Unconditionally stable symplectic integrators for the Navier-Stokes equations and other dissipative systems (2411.13569v1)

Abstract: Symplectic integrators offer vastly superior performance over traditional numerical techniques for conservative dynamical systems, but their application to \emph{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 applying existing symplectic integration schemes to dissipative systems, with particular emphasis on viscous fluids modeled by the Navier-Stokes equations. Two very simple such schemes are developed here. Not only are these schemes unconditionally stable for dissipative systems, they also outperform traditional methods with a similar degree of complexity in terms of accuracy for a given time step. For example, in the case of viscous flow between two infinite, flat plates, one of the schemes developed here is found to outperform both the implicit Euler method and the explicit fourth-order Runge-Kutta method in predicting the velocity profile. To the authors' knowledge, this is the very first time that a symplectic integration scheme has been applied successfully to the Navier-Stokes equations. We interpret the present success as direct empirical validation of the canonical Hamiltonian formulation of the Navier-Stokes problem recently published by Sanders~\emph{et al.} More sophisticated symplectic integration schemes are expected to exhibit even greater performance. It is hoped that these results will lead to improved numerical methods in computational fluid dynamics.

• The paper presents unconditionally stable symplectic schemes that extend traditional conservative methods to dissipative systems.
• It transforms the Navier-Stokes equations into a Hamiltonian framework, thereby enabling enhanced numerical accuracy.
• Numerical tests, including viscous flow and quadratic drag cases, demonstrate clear improvements over implicit Euler and RK4 methods.

An Overview of Unconditionally Stable Symplectic Integrators for Dissipative Systems

This paper addresses a nuanced challenge in numerical analysis that involves extending the applicability of symplectic integrators, traditionally renowned for their utility in conservative systems, to dissipative systems like those modeled by the Navier-Stokes equations. The authors propose a paradigm shift through a technique that effectively redefines non-Hamiltonian systems into a framework that accommodates the symplectic integration schemes typically reserved for Hamiltonian systems.

Key Contributions and Methodology

Symplectic integrators are distinguished for their ability to conserve the symplectic structure in Hamiltonian systems, thus extending their efficacy to dissipative systems presents a significant methodological advancement. The paper outlines a general technique exploiting the variational structure of higher-order dynamics. By transforming a dissipative system such as one governed by the Navier-Stokes equations into a higher-dimensional Hamiltonian framework, the authors deploy symplectic integrators on these newly formulated systems.

Of particular importance are the two simple symplectic schemes the authors developed. The schemes are not only unconditionally stable for dissipative systems, but also demonstrate superior accuracy relative to conventional methods like the implicit Euler and explicit fourth-order Runge-Kutta methods in specific scenarios. This is exemplified using a case paper of viscous flow between two infinite plates where the new schemes provide more precise predictions of velocity profiles.

Numerical Results and Comparative Analysis

The numerical experiments conducted reveal compelling results. Notably, in the test problem involving viscous flow, the symplectic integrator outperformed both the implicit Euler and RK4 methods when estimating velocity profiles. The implications are reinforced through further scenarios, such as the treatment of quadratic drag. Here the novel schemes continue to show a marked improvement over traditional methods, especially at larger time increments. The analysis articulates conditions under which these symplectic integrators maintain stability and accuracy.

Importantly, the paper expands upon the symplectic map framework to showcase that even simple symplectic integrators like MacKay’s method can be employed advantageously in non-Hamiltonian contexts if appropriately structured.

Implications for Computational Fluid Dynamics

The implications of this research are particularly pertinent to the field of computational fluid dynamics (CFD). Traditional CFD methods may face challenges concerning stability and accuracy when handling complex viscous flow problems. By utilizing symplectic integrators within this new framework, there is potential for significant improvements in computational performance. Future advancements could foreseeably integrate these methodologies into commercial CFD software, providing enhanced tools for scientific inquiry and engineering applications.

Theoretical Acknowledgments and Future Directions

Theoretically, the work substantiates the Hamiltonian reformulation of dissipative systems through empirical validation. Such validation remarks on the potential breadth of application for symplectic integrators beyond their conventional scope. Future research could further interrogate more sophisticated symplectic schemes and refine the balance between accuracy and computational complexity across diverse fluid dynamics scenarios.

In conclusion, the paper provides a substantive extension of symplectic integrators to dissipative systems. By reframing dynamics within a higher-dimensional symplectic space, it sets the stage for future computational advancements and broadens the theoretical understanding of fluid dynamics models.