompositions are reviewed by summarising prominent texts on each respective topic. We extend a finite volume method for convection equations to include diffusion terms and show that the method preserves integrals and positivity. The time discretisation uses an explicit Euler step that leads to a sequence of linear systems of equations defining a discrete approximation of the solution at some final time. The recurrence is stepped forwards in time by performing algebraic operations in the tensor train format. In some cases, this leads to a significant increase in computational efficiency.
We use our tensor train implementation of the finite volume method to approximate the allele frequency spectrum in three populations by solving the Wright-Fisher diffusion equation. Our method did not appear to outperform current methods for approximating the allele frequency spectrum. However, we develop some interesting and efficient tools for approximating the allele frequency spectrum if the solution to the Wright-Fisher diffusion equation is known in tensor train format.