On a Numerical Liapunov--Schmidt Spectral Method and Its Application to Biological Pattern Formation

Spectral expansions are used to provide a basis which preserves continuous symmetries. We show that spectral methods satisfy the conditions for convergence of numerical Liapunov--Schmidt methods. An explicit algorithm for the calculation of stationary bifurcation scenarios near primary instabilities...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 2002-01, Vol.40 (2), p.683-701
Main Authors: Böhmer, K., Geiger, C., Rodriguez, J. D.
Format: Article
Language:English
Subjects:
Algorithms
Methods
Navier-Stokes equations
Symmetry
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Summary:Spectral expansions are used to provide a basis which preserves continuous symmetries. We show that spectral methods satisfy the conditions for convergence of numerical Liapunov--Schmidt methods. An explicit algorithm for the calculation of stationary bifurcation scenarios near primary instabilities in general continuous symmetric equations is given. The above convergence is extended to $ \Gamma$-equivalent discrete and original bifurcation scenarios. The method is applied to a biologically motivated reaction-diffusion system with spherical symmetry forming patterns. A specific singularity of a generic steady state bifurcation is investigated in detail.
ISSN:0036-1429
1095-7170
DOI:10.1137/S0036142998339526