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

Abstract

Coarse grained, macroscale, spatial discretisations of nonlinear nonautonomous partial dif- ferential/difference equations are given novel support by centre manifold theory. Dividing the physical domain into overlapping macroscale elements empowers the approach to re- solve significant subgrid microscale structures and interactions between neighbouring ele- ments. The crucial aspect of this approach is that centre manifold theory organises the res- olution 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.