Symmetric Integration and Reversibility

Abstract

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.