Geometric integrator

In the mathematical field of numerical ODEs, 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 by$q(t)$ 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, is:$H(q,p)\; =\; T(p)+U(q)\; =\; frac\{1\}\{2\}p^2\; -\; cos\; q$,which gives Hamilton's equations:$(dot\; q,dot\; p)\; =\; (p,-sin\; q)$.It is natural to take the configuration space $Q$ of all $q$ to be the unitcircle $mathbb\; S^1$, so that $(q,p)$ lies on thecylinder $mathbb\; S^1\; imesmathbb\; R$. However, we will take$(q,p)inmathbb\; R^2$, simply because $(q,p)$-space isthen easier to plot. Define $z(t)\; =\; (q(t),p(t))^\{mathrm\; T\}$and $f(z)\; =\; (p,-sin\; q)^\{mathrm\; T\}$. Let us experiment byusing some simple numerical methods to integrate this system. As usual,we select a constant step-size $h$ and write$z\_k:=z(kh)$ for $kgeq\; 0$.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 theexplicit 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 $mathbb\; R^2$, the exacttrajectories and the numerical solutions of the system. For the explicitand implicit Euler methods we take $h=0.2$, and $z\_0\; =(0.5,0)$ and $(1.5,0)$ respectively; for the other twomethods we take $h=0.3$, and $z\_0\; =\; (0,0.7)$,$(0,1.4)$ and $(0,2.1)$.The explicit (resp. implicit) Euler method spirals out from (resp. into) the origin. The other two methods show the correct qualitativebehaviour, with the implicit midpoint rule agreeing with the exactsolution 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:$detfrac\{partialphi\_t\}\{partial\; (q\_0,p\_0)\}\; =\; 1$ for all $t$.This formula is easily verified by hand. For our pendulumexample we see that the numerical flow $Phi\_$mathrm{eE,h}:z_kmapsto z_{k+1} of the explicit Euler method is not area-preserving; viz.,:$detfrac\{partial\}\{partial\; (q\_0,p\_0)\}Phi\_$mathrm{eE,h}(z_0) = egin{vmatrix}1&h\-hcos q_0&1end{vmatrix} = 1+h^2cos q_0.A similar calculation can be carried out for the implicit Euler method,where the determinant is:$detfrac\{partial\}\{partial\; (q\_0,p\_0)\}Phi\_$mathrm{iE,h}(z_0) = (1+h^2cos q_1)^{-1}.However, the symplectic Euler method is area-preserving::mathrm{sE,h}(z_0) = egin{pmatrix}1&0\-hcos q_0&1end{pmatrix},thus $det(partialPhi\_$mathrm{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.

References

*Ernst Hairer, Christian Lubich and Gerhard Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations." Springer, Berlin, 2002. ISBN 3-540-43003-2.
*Ben Leimkuhler and Sebastian Reich, "Simulating Hamiltonian Dynamics." Cambridge University Press, 2005. ISBN 0-521-77290-7.