Smooth subgrid fields underpin rigorous closure in spatial discretisation of reaction-advection-diffusion PDEs

TL;DR

This paper introduces a holistic discretisation method that constructs a natural subgrid scale field for reaction-advection-diffusion PDEs, providing a systematic way to achieve exact closure and improved macroscale discretisation.

Contribution

It develops a new proof for the existence of exact closure in reaction-advection-diffusion PDEs and introduces a systematic approximation approach based on subgrid scale fields.

Findings

The method guarantees a natural cubic spline approximation of the field.

The discretisation maintains self-adjointness of the diffusion operator.

Demonstrated on Burgers' PDE with favorable stability properties.

Abstract

Finite difference/element/volume methods of discretising PDEs impose a subgrid scale interpolation on the dynamics. In contrast, the holistic discretisation approach developed herein constructs a natural subgrid scale field adapted to the whole system out-of-equilibrium dynamics. Consequently, the macroscale discretisation is fully informed by the underlying microscale dynamics. We establish a new proof that in principle there exists an exact closure of the dynamics of a general class of reaction-advection-diffusion PDEs, and show how our approach constructs new systematic approximations to the in-principle closure starting from a simple, piecewise-linear, continuous approximation. Under inter-element coupling conditions that guarantee continuity of several field properties, the holistic discretisation possesses desirable properties such as a natural cubic spline first-order approximation to the field, and the self-adjointness of the diffusion operator under periodic, Dirichlet and Neumann macroscale boundary conditions. As a concrete example, we demonstrate the holistic discretisation procedure on the well-known Burgers' PDE, and compare the theoretical and numerical stability of the resulting discretisation to other approximations. The approach developed here promises to be able to systematically construct automatically good, macroscale discretisations to a wide range of PDEs, including wave PDEs.