Resolution of subgrid microscale interactions enhances the discretisation of nonautonomous partial differential equations

• Published in: Applied mathematics and computation 2017-07, Vol.304, p.164-179
• Main Authors: Bunder, J.E., Roberts, A.J.
• Format: Article
• Language: English
• Subjects: Closure; Coarse graining; Multiscale modelling; Nonautonomous slow manifold; Nonlinear nonautonomous PDEs; Spatial discretisation
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2017.01.056

•We consider a general method for the macroscale modelling of nonlinear, nonautomomous partial differential equations or difference equations.•Our method is able to resolve the complex fluctuations which arise in nonliner, nonautonomous systems.•These fluctuations are not accounted for by popular multiscale averaging procedures and homogenization.•Our method is illustrated with the example of a forced Burgers’ equation. Coarse grained, macroscale, spatial discretisations of nonlinear nonautonomous partial differential/difference equations are given novel support by centre manifold theory. Dividing the physical domain into overlapping macroscale elements empowers the approach to resolve significant subgrid microscale structures and interactions between neighbouring elements. The crucial aspect of this approach is that centre manifold theory organises the resolution of the detailed subgrid microscale structure interacting via the nonlinear dynamics within and between neighbouring elements. The techniques and theory developed here may be applied to soundly discretise on a macroscale many dissipative nonautonomous partial differential/difference equations, such as the forced Burgers’ equation, adopted here as an illustrative example.