SIMD-vectorized implicit symplectic integrators can outperform explicit symplectic ones

Implicit Runge-Kutta schemes based on collocation with Gauss-Legendre nodes (IRKGL) possess attractive theoretical properties for long-term integration of Hamiltonian systems—they are symmetric, symplectic, and super-convergent—yet they have traditionally been considered less practical than explicit symplectic integrators. In this work, we challenge this conventional wisdom by introducing a stage-wise SIMD-vectorization approach that enables IRKGL schemes to outperform state-of-the-art explicit symplectic integrators for high-precision computations in double-precision floating-point arithmetic. Our approach reformulates the fixed-point iteration in terms of s-vectors (where s is the number of stages) to explicitly exploit Single Instruction Multiple Data (SIMD) capabilities of modern processors, significantly reducing computational overhead and enabling parallel evaluation of the right-hand side function across all stages. Additionally, we present a reformulation of IRKGL schemes for second-order ODEs that ensures exact symplecticity at the level of double-precision floating-point arithmetic, extending our previous approach for first-order systems to this setting. We demonstrate these ideas through IRKGL16, a Julia implementation of the 16th-order 8-stage IRKGL scheme that performs vectorization seamlessly and transparently to the user. Numerical experiments on several Hamiltonian problems with separable structure confirm the effectiveness of our approach.