Holistically discretise the Swift-Hohenberg equation on a scale larger than its spatial pattern
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arXiv
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| Format: | Preprint |
| Published: |
2001
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| _version_ | 1866908599966498816 |
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| author | Roberts, A. J. |
| author_facet | Roberts, A. J. |
| contents | I introduce an innovative methodology for deriving numerical models of systems of partial differential equations which exhibit the evolution of spatial patterns. The new approach directly produces a discretisation for the evolution of the pattern amplitude, has the rigorous support of centre manifold theory at finite grid size $h$, and naturally incorporates physical boundaries. The results presented here for the Swift-Hohenberg equation suggest the approach will form a powerful method in computationally exploring pattern selection in general. With the aid of computer algebra, the techniques may be applied to a wide variety of equations to derive numerical models that accurately and stably capture the dynamics including the influence of possibly forced boundaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0110153 |
| institution | arXiv |
| publishDate | 2001 |
| record_format | arxiv |
| spellingShingle | Holistically discretise the Swift-Hohenberg equation on a scale larger than its spatial pattern Roberts, A. J. Numerical Analysis I introduce an innovative methodology for deriving numerical models of systems of partial differential equations which exhibit the evolution of spatial patterns. The new approach directly produces a discretisation for the evolution of the pattern amplitude, has the rigorous support of centre manifold theory at finite grid size $h$, and naturally incorporates physical boundaries. The results presented here for the Swift-Hohenberg equation suggest the approach will form a powerful method in computationally exploring pattern selection in general. With the aid of computer algebra, the techniques may be applied to a wide variety of equations to derive numerical models that accurately and stably capture the dynamics including the influence of possibly forced boundaries. |
| title | Holistically discretise the Swift-Hohenberg equation on a scale larger than its spatial pattern |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/math/0110153 |