Geometric integrator explained

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.

Assume that we have a pendulum whose bob has mass

m =1 andwhose rod is massless of length \ell =1 . Take theacceleration due to gravity to be g =1 . Denote byq(t) the angular displacement of the rod from the vertical,and by p(t) the pendulum's momentum. The Hamiltonian ofthe system, the sum of its kinetic and potential energies, isH(q,p) = T(p) + U(q) = (1)/(2) p ^{2} - \cos q,

which gives Hamilton's equations

• ( • q, p) = (\partial H / \partial p, - \partial H / \partial q ) = (p, - \sin q). Q of all q to be the unitcircle S ^{1} , so that (q,p) lies on thecylinder S ^{1 x R} . However, we will take(q,p)\in R ^{2} , simply because (q,p) -space isthen easier to plot. Define z(t) = (q(t),p(t)) ^{T} and f(z) = (p, - \sin q) ^{T} . Let us experiment byusing some simple numerical methods to integrate this system. As usual,we select a constant step size, h , and for an arbitrary non-negative integer k we writez _{k:=z(kh)} .We use the following methods.z _{k+1} = z _{k} + hf(z _{k)} (explicit Euler),z _{k+1} = z _{k} + hf(z _{k+1} ) (implicit Euler),z _{k+1} = z _{k} + hf(q _{k,p} _{k+1} ) (symplectic Euler),z _{k+1} = z _{k} + hf((z _{k+1} + z _{k)/2)} (implicit midpoint rule).

(Note that the symplectic Euler method treats q by the explicit and

p by the implicit Euler method.)

The observation that

H is constant along the solutioncurves of the Hamilton's equations allows us to describe the exacttrajectories of the system: they are the level curves of p ^{2/2} - \cos q . We plot, in R ^{2} , the exacttrajectories and the numerical solutions of the system. For the explicitand implicit Euler methods we take h =0 . 2 , and z₀ = (0.5, 0) and (1.5, 0) respectively; for the other two methods we take h =0 . 3 , and z₀ = (0, 0.7), (0, 1.4) and (0, 2.1).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.

Recall that the exact flow

\phi _{t} of a Hamiltonian system with one degree of freedom isarea-preserving, in the sense that\det (\partial\phi _{t})/(\partial (q _{0,p} _{0)}) = 1 for all t .This formula is easily verified by hand. For our pendulumexample we see that the numerical flow \Phi _{{eE} ,h}:z _{k\mapsto} z _{k+1} of the explicit Euler method is not area-preserving; viz.,\det (\partial)/(\partial (q _{0,p} _{0)}) \Phi _{{eE} ,h}(z _{0)} = \begin{vmatrix} 1 &h\\ - h\cos q _{0&1\end{vmatrix}} = 1+ h ^{2\cos} q _{0.}

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

\det (\partial)/(\partial (q _{0,p} _{0)}) \Phi _{{iE} ,h}(z _{0)} = ( 1+ h ^{2\cos} q ^{-1} _{1 )} .

However, the symplectic Euler method is area-preserving:

\begin{pmatrix} 1 & - h\\ 0 & 1 \end{pmatrix} (\partial)/(\partial (q _{0,p} _{0)}) \Phi _{{sE} ,h}(z _{0)} = \begin{pmatrix} 1 & 0 \\ - h\cos q _{0&1\end{pmatrix},}

thus

\det(\partial\Phi _{{sE} ,h}/\partial (q _{0,p} _{0))} = 1 . The implicit midpoint rule has similar geometric properties.

To summarize: the pendulum example shows that, besides the explicit andimplicit Euler methods not being good choices of method to solve theproblem, the symplectic Euler method and implicit midpoint rule agreewell with the exact flow of the system, with the midpoint rule agreeingmore 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.^[1]

See also

• Energy drift
• Mimesis (mathematics)

Further reading

• Book: Ernst . Hairer . Christian . Lubich . Gerhard . Wanner . Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations . 2002 . Springer-Verlag . 3-540-43003-2 .
• Book: Leimkuhler . Ben . Reich . Sebastian . Simulating Hamiltonian Dynamics . 2005 . Cambridge University Press . 0-521-77290-7 .
• News: C.J. . Budd . M.D. . Piggott . Geometric Integration and its Applications . Handbook of Numerical Analysis . 11 . 35–139 . 2003 . Elsevier . 10.1016/S1570-8659(02)11002-7 . 9780444512475 .
• News: Pilwon . Kim . Invariantization of Numerical Schemes Using Moving Frames . BIT Numerical Mathematics . 47 . 525–546 . 2007 . 3 . Springer . 10.1007/s10543-007-0138-8 .

Notes and References

1. Pilwon Kim (2006), " Invariantization of Numerical Schemes Using Moving Frames"