Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In the mathematical field of numerical ordinary differential equations, a geometric integrator is a numerical method that preserves geometric properties of the exact flow of a differential equation.To summarize: the pendulum example shows that, besides the explicit and implicit Euler methods not being good choices of method to solve the problem, the symplectic Euler method and implicit midpoint rule agree well with the exact flow of the system, with the midpoint rule agreeing more closely. Furthermore, these latter two methods are area-preserving, just as the exact flow is; they are two examples of geometric (in fact, symplectic) integrators.
