Accurate macroscale modelling of spatial dynamics in multiple dimensions
Technical report
Roberts, A. J., MacKenzie, Tony and Bunder, J. E.. 2012. Accurate macroscale modelling of spatial dynamics in multiple dimensions. Adelaide, Australia. University of Adelaide.
| Title | Accurate macroscale modelling of spatial dynamics in multiple dimensions |
|---|---|
| Report Type | Technical report |
| Authors | Roberts, A. J. (Author), MacKenzie, Tony (Author) and Bunder, J. E. (Author) |
| Institution of Origin | University of Adelaide |
| Number of Pages | 58 |
| Year | 2012 |
| Publisher | University of Adelaide |
| Place of Publication | Adelaide, Australia |
| Web Address (URL) | http://arxiv.org/abs/1103.1187 |
| Abstract | Developments in dynamical systems theory provides new support for the macroscale modelling of pdes and other microscale systems such as Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically resolving subgrid microscale dynamics the dynamical systems approach constructs accurate closures of macroscale discretisations of the microscale system. Here we specifically explore reaction-diffusion problems in two spatial dimensions as a prototype of generic systems in multiple dimensions. Our approach unifies into one the modelling of systems by a type of finite elements, and the 'equation free' macroscale modelling of microscale simulators efficiently executing only on small patches of the spatial domain. Centre manifold theory ensures that a closed model exist on the macroscale grid, is emergent, and is systematically approximated. Dividing space either into overlapping finite elements or into spatially separated small patches, the specially crafted inter-element/patch coupling also ensures that the constructed discretisations are consistent with the microscale system/PDE to as high an order as desired. Computer algebra handles the considerable algebraic details as seen in the specific application to the Ginzburg--Landau PDE. However, higher order models in multiple dimensions require a mixed numerical and algebraic approach that is also developed. The modelling here may be straightforwardly adapted to a wide class of reaction-diffusion PDEs and lattice equations in multiple space dimensions. When applied to patches of microscopic simulations our coupling conditions promise efficient macroscale simulation. |
| Keywords | dynamical systems theory; mathematical modelling |
| ANZSRC Field of Research 2020 | 490199. Applied mathematics not elsewhere classified |
| 490302. Numerical analysis | |
| 490409. Ordinary differential equations, difference equations and dynamical systems | |
| Public Notes | Files associated with this item cannot be displayed due to copyright restrictions. |
| Byline Affiliations | University of Adelaide |
| Department of Mathematics and Computing |
Permalink -
https://research.usq.edu.au/item/q1639/accurate-macroscale-modelling-of-spatial-dynamics-in-multiple-dimensions
1633
total views6
total downloads1
views this month0
downloads this month
Export as
Related outputs
Co-ordinate transforms underpin multiscale modelling and reduction in deterministic and stochastic systems
Roberts, A. J.. 2008. "Co-ordinate transforms underpin multiscale modelling and reduction in deterministic and stochastic systems." Abbott, Derek, Aste, Tomaso, Batchelor, Murray and Dewar, Robert (ed.) Complex Systems II: SPIE Symposium on Microelectronics, MEMS, and Nanotechnology 2007. Canberra, Australia 05 - 07 Dec 2007 United States. SPIE. https://doi.org/10.1117/12.767596A dynamical systems approach to simulating macroscale spatial dynamics in multiple dimensions
Roberts, A. J., MacKenzie, T. and Bunder, J. E.. 2014. "A dynamical systems approach to simulating macroscale spatial dynamics in multiple dimensions." Journal of Engineering Mathematics. 86 (1), pp. 175-207. https://doi.org/10.1007/s10665-013-9653-6Computer algebra derives the slow manifold of patch or element dynamics on lattices in two dimensions
MacKenzie, Tony and Roberts, A. J.. 2014. Computer algebra derives the slow manifold of patch or element dynamics on lattices in two dimensions. Adelaide, Australia. University of Adelaide.Branching behavior of standing waves - the signatures of resonance
Smith, D. H. and Roberts, A. J.. 1999. "Branching behavior of standing waves - the signatures of resonance." Physics of Fluids. 11 (5), pp. 1051-1064. https://doi.org/10.1063/1.869976The style files: use the most informative synonym
Roberts, Tony. 2007. "The style files: use the most informative synonym." Gazette of the Australian Mathematical Society. 34 (4), pp. 208-209.Modelling the dynamics of turbulent floods
Mei, Z., Roberts, A. J. and Li, Zhenquan. 2002. "Modelling the dynamics of turbulent floods." SIAM Journal on Applied Mathematics. 63 (2), pp. 423-458. https://doi.org/10.1137/S0036139999358866Reproductive pair correlations and the clustering of organisms
Young, W. R., Roberts, A. J. and Stuhne, Gordan. 2001. "Reproductive pair correlations and the clustering of organisms." Nature. 412 (6844), pp. 328-331. https://doi.org/10.1038/35085561The style files: write what you mean
Roberts, Tony. 2007. "The style files: write what you mean." Gazette of the Australian Mathematical Society. 34 (3), pp. 156-157.General tooth boundary conditions for equation free modeling
Roberts, A. J. and Kevrekidis, I. G.. 2007. "General tooth boundary conditions for equation free modeling." SIAM Journal on Scientific Computing. 29 (4), pp. 1495-1510. https://doi.org/10.1137/060654554The style files: appearance affects communication; but not necessarily as you like
Roberts, Tony. 2007. "The style files: appearance affects communication; but not necessarily as you like." Gazette of the Australian Mathematical Society. 34 (2), pp. 78-80.Two-zone model of shear dispersion in a channel using centre manifolds
Roberts, A. J. and Strunin, Dmitry V.. 2004. "Two-zone model of shear dispersion in a channel using centre manifolds." The Quarterly Journal of Mechanics and Applied Mathematics. 57 (3), pp. 363-378. https://doi.org/10.1093/qjmam/57.3.363The style files: omit redundant words
Roberts, Tony. 2007. "The style files: omit redundant words." Gazette of the Australian Mathematical Society. 34 (1), pp. 20-21.Accurately model the Kuramoto-Sivashinsky dynamics with holistic discretisation
MacKenzie, T. and Roberts, A. J.. 2006. "Accurately model the Kuramoto-Sivashinsky dynamics with holistic discretisation." SIAM Journal on Applied Dynamical Systems. 5 (3), pp. 365-402. https://doi.org/10.1137/050627733Low-dimensional modelling of a generalised Burgers equation
Li, Zhenquan and Roberts, A. J.. 2007. "Low-dimensional modelling of a generalised Burgers equation." Global Journal of Pure and Applied Mathematics. 3 (3), pp. 203-218.A flexible error estimate for the application of centre manifold theory
Li, Zhenquan and Roberts, A. J.. 2007. "A flexible error estimate for the application of centre manifold theory." Global Journal of Pure and Applied Mathematics. 3 (3), pp. 241-249.Fractal landscape method: an alternative approach to measuring area-restricted searching behavior
Tremblay, Yann, Roberts, Antony J. and Costa, Daniel P.. 2007. "Fractal landscape method: an alternative approach to measuring area-restricted searching behavior." The Journal of Experimental Biology. 210 (6), pp. 935-945. https://doi.org/10.1242/jeb.02710A normal form of thin fluid film equations solves the transient paradox
Roberts, A. J.. 2006. "A normal form of thin fluid film equations solves the transient paradox." Physica D: Nonlinear Phenomena. 223 (1), pp. 69-81. https://doi.org/10.1016/j.physd.2006.08.018Low-dimensional modelling of dynamical systems applied to some dissipative fluid mechanics
Roberts, A. J.. 2003. "Low-dimensional modelling of dynamical systems applied to some dissipative fluid mechanics." Ball, Rowena and Akhmediev, Nial (ed.) Nonlinear dynamics from lasers to butterflies: selected lectures from the 15th Canberra International Physics Summer School. World Scientific Publishing. pp. 257-313Modelling nonlinear dynamics of shape-memory-alloys with approximate models of coupled thermoelasticity
Melnik, Roderick V. N. and Roberts, A. J.. 2003. "Modelling nonlinear dynamics of shape-memory-alloys with approximate models of coupled thermoelasticity." Journal of Applied Mathematics and Mechanics (ZAMM). 83 (2), pp. 93-104. https://doi.org/10.1002/zamm.200310009Modelling turbulent flow from dam break using slow manifolds
Georgiev, D. J., Roberts, A. J. and Strunin, D. V.. 2009. "Modelling turbulent flow from dam break using slow manifolds ." Australian and New Zealand Industrial and Applied Mathematics (ANZIAM) Journal. 50 (S), pp. C1033-C1051.Low Prandtl number fluid convection modelled using symbolic algebra (REDUCE) and Matlab
Passmore, Tim and Roberts, A. J.. 2003. "Low Prandtl number fluid convection modelled using symbolic algebra (REDUCE) and Matlab." ANZIAM 2003: Computational Techniques and Applications. Toowoomba, Australia 16 - 18 Jul 2003 Cambridge, United Kingdom .Low-dimensional boundary-layer model of turbulent dispersion in a channel
Strunin, Dmitry V. and Roberts, Anthony J.. 2009. "Low-dimensional boundary-layer model of turbulent dispersion in a channel." Ao, S. I., Gelman, Len, Hukins, David W. L., Hunter, Andrew and Korsunsky, A. M. (ed.) WCE 2009: World Congress of Engineering . London, United Kingdom 01 - 03 Jul 2009 London, UK.A mathematical model to account for quantitative iterative effect of TNF-α positive feedback mechanism on inflammatory cascade in TLR-4 mediated TNF-α signaling pathway
Roberts, Tony, Essien, Bryan, Kotiw, Michael, Butler, Harry and Strunin, Dmitry. 2008. "A mathematical model to account for quantitative iterative effect of TNF-α positive feedback mechanism on inflammatory cascade in TLR-4 mediated TNF-α signaling pathway." Ragan, Mark (ed.) 19th International Conference on Genome Informatics (GIW 2008). Gold Coast, Australia 01 - 03 Dec 2008 London, United Kingdom.Advection-dispersion in symmetric field-flow fractionation channels
Suslov, Sergey A. and Roberts, A. J.. 1998. Advection-dispersion in symmetric field-flow fractionation channels. Toowoomba, Australia. University of Southern Queensland.The style files: teach explicit skills with feedback
Roberts, Tony. 2008. "The style files: teach explicit skills with feedback." Gazette of the Australian Mathematical Society. 35 (3), pp. 156-157.The style files: write to read breadth first, not depth first
Roberts, Tony. 2008. "The style files: write to read breadth first, not depth first." Gazette of the Australian Mathematical Society. 35 (1), pp. 17-19.Computer algebra derives discretisations via self-adjoint multiscale modelling
Roberts, A. J.. 2008. "Computer algebra derives discretisations via self-adjoint multiscale modelling." Unpublihsed.Computer algebra describes flow of turbulent floods via the Smagorinski large eddy closure
Roberts, A. J.. 2008. Computer algebra describes flow of turbulent floods via the Smagorinski large eddy closure. Toowoomba, Australia. University of Southern Queensland.The inertial dynamics of thin film flow of non-Newtonian fluids
Roberts, A. J.. 2008. "The inertial dynamics of thin film flow of non-Newtonian fluids." Physics Letters A. 372 (10), pp. 1607-1611. https://doi.org/10.1016/j.physleta.2007.10.014Normal form transforms separate slow and fast modes in stochastic dynamical systems
Roberts, A. J.. 2008. "Normal form transforms separate slow and fast modes in stochastic dynamical systems." Physica A: Statistical Mechanics and its Applications. 387 (1), pp. 12-38. https://doi.org/10.1016/j.physa.2007.08.023Computer algebra models dynamics on a multigrid across multiple length and time scales
Roberts, A. J.. 2007. Computer algebra models dynamics on a multigrid across multiple length and time scales.Nonlinear dynamics on centre manifolds describing turbulent floods: k-\omega model
Georgiev, Dian J., Roberts, A. J. and Strunin, Dmitry V.. 2007. "Nonlinear dynamics on centre manifolds describing turbulent floods: k-\omega model." Discrete and Continuous Dynamical Systems Series A.The dynamics of the vertical structure of turbulence in flood flows
Georgiev, D. J., Roberts, A. J. and Strunin, D. V.. 2007. "The dynamics of the vertical structure of turbulence in flood flows." Australian and New Zealand Industrial and Applied Mathematics (ANZIAM) Journal. 48, pp. C573-C590. https://doi.org/10.0000/anziamj.v48i0.124Mathematical and numerical analysis of Falk-Konopka-type models for shape-memory alloys
Melnik, R. V. N., Roberts, A. J. and Thomas, K. A.. 2000. "Mathematical and numerical analysis of Falk-Konopka-type models for shape-memory alloys." International Journal of Differential Equations and Applications. 1A (3), pp. 291-300.Computing dynamics of copper-based SMA via center manifold reduction of 3D models
Melnik, R. V. N., Roberts, A. J. and Thomas, K. A.. 2000. "Computing dynamics of copper-based SMA via center manifold reduction of 3D models." Computational Materials Science. 18 (3/4), pp. 255-268.Models encompassing hydraulic jumps in radial flows over a horizontal plate
Strunin, Dmitry and Roberts, Tony. 2001. "Models encompassing hydraulic jumps in radial flows over a horizontal plate." Kluev, Vitaly and Mastorakis, Nikos (ed.) 2nd WSEAS Multiconference on Applied and Theoretical Mathematics (WSEAS 2001). Cairns, Australia 17 - 23 Dec 2001 Greece.Dip coating process for hot metal castings
McGuinness, M. and Roberts, Anthony. 2000. "Dip coating process for hot metal castings." Hewitt, John (ed.) 16th Australian and New Zealand Mathematics In Industry Study Group (MISG 1999). Brisbane, Australia 01 - 05 Feb 1999 Adelaide, Australia.Coupled thermomechanical waves in hyperbolic thermoelasticity
Strunin, D. V., Melnik, R. V. N. and Roberts, A. J.. 2001. "Coupled thermomechanical waves in hyperbolic thermoelasticity." Journal of Thermal Stresses. 24 (2), pp. 121-140. https://doi.org/10.1080/01495730150500433A corrected quadrature formula and applications
Ujevic, Nenad and Roberts, A. J.. 2004. "A corrected quadrature formula and applications." Australian and New Zealand Industrial and Applied Mathematics (ANZIAM) Journal. 45 (E), pp. E41-E56.Phase transitions in shape memory alloys with hyperbolic heat conduction and differential-algebraic models
Melnik, R. V. N., Roberts, A. J. and Thomas, K. A.. 2002. "Phase transitions in shape memory alloys with hyperbolic heat conduction and differential-algebraic models." Computational Mechanics. 29 (1), pp. 16-26. https://doi.org/10.1007/s00466-002-0311-5Coupled thermomechanical dynamics of phase transitions in shape memory alloys and related hysteresis phenomena
Melnik, R. V. N., Roberts, A. J. and Thomas, K. A.. 2001. "Coupled thermomechanical dynamics of phase transitions in shape memory alloys and related hysteresis phenomena." Mechanics Research Communications. 28 (6), pp. 637-651. https://doi.org/10.1016/S0093-6413(02)00216-1A holistic finite difference approach models linear dynamics consistently
Roberts, A. J.. 2003. "A holistic finite difference approach models linear dynamics consistently." Mathematics of Computation. 72 (241), pp. 247-262. https://doi.org/10.1090/S0025-5718-02-01448-5Holistic discretisation ensures fidelity to Burger's equation
Roberts, A. J.. 2001. "Holistic discretisation ensures fidelity to Burger's equation." Applied Numerical Mathematics. 37 (3), pp. 371-396. https://doi.org/10.1016/S0168-9274(00)00053-2A lubrication model of coating flows over a curved substrate in space
Roy, R. Valery, Roberts, A. J. and Simpson, M. E.. 2002. "A lubrication model of coating flows over a curved substrate in space." Journal of Fluid Mechanics. 454 (1), pp. 235-261. https://doi.org/10.1017/S0022112001007133Computer algebra models the inertial dynamics of a thin film flow of power law fluids and other non-Newtonian fluids
Roberts, A. J.. 2007. Computer algebra models the inertial dynamics of a thin film flow of power law fluids and other non-Newtonian fluids . University of Southern Queensland.Computer algebra derives normal forms of stochastic differential equations
Roberts, A. J.. 2007. Computer algebra derives normal forms of stochastic differential equations . Toowoomba, Australia. University of Southern Queensland.Create accurate numerical models of complex spatio-temporal dynamical systems with holistic discretisation
MacKenzie, Tony. 2005. Create accurate numerical models of complex spatio-temporal dynamical systems with holistic discretisation. PhD Thesis Doctor of Philosophy. University of Southern Queensland.Computer algebra derives discretisations of the stochastically forced Burgers' partial differential equation
Roberts, A. J.. 2006. Computer algebra derives discretisations of the stochastically forced Burgers' partial differential equation. Toowoomba, Australia. University of Southern Queensland.The style files: favour the present tense
Roberts, Tony. 2006. "The style files: favour the present tense." Gazette of the Australian Mathematical Society. 33 (5), pp. 313-314.The style files: explicitly avoid false conditionals
Roberts, Tony. 2006. "The style files: explicitly avoid false conditionals." Gazette of the Australian Mathematical Society. 33 (4), pp. 241-242.The style files: inform with titles, abstracts and introductions
Roberts, Tony. 2006. "The style files: inform with titles, abstracts and introductions." Gazette of the Australian Mathematical Society. 33 (3), pp. 169-170.The style files: clarify this
Roberts, Tony. 2006. "The style files: clarify this." Gazette of the Australian Mathematical Society. 33 (2), pp. 104-105.The style files: prefer active writing to passive
Roberts, Tony. 2006. "The style files: prefer active writing to passive." Gazette of the Australian Mathematical Society. 33 (1), pp. 22-23.Computer algebra resolves a multitude of microscale interactions to model stochastic partial differential equations
Roberts, A. J.. 2005. Computer algebra resolves a multitude of microscale interactions to model stochastic partial differential equations. Toowoomba, Australia. University of Southern Queensland.