Generalized Symplectic Integrators

Abstract

Abstract. Symplectic integrators offer vastly superior performance over traditional numerical techniques for conservative dynamical systems, but their application to dissipative systems is inherently difficult due to dissipative systems' lack of symplectic structure. Leveraging the intrinsic variational structure of higher-order dynamics, this paper presents a general technique for adapting existing symplectic integration schemes to arbitrary dynamical systems (conservative or dissipative). Utilizing the present method, any existing symplectic integrator can be generalized and made to incorporate any number of tuning parameters, which can be tailored for optimal performance. Indeed, for the linear test problem considered here, the tuning parameters can be chosen to achieve zero numerical error at every time-step for any given step size. This “tunability” emerges organically out of the intrinsic symplectic structure of the higher-order formulation. Another interesting result is that the process of introducing tuning parameters automatically circumvents the need to supply additional initial conditions for the higher-order formulation. That is, doubling the order of the equation does not actually introduce any additional complexity from a numerical perspective. For illustration, a simple scheme involving two tuning parameters is proposed, and the optimal parameter values are computed in general for two limit cases. In both cases, the resulting scheme is unconditionally stable and outperforms the implicit Euler method. These results open the door to an entirely new class of symplectic integrators that can be tailored to suit specific problems. Future work will focus on tailoring such schemes to viscous flows modeled by the Navier–Stokes equations.