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Grid adaption, order selection, and complexity

Gustav Söderlind

Abstract

Differential equation solvers use grid adaption and sometimes variable order to increase computational efficiency. Grid and order control algorithms have often been heuristic, but today these algorithms can be designed and analyzed using mathematical principles. This has many important consequences. Examples will be given to illustrate, for ordinary differential equations:

1. Step size control affects computational stabillity

2. Hamiltonian systems can be solved with both energy conservation and adaptive step size selection; as a result, both accuracy and efficiency increase

3. Complexity is usually claimed to be exponential in the required precision, but S. Ilie has recently shown theoretically that with variable order complexity can be lower. We will demonstrate a new order control algorithm that supports Ilie's claim, when implemented in DASPK and solving real problems.

4. Grid refinement (or moving meshes) can be constructed based on control theory