Create accurate numerical models of complex spatio-temporal dynamical systems with holistic discretisation

Abstract

This dissertation focuses on the further development of creating accurate numerical models of complex dynamical systems using the holistic discretisation technique [Roberts, Appl. Num. Model., 37:371-396, 2001]. I extend the application from second to fourth order systems and from only one spatial dimension in all previous work to two dimensions (2D). We see that the holistic technique provides useful and accurate numerical discretisations on coarse grids. We explore techniques to model the evolution of spatial patterns governed by pdes such as the Kuramoto-Sivashinsky equation and the real-valued Ginzburg-Landau equation. We aim towards the simulation of fluid flow and convection in three spatial dimensions. I show that significant steps have been taken in this dissertation towards achieving this aim. Holistic discretisation is based upon centre manifold theory [Carr, Applications of centre manifold theory, 1981] so we are assured that the numerical discretisation accurately models the dynamical system and may be constructed systematically. To apply centre manifold theory the domain is divided into elements and using a homotopy in the coupling parameter, subgrid scale fields are constructed consisting of actual solutions of the governing partial differential equation(pde). These subgrid scale fields interact through the introduction of artificial internal boundary conditions. View the centre manifold (macroscale) as the union of all states of the collection of subgrid fields (microscale) over the physical domain. Here we explore how to extend holistic discretisation to the fourth order Kuramoto-Sivashinsky pde. I show that the holistic models give impressive accuracy for reproducing the steady states and time dependent phenomena of the Kuramoto-Sivashinsky equation on coarse grids. The holistic method based on local dynamics compares favourably to the global methods of approximate inertial manifolds. The excellent performance of the holistic models shown here is strong evidence in support of the holistic discretisation technique. For shear dispersion in a 2D channel a one-dimensional numerical approximation is generated directly from the two-dimensional advection-diffusion dynamics. We find that a low order holistic model contains the shear dispersion term of the Taylor model [Taylor, IMA J. Appl. Math., 225:473-477, 1954]. This new approach does not require the assumption of large x scales, formerly absolutely crucial in deriving the Taylor model. I develop holistic discretisation for two spatial dimensions by applying the technique to the real-valued Ginzburg-Landau equation as a representative example of second order pdes. The techniques will apply quite generally to second order reaction-diffusion equations in 2D. This is the first study implementing holistic discretisation in more than one spatial dimension. The previous applications of holistic discretisation have developed algebraic forms of the subgrid field and its evolution. I develop an algorithm for numerical construction of the subgrid field and its evolution for 1D and 2D pdes and explore various alternatives. This new development greatly extends the class of problems that may be discretised by the holistic technique. This is a vital step for the application of the holistic technique to higher spatial dimensions and towards discretising the Navier-Stokes equations.