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DSWeb Dynamical Systems Software aims to collect all available software on dynamical systems theory. This project was originally launched during the special year Emerging Applications of Dynamical Systems, 1997/1998, at the Institute for Mathematics and its Applications. The information here includes functionality, platforms, languages, references, and contacts.

Please note that DSWeb is not responsible for any direct, indirect, special, incidental, or consequential damages arising from the use of the content provided here.

Holistic discretisation of dynamical PDEs

By Tony Roberts
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Via this web page you obtain the spatial discretisation of a dynamical partial differential equation (PDE) using dynamical systems theory. The web page not only returns the algebraic expressions for your consideration, but also an optimised Matlab/Octave function ready for standard numerical integrators. The technique not only ensures consistency of the discretisation, but remarkably theory ensures the exponentially quick relevance of the discretisation at finite grid spacing h. Theory also suggests the numerical disretisation should have good stability properties on a coarse spatial grid. I use a generalised Burgers' equation as an example:
du
dt
 +u du
dx
 = d2u
dx2
 +ru3.
A companion web page discretises up to three coupled PDEs in one space dimension, such as the complex Ginzburg--Landau equation.
KeywordsOther
Model
  • PDEs
Software Type
  • Package
Language
  • Other
Platform
  • Unix
  • Linux
  • Windows
  • MacOS
Availability
Contact Person
Tony Roberts, University of Adelaide, [email protected]
References to Papers

T. Mackenzie and A. J. Roberts. Holistic finite differences accurately model the dynamics of the Kuramoto--Sivashinsky equation. ANZIAM J., 42(E):C918--C935, 2000. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/628.

T. MacKenzie and A. J. Roberts. Accurately model the Kuramoto--Sivashinsky dynamics with holistic discretisation. SIAM J. Applied Dynamical Systems, 5(3):365--402, 2006.

A. J. Roberts. Holistic discretisation ensures fidelity to Burgers’ equation. Applied Numerical Modelling, 37:371--396, 2001. http://dx.doi.org/10.1016/S0168-9274(00)00053-2.

A. J. Roberts. Holistic projection of initial conditions onto a finite difference approximation. Computer Phys. Comm., 142:316--321, 2001.

A. J. Roberts. A holistic finite difference approach models linear dynamics consistently. Mathematics of Computation, 72:247--262, 2003.

A. J. Roberts, T. MacKenzie, and J. Bunder. Accurate macroscale modelling of spatial dynamics in multiple dimensions. J. Engineering Mathematics, accepted 2013-03-28.

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