Derive boundary conditions for holistic discretisations of Burgers' equation

TL;DR

The paper addresses deriving boundary conditions for holistic discretisations of PDEs on bounded domains, using Burgers' equation as a test case. It applies centre manifold theory with inter-element boundary conditions to produce high-order, constant-bandwidth discretisations that preserve the self-adjoint diffusion operator and account for subgrid interactions, including boundary-forcing terms and time-derivative effects. The authors demonstrate Dirichlet conditions at a grid point and Neumann conditions at a boundary midpoint, deriving explicit near-boundary discretisations with matrices that mirror the interior structure and ensure nonlinear consistency. This framework extends holistic discretisation to bounded domains and supports accurate, stable coarse-grid simulations of nonlinear spatio-temporal dynamics on general domains.

Abstract

I previously used Burgers' equation to introduce a new method of numerical discretisation of \pde{}s. The analysis is based upon centre manifold theory so we are assured that the discretisation accurately models all the processes and their subgrid scale interactions. Here I show how boundaries to the physical domain may be naturally incorporated into the numerical modelling of Burgers' equation. We investigate Neumann and Dirichlet boundary conditions. As well as modelling the nonlinear advection, the method naturally derives symmetric matrices with constant bandwidth to correspond to the self-adjoint diffusion operator. The techniques developed here may be used to accurately model the nonlinear evolution of quite general spatio-temporal dynamical systems on bounded domains.