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Create accurate numerical models of complex

spatio-temporal dynamical systems with holistic

discretisation

for the degree of Doctor of Philosophy

Tony MacKenzie, BIT, BSc(Hons)

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Abstract

This dissertation focuses on the further development of creating accurate numerical models of complex dynamical systems using the holistic discreti-sation technique [Roberts, Appl. Num. Model., 37:371–396, 2001]. I extend the application from second to fourth order systems and from only one spa-tial 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 pat-terns governed bypdes 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, Ap-plications of centre manifold theory, 1981] so we are assured that the nu-merical 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 γ, subgrid scale fields are constructed consisting of actual solutions of the gov-erning partial differential equation(pde). These subgrid scale fields interact

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iv Abstract

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 ap-proximate inertial manifolds. The excellent performance of the holistic mod-els shown here is strong evidence in support of the holistic discretisation technique.

For shear dispersion in a 2Dchannel a one-dimensional numerical approx-imation is generated directly from the two-dimensional advection-diffusion dynamics. We find that a low order holistic model contains the shear disper-sion 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 alge-braic 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

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Certification of dissertation

The work contained in this dissertation is the bonafide work of the author; the work has not been previously submitted for an award; and to the best of my knowledge and belief, the dissertation contains no material previously published or written by another person except where due acknowledgement and reference is made in the dissertation to that work.

Signed: Date:

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Acknowledgements

This dissertation was prepared while on an Australian Postgraduate Award. I also acknowledge the receipt of a University of Southern Queensland post-graduate research scholarship for the first 12 months of my candidature. I also acknowledge the support of Queensland Treasury Corporation.

I would like to express my sincere thanks to the following people:

• My supervisor, Professor Tony Roberts for his guiding hand, invaluable expertise and the stimulus for this dissertation. I thank Tony for his commitment to this project and his perseverance. Tony’s mentoring through my candidature has added value far beyond the scope of this dissertation and the legacy of his unique approach to applied mathe-matics is embedded in my day to day professional life.

• Dr Chris Harman my associate supervisor for his direction, suggestions and proof reading of this dissertation.

• Dr Dmitry Strunin and Dr Sergey Suslov for their comments, sugges-tions and valued mentoring throughout this learning experience.

• Dr Mark Thompson, Dr Glen Lochhead, Tanya and my mum for their constant support through the final stages of writing this dissertation.

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Contents

Abstract iii

Certification of dissertation vii

Acknowledgements ix

1 Introduction 1

1.1 The scope of the dissertation . . . 6

2 Discretise the Kuramoto–Sivashinsky equation 11 2.1 Apply a homotopy in the inter-element coupling parameter . . 15

2.1.1 The local IBCs . . . 17

2.1.2 The non-local IBCs . . . 19

2.2 Centre manifold theory provides the basis for the analysis . . . 22

2.2.1 There exists a centre manifold . . . 22

2.2.2 The holistic model is relevant to the PDE . . . 24

2.2.3 Approximate the shape of the centre manifold . . . 25

2.3 Computer algebra handles the details . . . 26

2.3.1 Solve symbolic equations for the centre manifold . . . . 27

2.3.2 Various holistic models . . . 28

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xii Contents

2.4 Illustration of subgrid field enhances our view . . . 34

2.5 Holistic models are consistent as h→0 . . . 40

2.6 Summary . . . 44

3 Holistic models are accurate for steady states of the KSE 47 3.1 Bifurcation diagrams show steady states . . . 51

3.2 Accurate steady state solutions . . . 53

3.2.1 Conventions for the bifurcation diagrams . . . 53

3.2.2 Explore some steady state solutions . . . 57

3.3 Holistic models are accurate on coarse grids . . . 57

3.3.1 Bifurcation diagrams show success . . . 59

3.4 Non-local IBCs are superior . . . 63

3.4.1 Bifurcation diagrams of low order holistic models . . . 64

3.4.2 Higher order models confirm non-local IBCs better . . 65

3.4.3 Holistic models outperform centered differences . . . . 68

3.4.4 Grid refinement improves accuracy . . . 71

3.5 Comparison to Galerkin approximations . . . 74

3.5.1 The Galerkin approximations . . . 75

3.5.2 Bifucation diagrams for the Galerkin approximations . 75 3.6 Coarse grid allows larger time steps . . . 79

3.7 Summary . . . 83

4 Holistic models are accurate for time dependent phenomena 85 4.1 Dynamics near the steady states are reproduced . . . 88

4.1.1 Compare eigenvalues along the bimodal branch . . . . 89

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Contents xiii

4.2 Extend the Hopf bifurcations . . . 93

4.2.1 Investigate the first Hopf bifurcation . . . 95

4.2.2 Period doubling is more accurately modelled . . . 98

4.3 Investigate travelling wave solutions . . . 100

4.3.1 Good performance for holistic models at low α . . . 104

4.3.2 Good performance for more complex behaviour . . . . 106

4.4 Summary . . . 116

5 Shear dispersion is modelled by holistic discretisation 119 5.1 The cross-channel advection velocity and diffusion profile . . . 123

5.2 The domain is divided into elements . . . 124

5.2.1 Use a 2D version of the non-local IBCs . . . 125

5.3 Centre manifold theory provides the justification . . . 126

5.4 Shear dispersion appears with a low order approximation . . . 128

5.4.1 The O(γ3,P3) holistic model . . . 129

5.4.2 View the subgrid field . . . 131

5.5 Inlet and Outlet boundary conditions are easily incorporated . 132 5.5.1 The holistic models near the boundaries . . . 134

5.6 Summary . . . 136

6 The Ginzburg–Landau equation 139 6.1 The Ginzburg–Landau equation with real coefficients . . . 142

6.1.1 The application is similar to Burgers’ equation . . . 142

6.2 The iteration scheme . . . 145

6.3 Computer algebra handles the details . . . 149

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xiv Contents

6.4.1 The accurate bifurcation diagram . . . 150

6.4.2 The O(γ2, α2) holistic model underperforms . . . 152

6.4.3 Higher order models improve performance . . . 153

6.5 Summary . . . 156

7 Extension to two spatial dimensions provides challenges 159 7.1 Divide the domain into square elements . . . 162

7.1.1 Extend the non-local IBCs to 2D . . . 163

7.2 Centre manifold theory is applied . . . 163

7.3 The dynamics on the manifold form the discretisation . . . 166

7.3.1 The O(γ2, α2) holistic model . . . 167

7.3.2 The holistic model has dual justification . . . 167

7.4 Illustration of the subgrid field in 2D . . . 167

7.5 The O(γ2, α2) holistic model is poor . . . 169

7.6 Higher order models need numerical construction . . . 171

7.7 Summary . . . 172

8 Generally compute 2D subgrid fields numerically 173 8.1 Discretise the subgrid field structure . . . 178

8.1.1 A low resolution subgrid illustrates the iteration scheme 180 8.1.2 Coefficients converge to analytic holistic model . . . 187

8.2 Extrapolation improves accuracy . . . 189

8.3 Low resolution subgrids are accurate . . . 190

8.4 An efficient computer algebra algorithm is the key . . . 196

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Contents xv

8.4.3 The REDUCE implementation in 1D . . . 200

8.5 Discretise the subgrid field structure in two spatial dimensions 205 8.5.1 The algorithm is extended to 2D . . . 206

8.5.2 Low resolution subgrids are accurate in 2D . . . 207

8.5.3 Higher order holistic models in 2D . . . 208

8.6 Summary . . . 211

9 Conclusions 213 9.1 Summary of results . . . 214

9.2 Future directions . . . 217

A REDUCE programs to construct holistic models 221 A.1 1D Analytical subgrid fields . . . 221

A.1.1 Second order dissipative PDEs . . . 221

A.1.2 Fourth order dissipative PDEs . . . 223

A.2 2D Analytical subgrid fields . . . 224

A.2.1 Second order dissipative PDEs . . . 224

A.2.2 Shear dispersion in a 2D channel . . . 226

A.3 1D Numerical subgrid fields . . . 230

A.3.1 Second order dissipative PDEs . . . 230

A.3.2 Fourth order dissipative PDEs . . . 232

A.3.3 MATHEMATICA code for second order PDEs . . . 235

A.4 2D Numerical subgrid fields . . . 237

A.4.1 Second order dissipative PDEs . . . 237

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