Adaptivity and computational complexity in the numerical solution of ODEs

Authors:
Silvana Ilie;Gustaf Söderlind;Robert M. Corless
Affiliations:
Centre for Mathematical Sciences, Numerical Analysis, Lund University, Box 118 SE-221 00 Lund, Sweden;Centre for Mathematical Sciences, Numerical Analysis, Lund University, Box 118 SE-221 00 Lund, Sweden;Department of Applied Mathematics, University of Western Ontario London, Ont., Canada N6A 5B7
Venue:
Journal of Complexity
Year:
2008

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Abstract

In this paper we analyze the problem of adaptivity for one-step numerical methods for solving ODEs, both IVPs and BVPs, with a view to generating grids of minimal computational cost for which the local error is below a prescribed tolerance (optimal grids). The grids are generated by introducing an auxiliary independent variable @t and finding a grid deformation map, t=@Q(@t), that maps an equidistant grid {@t"j} to a non-equidistant grid in the original independent variable, {t"j}. An optimal deformation map @Q is determined by a variational approach. Finally, we investigate the cost of the solution procedure and compare it to the cost of using equidistant grids. We show that if the principal error function is non-constant, an adaptive method is always more efficient than a non-adaptive method.