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Related Experiment Video

Updated: Dec 25, 2025

Merging Ion Concentration Polarization between Juxtaposed Ion Exchange Membranes to Block the Propagation of the Polarization Zone
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Structure-preserving integrators for dissipative systems based on reversible- irreversible splitting.

Xiaocheng Shang1, Hans Christian Öttinger2

  • 1School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences
|March 24, 2020
PubMed
Summary
This summary is machine-generated.

We developed structure-preserving numerical integrators for dissipative systems using the General Equation for Nonequilibrium Reversible-Irreversible Coupling (GENERIC) framework. These integrators improve accuracy and preserve essential thermodynamic properties.

Keywords:
(conformal) symplecticGENERICdiscrete gradient methodsdissipative systemsmetriplecticstructure-preserving integrators

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Area of Science:

  • Computational Physics
  • Thermodynamics
  • Numerical Analysis

Background:

  • Dissipative systems possess an underlying thermodynamic structure governed by the General Equation for Nonequilibrium Reversible-Irreversible Coupling (GENERIC).
  • Standard numerical integrators often fail to preserve these crucial thermodynamic properties, leading to inaccuracies.
  • Designing integrators that respect the GENERIC structure is essential for reliable simulations of complex physical phenomena.

Purpose of the Study:

  • To develop a framework for constructing optimal structure-preserving numerical integrators for dissipative systems described by GENERIC.
  • To enhance the accuracy and stability of numerical simulations by preserving the system's inherent thermodynamic structure.
  • To provide a method for accurately controlling energy conservation and entropy production.

Main Methods:

  • A novel framework for constructing integrators by splitting the system into reversible and irreversible dynamics.
  • Utilizing symplectic methods (e.g., Verlet) for the reversible part, incorporating backward error analysis to derive a modified Hamiltonian and energy.
  • Constructing a modified friction matrix for the irreversible part to satisfy a modified degeneracy condition, solved via explicit midpoint methods when necessary.

Main Results:

  • The proposed structure-preserving integrators demonstrate superior accuracy in controlling energy conservation and entropy production compared to alternative schemes.
  • These integrators effectively preserve the conformal symplectic structure, particularly in linearly damped systems.
  • Numerical experiments validate the theoretical framework and highlight the practical advantages of the developed methods.

Conclusions:

  • The developed framework provides an effective approach for designing numerical integrators that preserve the thermodynamic structure of dissipative systems.
  • Structure-preserving integrators offer significant improvements in simulation accuracy and reliability for systems governed by GENERIC.
  • This work contributes to the advancement of computational methods in nonequilibrium thermodynamics and statistical physics.