Physics of symplectic integrators: Perihelion advances and symplectic corrector algorithms

Symplectic integrators evolve dynamical systems according to modified Hamiltonians whose error terms are also well-defined Hamiltonians. The error of the algorithm is the sum of each error Hamiltonian’s perturbation on the exact solution. When symplectic integrators are applied to the Kepler problem, these error terms cause the orbit to precess. In this work, by developing a general method of computing the perihelion advance via the Laplace-Runge-Lenz vector even for nonseparable Hamiltonians, I show that the precession error in symplectic integrators can be computed analytically. It is found that at leading order, each paired error Hamiltonians cause the orbit to precess oppositely by exactly the same amount after each period. Hence, symplectic corrector, or process integrators, which have equal coefficients for these paired error terms, will have their precession errors cancel at that order after each period. With the use of correctable algorithms, both the energy and precession error are of effective order $n+2$ where $n$ is the nominal order of the algorithm. Thus the physics of symplectic integrators determines the optimal algorithm for integrating long-time periodic motions.

Figure 1

Rotation of the Laplace-Runge-Lenz vector due to second-order error Hamiltonian $-H_{VTV}$ and $H_{TTV}$ in algorithms I and II. Each algorithm rotates the LRL vector differently in time, but by exactly the same amount after one period. Most of the rotation takes place near the midperiod. The solid line gives the theoretical value of $-45.33318$.

Figure 2

Rotation of the Laplace-Runge-Lenz vector due to fourth-order error Hamiltonians $H_{VTTTV}$ and $-H_{TTTTV}$. Because the fourth-order error terms are more singular, the rotation takes place over a narrower range near midperiod. The solid line gives the theoretical value of $-5933.72$

Figure 3

Rotation of the Laplace-Runge-Lenz vector for three second-order symplectic algorithms: velocity-Verlet (VV), Takahashi-Imada (TI), and the nonforward corrector algorithm (NF). The solid line gives the theoretical rotation value of the VV algorithm: $\Delta {\theta }_{TTV}∕24=-1.8888$.

Figure 4

Rotation of the Laplace-Runge-Lenz vector for three fourth-order integrators: algorithm $C$, algorithm $C^{\prime }$ with added gradient term to force the rotation angle back to zero, and the true symplectic corrector algorithm $4S$. As with most integrators, algorithm $C$’s precession error does not return to zero.

Figure 5

Energy error functions of algorithms $C$, $C^{\prime }$, and $4S$. Algorithm $4S$’s maximum error is three times smaller than that of $C$.