Self-scaling collocation methods preserving constants of motion

We present a class of self-scaling collocation integrators which, unlike explicit Runge-Kutta integrators, will automatically preserve quadratic constants of motion such as energy, linear momentum, and angular momentum, without the need for small stepsizes. However, collocation integrators in general require numerical optimization to solve an implicit set of equations which are difficult and computationally expensive to solve when the problem’s Jacobian becomes ill-conditioned. Therefore, the integrators we present automatically scale the collocation conditions using a generalized eigenvalue problem (GEVP) approach, and exploit the Jacobian’s structure to improve the optimization algorithm’s precision and speed. Numerical results show that the presented integrators preserve invariants in a broader class of integration problems, with an improvement of approximately one order of magnitude over traditional Runge-Kutta 4(5) schemes.