The long-time integration of systems of differential equations by many numerical integrators is not reliable because of the buildup of error, which can lead to physically incorrect solutions. Symplectic integrators, when they can be constructed, do a better job of preserving important physical properties such as energy.

This paper considers a particular three-dimensional system of differential equations that is a simplified case of the streamlined equations for three-dimensional Arnold-Beltrami-Childress flows. Tippett observes that the differential equations may be rewritten in a two-dimensional Hamiltonian form. Using Yohida’s approach, a family of symplectic integrators is constructed, and several symplectic integrators are implemented. This involves the evaluation of elliptic functions and integrals. Computational experiments are performed, and the method is compared to a fourth-order Runge-Kutta method.

The paper is interesting as an academic special case and because the symplectic integrators appear to perform well even though they have variable step sizes. However, the methods are computationally intensive, and the integrators derived here can only be used on this problem. The derivations given would be hard to apply to a larger, more complex problem. The only comparison to other methods is to a fourth-order Runge-Kutta method. No justification is given for the use of this method or this implementation.