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

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  • Bunder, J.E.
  • Roberts, A.J.

Abstract

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.

Suggested Citation

  • Bunder, J.E. & Roberts, A.J., 2017. "Resolution of subgrid microscale interactions enhances the discretisation of nonautonomous partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 304(C), pages 164-179.
    • Handle: RePEc:eee:apmaco:v:304:y:2017:i:c:p:164-179
    DOI: 10.1016/j.amc.2017.01.056
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    References listed on IDEAS

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    1. Xu, Chao & Roberts, A.J., 1996. "On the low-dimensional modelling of Stratonovich stochastic differential equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 225(1), pages 62-80.
    2. Roberts, A.J., 2008. "Normal form transforms separate slow and fast modes in stochastic dynamical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(1), pages 12-38.
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