1. Adaptivity and computational complexity in the numerical solution of ODEs
- Author
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Ilie, Silvana, Söderlind, Gustaf, and Corless, Robert M.
- Subjects
- *
MATHEMATICAL analysis , *COMPUTATIONAL complexity , *NUMERICAL solutions to differential equations , *ERROR functions - Abstract
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 and finding a grid deformation map, , that maps an equidistant grid to a non-equidistant grid in the original independent variable, . An optimal deformation map 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. [Copyright &y& Elsevier]
- Published
- 2008
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