Skip to main content Accessibility help
×
Home

Resolving the Multitude of Microscale Interactions Accurately Models Stochastic Partial Differential Equations

  • A. J. Roberts (a1)

Abstract

Constructing numerical models of noisy partial differential equations is a very delicate task. Our long-term aim is to use modern dynamical systems theory to derive discretisations of dissipative stochastic partial differential equations. As a second step, we consider here a small domain, representing a finite element, and derive a one-degree-of-freedom model for the dynamics in the element; stochastic centre manifold theory supports the model. The approach automatically parametrises the microscale structures induced by spatially varying stochastic noise within the element. The crucial aspect of this work is that we explore how a multitude of microscale noise processes may interact in nonlinear dynamical systems. The analysis finds that noise processes with coarse structure across a finite element are the significant noises for the modelling. Further, the nonlinear dynamics abstracts effectively new noise sources over the macroscale time-scales resolved by the model.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Resolving the Multitude of Microscale Interactions Accurately Models Stochastic Partial Differential Equations
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Resolving the Multitude of Microscale Interactions Accurately Models Stochastic Partial Differential Equations
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Resolving the Multitude of Microscale Interactions Accurately Models Stochastic Partial Differential Equations
      Available formats
      ×

Copyright

References

Hide All
1. Arnold, Ludwig, Random dynamical systems, Springer Monogr. Malh. (Springer, 2003) ISBN 3–540–63758–3.
2. Arnold, L., Sri Namachchivaya, N. and Schenk-Hoppé, K. R., ‘Toward an understanding of stochastic Hopf bifurcation: a case study’, Internal. J. Bifur. Chaos Appl. Sci. Engrg. 6 (1996) 19471975.
3. Baxter, Martin and Rennie, Andrew, Financial calculus: An introduction to derivative pricing (Cambridge University Press, 1996).
4. Bensoussan, A. and Flandoli, F., ‘Stochastic inertial manifold’. Stochastics Stochastics Rep. 53 (1995) 1339.
5. Berglund, Nils and Gentz, Barbara, ‘Geometric singular perturbation theory for stochastic differential equations’, J. Differential Equations 191 (2003) 154, http://dx.doi.org/10.1016/S0022-0396(03)0002 0-2.
6. Blomker, D., Hairer, M. and Pavliotis, G. A., ‘Modulation equations: stochastic bifurcation in large domains’, Comm. Math. Phys. 258 (2005) 479512, http://dx.doi.org/10.1007/s00220-005-1368-8.
7. Boxler, P., ‘A stochastic version of the centre manifold theorem’. Probab. Theory Related Fields 83 (1989) 509545.
8. Boxler, P., How to construct stochastic center manifolds on the level of vector fields, Lecture Notes in Math. 1486 (Springer, 1991) 141158.
9. Caraballo, Tomas, Langa, Jose A. and Robinson, James C., ‘A stochastic pitchfork bifurcation in a reaction-diffusion equation’, Proc. Roy. Soc. Lond. A 457 (2001) 20412061. http://dx.doi.org/10.1098/rspa.2001.0819.
10. Chao, Xu and Roberts, A. J., ‘On the low-dimensional modelling of Stratonovich stochastic differential equations’, Physica A 225 (1996) 6280, http://dx.doi.org/10.1016/0378-4371(95)00387-8.
11. Chatwin, P. C., ‘The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe’. J. Fluid Mech. 43 1970) 321’352.
12. Chicone, C. and Latushkin, Y., ‘Center manifolds for infinite dimensional non-autonomous differential equations’, J. Differential Equations 141 (1997) 356399. http://www.ingentaconnect.com/content/ap/de/1997/00000141/00000002/art03343.
13. Coullet, P. H., Elphick, C. and Tirapegui, E., ‘Normal form of a Hopf bifurcation with noise’, Phys. Lett. 111 A (1985) 277282.
14. Drolet, Francois and Vinals, Jorge, ‘Adiabatic reduction near a bifurcation in stochastically modulated systems’, Phys. Rev. E 57 (1998) 50365043. http://link.aps.org/abstract/PRE/v57/p5036.
15. Drolet, Francois and Vinals, Jorge, ‘Adiabatic elimination and reduced probability distribution functions in spatially extended systems with a fluctuating control parameter’, Phys. Rev. E 64 (2001) 026120, http://link.aps.org/abstract/PRE/v64/e026120.
16. Duan, Jinqiao, Lu, Kening and Schmalfuss, Bjorn, Invariant manifolds for stochastic partial differential equations’, Ann. Probab. 31 (2003) 21092135.
17. Gallay, Th., ‘A center-stable manifold theorem for differential equations in Banach spaces’. Comm. Math. Phys 152 (1993) 249268.
18. Grecksch, W. and Kloeden, P. E., ‘Time-discretised Galerkin approximations of parabolic stochastics PDEs’, Bull. Austral Math. Soc. 54 (1996) 7985.
19. Just, Wolfram, Kantz, Holger, Rodenbeck, Christian and Helm, Mario, ‘Stochastic modelling: replacing fast degrees of freedom by noise’. J. Phys. A: Math. Gen. 34 (2001) 31993213.
20. Kabanov, Yuri and Pergamenshchikov, Sergei, Two-scale stochastic systems, Applications of Mathematics: Stochastic Modelling and Applied Probability 49 (Springer. 2003).
21. Kloeden, P. E. and Platen, E., Numerical solution of stochastic differential equations, Appl. Math. 23 (Springer. 1992).
22. Knobloch, E. and Wiesenfeld, K. A., ‘Bifurcations in fluctuating systems: The centre manifold approach’. J. Statist. Phys 33 (1983) 611637.
23. MacKenzie, T. and Roberts, A. J., ‘Holistic finite differences accurately model the dynamics of the Kuramoto-Sivashinsky equation’, ANZ1AM J. 42 (2000) C918–C935. http://anziamj.austms.org.au/V42/CTAC99/Mack.
24. MacKenzie, T. and Roberts, A. J., ‘Holistic discretisation of shear dispersion in a two-dimensional channel’, Proc, 10th Computational Techniques and Applications Conference CTAC-2001. ed. Burrage, K. and Sidje, Roger B., ANZIAM J. 44C (2003) C512–C530, http://anziamj.austms.org.au/V44/CTAC2001/Mack.
25. Metzler, R., ‘Non-homogeneous random walks, generalised master equations, fractional Fokker-Planck equations, and the generalised Kramers-Moyal expansion’, Eur. Phys. J. B 19 (2001) 249258. http://www.edpsciences.org/articles/epjb/pdf/2001/02/b0331.pdf.
26. Naert, A., Friedrich, R. and Peinke, J., ‘Fokker-Planck equation for the energy cascade in turbulence’, Phys. Rev. E 56 (1997) 67196722.
27. Pollett, P. K. and Roberts, A.J., ‘A description of the long-term behaviour of absorbing continuous time Markov chains using a centre manifold’. Adv. in Appl. Probab. 22 (1990) 111‘128.
28. Roberts, A. J., ‘The application of centre manifold theory to the evolution of systems which vary slowly in space’. J. Austral. Math. Soc. B 29 (1988) 480500.
29. Roberts, A. J., ‘Low-dimensional modelling of dynamics via computer algebra’, Comput. Phys. Comm. 100 (1997) 215230.
30. Roberts, A. J., ‘Holistic discretisation ensures fidelity to Burgers' equation’. Appl. Numer. Modelling 37 (2001) 371396. http://arXiv.org/abs/chao-dyn/9901011.
31. Roberts, A. J., ‘Holistic projection of initial conditions onto a finite difference approximation’. Comput. Phys. Comm. 142 (2001) 316321, http://arXiv.org/abs/math.NA/0101205.
32. Roberts, A. J., ‘A holistic finite difference approach models linear dynamics consistently’, Math. Comp. 72 (2002) 247262. http://arXiv.org/abs/math.NA/0003135.
33. Roberts, A. J., ‘Derive boundary conditions for holistic discretisations of Burgers' equation’, Proc, 10th Computational Techniques and Applications Conference CTAC-2001, ed. Burrage, K. and Sidje, Roger B., ANZIAM J. 44C (2003) C664C686. http://anziamj.austms.org.au/V44/CTAC2001/Robe.
34. Roberts, A. J., ‘Low-dimensional modelling of dynamical systems applied to some dissipative fluid mechanics’, Nonlinear dynamics from lasers to butterflies, ed.Ball, Rowena and Akhmediev, Nail, Lecture Notes in Complex Systems I (World Scientific, 2003) 257313.
35. Roberts, A. J., ‘A step towards holistic discretisation of stochastic partial differential equations’. Proc, 11th Computational Techniques and Applications Conference CTAC-2003 (Dec. 2003). ed. Crawford, Jagoda and Roberts, A. J., ANZIAM J. 45 (2004) C1–C15, http://anziamj.austms.org.au/V45/CTAC2003/Robe.
36. Roberts, A. J., ‘Computer algebra resolves a multitude of microscale interactions to model stochastic partial differential equations’. Technical report. University of Southern Queensland. December 2005. http://www.sci.usq.edu.au/staff/robertsa/CA/multinoise.pdf.
37. Robinson, J. C., ‘The asymptotic completeness of inertial manifolds’, Nonlinearity 9 (1996) 13251340, http://www.iop.org/EJ/abstract/0951-7715/9/5/013.
38. Schöner, G. and Haken, H., ‘The slaving principle for Stratonovich stochastic differential equations’, Z Phys. B — Condensed matter 63 (1986) 493504.
39. Sri Namachchivaya, N. and Lin, Y. K., ‘Method of stochastic normal forms’, Int. J. Nonlinear Mechanics 26 (1991) 931943.
40. Tutkun, M. and Mydlarski, L., ‘Markovian properties of passive scalar increments in grid-generated turbulence’, New J. Phys. 6 (2004) article 49, 1–24. http://dx.doi.org/10.1088/1367-2360/6/1/049.
41. Vanden-Eijnden, Eric, ‘Asymptotic techniques for SDEs’. Fast times and fine scales: Proc, 2005 Program in Geophysical Fluid Dynamics (Woods Hole Oceanographic Institution, 2005), http://gfd.whoi-edu/proceedings/2005/PDFvol2005.html.
42. Werner, M. J. and Drummond, P. D., ‘Robust algorithms for solving stochastic partial differential equations’. J. Comput. Phys. 132 (1997) 312326.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Related content

Powered by UNSILO

Resolving the Multitude of Microscale Interactions Accurately Models Stochastic Partial Differential Equations

  • A. J. Roberts (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.