IMA Journal of Numerical Analysis Advance Access originally published online on March 7, 2005
IMA Journal of Numerical Analysis 2005 25(4):698-725; doi:10.1093/imanum/dri013
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Computation of non-smooth local centre manifolds
1 Department of Mathematics, Indiana University, Bloomington, IN 47405, USA, 2 Instituto de Matemática, Caixa Postal 68530, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ 21954-970, Brazil
** Email: msjolly{at}indiana.edu
*** Email: rrosa{at}ufrj.br
An iterative Lyapunov–Perron algorithm for the computation of inertial manifolds is adapted for centre manifolds and applied to two test problems. The first application is to compute a known non-smooth manifold (once, but not twice differentiable), where a Taylor expansion is not possible. The second is to a smooth manifold arising in a porous medium problem, where rigorous error estimates are compared to both the correction at each iteration and the addition of each coefficient in a Taylor expansion. While in each case the manifold is 1D, the algorithm is well-suited for higher dimensional manifolds. In fact, the computational complexity of the algorithm is independent of the dimension, as it computes individual points on the manifold independently by discretising the solution through them. Summations in the algorithm are reformulated to be recursive. This acceleration applies to the special case of inertial manifolds as well.
Key Words: computation of invariant manifolds; centre manifolds
Received on 29 August 2003. revised on 28 November 2004.