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Wikipedia / Geometric integrator

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.

Pendulum example

We can motivate the study of geometric integrators by considering the motion of a pendulum.

the angular displacement the pendulum's momentum. The Hamiltonian of the system, the sum of its kinetic and potential energies, is

which gives Hamilton's equations

It is natural to take the configuration space. We use the following methods.

  1. (explicit Euler),

  2. (implicit Euler),

  3. (symplectic Euler),

  4. (implicit midpoint rule).

(Note that the symplectic Euler method treats q by the implicit Euler method.)

is constant along the solution curves of the Hamilton's equations allows us to describe the exact trajectories of the system: they are the level curves, and and , and , and . The explicit (resp. implicit) Euler method spirals out from (resp. in to) the origin. The other two methods show the correct qualitative behaviour, with the implicit midpoint rule agreeing with the exact solution to a greater degree than the symplectic Euler method.

of a Hamiltonian system of the explicit Euler method is not area-preserving; viz.,

A similar calculation can be carried out for the implicit Euler method, where the determinant is

However, the symplectic Euler method is area-preserving:

. The implicit midpoint rule has similar geometric properties.

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.

Moving frame method

The moving frame method can be used to construct numerical methods which preserve Lie symmetries of the ODE. Existing methods such as Runge-Kutta can be modified using moving frame method to produce invariant versions.

See also

References

Further reading