Subgrid and interelement interactions affect discretisations of stochastically forced diffusion

A. J. Roberts

Abstract


Constructing discrete models of stochastic partial differential equations is very delicate. I apply dynamical systems theory to support spatial discretisations of the stochastically forced diffusion equation. To apply stochastic centre manifold theory, divide the physical domain into finite sized elements by introducing insulating internal boundaries which are later removed to fully couple the dynamical interactions between neighbouring elements. The approach automatically parametrises the stochastically forced microscale, subgrid structures within each element. The crucial aspect of this work is that we explore how a multitude of noise processes interact within and between neighbouring elements. Noise processes with coarse structure across a finite element are the most significant noises for the discrete model. Their influence also diffuses away to weakly correlate the noise in a spatial discretisation.

References
  • Jinqiao Duan, Kening Lu, and Bjorn Schmalfuss. Invariant manifolds for stochastic partial differential equations. The Annals of Probability, 31:2109--2135, 2003. doi:10.1214/aop/1068646380.
  • E. Knobloch and K. A. Wiesenfeld. Bifurcations in fluctuating systems: The centre manifold approach. J. Stat Phys, 33:611--637, 1983.
  • T. MacKenzie and A. J. Roberts. Holistic discretisation of shear dispersion in a two-dimensional channel. In K. Burrage and Roger B. Sidje, editors, Proc. of 10th Computational Techniques and Applications Conference CTAC-2001, volume 44, pages C512--C530, March 2003. http://anziamj.austms.org.au/V44/CTAC2001/Mack.
  • A. J. Roberts. Low-dimensional modelling of dynamics via computer algebra. Computer Phys. Comm., 100:215--230, 1997.
  • A. J. Roberts. Holistic discretisation ensures fidelity to {Burgers'} equation. Applied Numerical Modelling, 37:371--396, 2001.
  • A. J. Roberts. A holistic finite difference approach models linear dynamics consistently. Mathematics of Computation, 72:247--262, 2002. http://www.ams.org/mcom/2003-72-241/S0025-5718-02-01448-5.
  • A. J. Roberts. Derive boundary conditions for holistic discretisations of {Burgers'} equation. In K. Burrage and Roger B. Sidje, editors, Proc. of 10th Computational Techniques and Applications Conference CTAC-2001, volume 44, pages C664--C686, March 2003. http://anziamj.austms.org.au/V44/CTAC2001/Robe.
  • A. J. Roberts. A step towards holistic discretisation of stochastic partial differential equations. In Jagoda Crawford and A. J. Roberts, editors, Proc. of 11th Computational Techniques and Applications Conference CTAC-2003, volume 45, pages C1--C15, December 2003. [Online] http://anziamj.austms.org.au/V45/CTAC2003/Robe [December 14, 2003].
  • J. C. Robinson. The asymptotic completeness of inertial manifolds. Nonlinearity, 9:1325--1340, 1996. http://www.iop.org/EJ/abstract/0951-7715/9/5/013.
  • N. {Sri Namachchivaya} and Y. K. Lin. Method of stochastic normal forms. Int. J. Nonlinear Mechanics, 26:931--943, 1991.
  • D. Blomker, M. Hairer, and G. A. Pavliotis. Modulation equations: stochastic bifurcation in large domains. Communications in Mathematical Physics, 258:479--512, 2005. doi:10.1007/s00220-005-1368-8.
  • P. Boxler. A stochastic version of the centre manifold theorem. Probab. Th. Rel. Fields, 83:509--545, 1989.
  • Xu Chao and A. J. Roberts. On the low-dimensional modelling of {Stratonovich} stochastic differential equations. Physica A, 225:62--80, 1996. doi:10.1016/0378-4371(95)00387-8.
  • P. H. Coullet, C. Elphick, and E. Tirapegui. Normal form of a {Hopf} bifurcation with noise. Physics Letts, 111A(6):277--282, 1985.

Full Text:

PDF BibTeX


DOI: http://dx.doi.org/10.21914/anziamj.v48i0.36



Remember, for most actions you have to record/upload into this online system
and then inform the editor/author via clicking on an email icon or Completion button.
ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.