A holistic finite difference approach models linear dynamics consistently
Article
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/S0025-5718-02-01448-5
| Article Title | A holistic finite difference approach models linear dynamics consistently |
|---|---|
| ERA Journal ID | 326 |
| Article Category | Article |
| Authors | |
| Author | Roberts, A. J. |
| Journal Title | Mathematics of Computation |
| Journal Citation | 72 (241), pp. 247-262 |
| Year | 2003 |
| Place of Publication | United States |
| ISSN | 0025-5718 |
| 1088-6842 | |
| Digital Object Identifier (DOI) | https://doi.org/10.1090/S0025-5718-02-01448-5 |
| Web Address (URL) | http://www.ams.org/mcom/2003-72-241/S0025-5718-02-01448-5/home.html |
| Abstract | [Abstract]: I prove that a centre manifold approach to creating finite difference models will consistently model linear dynamics as the grid spacing becomes small. Using such tools of dynamical systems theory gives new assurances about the quality of finite difference models under nonlinear and other perturbations on grids with finite spacing. For example, the linear advection-diffusion equation is found to be stably modelled for all advection speeds and all grid spacings. The theorems establish an extremely good form for the artificial internal boundary conditions that need to be introduced to apply centre manifold theory. When numerically solving nonlinear partial differential equations, this approach can be used to systematically derive finite difference models which automatically have excellent characteristics. Their good performance for finite grid spacing implies that fewer grid points may be used and consequently there will be less difficulties with stiff rapidly decaying modes in continuum problems. |
| Keywords | centre manifold approach; finite difference models; linear dynamics |
| ANZSRC Field of Research 2020 | 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 | Department of Mathematics and Computing |
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