B-Series and Geometric Integration

Abstract

Although it has not been possible to survey all aspects of the burgeoning subject of Geometric Integration, symplectic Runge–Kutta methods and their generalization to general linear methods are introduced to the extent that their main properties are studied and explained. It is perhaps surprising that G-symplectic methods perform well over millions of time steps, even though they will eventually fail because of round-off errors. Energy preserving methods are briefly surveyed, based on Continuous Stage Runge–Kutta methods, which were previously introduced in Chapter 4, as examples of integration methods.