# Pseudospectral methods for density functional theory in bounded and unbounded domains

## TL;DR

This paper introduces a novel pseudo-spectral collocation scheme for density functional theory that efficiently computes non-local terms with high accuracy, enabling better simulations of fluid phenomena in various domains.

## Contribution

The paper presents an efficient pseudo-spectral method with specialized quadrature for DFT, reducing computational nodes needed and improving accuracy in complex fluid simulations.

## Key findings

- Accurately computes non-local terms in DFT using fewer grid points.
- Demonstrates effectiveness on two-dimensional fluid models with various interactions.
- Results satisfy statistical mechanical sum rules.

## Abstract

Classical Density Functional Theory (DFT) is a statistical-mechanical framework to analyze fluids, which accounts for nanoscale fluid inhomogeneities and non-local intermolecular interactions. DFT can be applied to a wide range of interfacial phenomena, as well as problems in adsorption, colloidal science and phase transitions in fluids. Typical DFT equations are highly non-linear, stiff and contain several convolution terms. We propose a novel, efficient pseudo-spectral collocation scheme for computing the non-local terms in real space with the help of a specialized Gauss quadrature. Due to the exponential accuracy of the quadrature and a convenient choice of collocation points near interfaces, we can use grids with a significantly lower number of nodes than most other reported methods. We demonstrate the capabilities of our numerical methodology by studying equilibrium and dynamic two-dimensional test cases with single- and multispecies hard-sphere and hard-disc particles modelled with fundamental measure theory, with and without van der Waals attractive forces, in bounded and unbounded physical domains. We show that our results satisfy statistical mechanical sum rules.