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

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@article{1d8b9d780990448d9c5fc13520b33dda,
title = "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.",
keywords = "(conformal) symplectic, Discrete gradient methods, Dissipative systems, GENERIC, Metriplectic, Structure-preserving integrators",
author = "Xiaocheng Shang and {\"O}ttinger, {Hans Christian}",
year = "2020",
month = feb,
day = "12",
doi = "10.1098/rspa.2019.0446",
language = "English",
volume = "476",
journal = "Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences",
issn = "0080-4630",
publisher = "The Royal Society",
number = "2234",

}

RIS

TY - JOUR

T1 - Structure-preserving integrators for dissipative systems based on reversible-irreversible splitting

AU - Shang, Xiaocheng

AU - Öttinger, Hans Christian

PY - 2020/2/12

Y1 - 2020/2/12

N2 - 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.

AB - 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.

KW - (conformal) symplectic

KW - Discrete gradient methods

KW - Dissipative systems

KW - GENERIC

KW - Metriplectic

KW - Structure-preserving integrators

UR - http://www.scopus.com/inward/record.url?scp=85082030548&partnerID=8YFLogxK

U2 - 10.1098/rspa.2019.0446

DO - 10.1098/rspa.2019.0446

M3 - Article

VL - 476

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0080-4630

IS - 2234

M1 - 20190446

ER -