Holistic discretisation ensures fidelity to dynamics in two spatial dimensions
This work provides a theoretically grounded discretisation framework for reaction-diffusion PDEs, but it is incremental as it extends existing holistic discretisation ideas to 2D with a specific application.
The authors develop a holistic discretisation method for reaction-diffusion PDEs in two spatial dimensions that ensures fidelity to the dynamics by systematically resolving subgrid microscale dynamics. They demonstrate the approach on the Ginzburg-Landau equation and show it can achieve arbitrarily high order accuracy, though higher-order models require a mixed numerical-algebraic approach.
Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate, macroscale closures of discrete models of the \pde. Here we explore reaction-diffusion problems in two spatial dimensions. Centre manifold theory ensures that slow manifold, holistic, discretisations exists, are quickly attractive, and are systematically approximated. Special coupling of the finite elements ensures that the resultant discretisations are consistent with the \pde to as high an order as desired. Computer algebra handles the enormous algebraic details as seen in the specific application to the Ginzburg--Landau equation. However, higher order models in 2D appear to require a mixed numerical and algebraic approach that is also developed. Being driven by the residuals of the equations, the modelling here may be straightforwardly adapted to a wide class of reaction-diffusion differential and lattice equations in multiple space dimensions.