Midpoint rule as a variational-symplectic integrator: Hamiltonian systems

Numerical algorithms based on variational and symplectic integrators exhibit special features that make them promising candidates for application to general relativity and other constrained Hamiltonian systems. This paper lays part of the foundation for such applications. The midpoint rule for Hamilton’s equations is examined from the perspectives of variational and symplectic integrators. It is shown that the midpoint rule preserves the symplectic form, conserves Noether charges, and exhibits excellent long-term energy behavior. The energy behavior is explained by the result, shown here, that the midpoint rule exactly conserves a phase space function that is close to the Hamiltonian. The presentation includes several examples.

Figure 1

Discretization in time. The nodes are labeled $n=0,\dots ,N$ and the zones (or time intervals) are labeled $n=1,\dots ,N$. The coordinates and momenta are node-centered, the Hamiltonian function is zone-centered.

Figure 2

The amplitude $x$ for the coupled harmonic oscillator as a function of time. The VI simulation produces the solid curve that tracks the exact solution (dashed curve) fairly closely. The other solid curve is obtained from RK2.

Figure 3

Energy error for the coupled harmonic oscillator for VI. The solid curve has timestep $\Delta t=0.01$. The dashed curve shows the error divided by 100 for $\Delta t=0.1$. The results are displayed for the first and last $\sim 12$ time units; the total run time was $t=1\times {10}^6$.

Figure 4

Energy error for the coupled harmonic oscillator for RK2. The solid curve has timestep $\Delta t=0.01$. The dashed curve shows the error divided by 8 for $\Delta t=0.02$.

Figure 5

Energy error for the coupled harmonic oscillator for RK4. The solid curve has timestep $\Delta t=0.01$. The dashed curve shows the error divided by 32 for $\Delta t=0.02$.

Figure 6

Phase space diagram for the pendulum. The initial data occupy a circle around $x=\pi /2$, $p=0$, and are evolved for just under 10 oscillation periods. Final data is shown for VI and RK2, for low and high resolutions.

Figure 7

Energy as a function of time for a particle in a one-dimensional potential, $V(x)=-x^{6/5}$, obtained with VI. As expected, the error in energy grows linearly with time.

Figure 8

Angular momentum as a function of time for motion in a central potential. The solid curve is obtained from RK2. The constant, dashed line is obtained with the variational integrator.