Structure-preserving integrators for dissipative systems based on reversible-irreversible splitting

Abstract

We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible-irreversible coupling). We present a frame-work to construct structure-preserving integrators by splitting the system into reversible and irreversible dynamics. The reversible part, which is often degenerate and reduces to a Hamiltonian form on its symplectic leaves, is solved by using a symplectic method (e.g., Verlet) with degenerate variables being left unchanged, for which an associated modified Hamiltonian (and subsequently a modified energy) in the form of a series expansion can be obtained by using backward error analysis. The modified energy is then used to construct a modified friction matrix associated with the irreversible part in such a way that a modified degeneracy condition is satisfied. The modified irreversible dynamics can be further solved by an explicit midpoint method if not exactly solvable. Our findings are verified by various numerical experiments, demonstrating the superiority of structure-preserving integrators over alternative schemes in terms of not only the accuracy control of both energy conservation and entropy production but also the preservation of the conformal symplectic structure in the case of linearly damped systems.

References in corpus (12)

• Robust and efficient configurational molecular sampling via Langevin Dynamics
• The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics
• Structure and structure-preserving algorithms for plasma physics
• On the numerical treatment of dissipative particle dynamics and related systems
• Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation
• Metriplectic Integrators for the Landau Collision Operator
• Adaptive Thermostats for Noisy Gradient Systems
• How to improve the accuracy of the discrete gradient method in the one-dimensional case
• Pairwise adaptive thermostats for improved accuracy and stability in dissipative particle dynamics
• Time-symmetric integration in astrophysics
• Assessing numerical methods for molecular and particle simulation
• Variational principles for dissipative (sub)systems, with applications to the theory of linear dispersion and geometrical optics

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• Exergetic Port-Hamiltonian Systems: Modelling Basics
• Metriplectic foundations of gyrokinetic Vlasov-Maxwell-Landau theory
• Stochastic Norton dynamics: An alternative approach for the computation of transport coefficients in dissipative particle dynamics