Symmetric Integration and Reversibility
Symmetric methods of this chapter and symplectic methods of the next chapter play a central role in the geometric integration of differential equations. We discuss reversible differential equations and reversible maps, and we explain how symmetric integrators are related to them. We study symmetric Runge-Kutta and composition methods, and we show how standard approaches for solving differential equations on manifolds can be symmetrized. A theoretical explanation of the excellent longtime behaviour of symmetric methods applied to reversible differential equations will be given in Chap. XI.
KeywordsKepler Problem Composition Method Discrete Gradient Left Picture Symmetric Method
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- 1.The study of symmetric methods has its origin in the development of extrapolation methods (Gragg 1965, Stetter 1973), because the global error admits an asymptotic expansion in even powers of h. The notion of time-reversible methods is more common in the Computational Physics literature (Buneman 1967).Google Scholar
- 2.For irreducible Runge-Kutta methods, the condition (2.4) is also necessary for symmetry (after a suitable permutation of the stages).Google Scholar
- 3.The authors are grateful to S. Blanes for this reference.Google Scholar