A thermodynamically consistent discretization of 1D thermal-fluid models using their metriplectic 4-bracket structure

Barham, William; Morrison, Philip J.; Zaidni, Azeddine

doi:10.1016/j.cnsns.2025.108683

A thermodynamically consistent discretization of 1D thermal-fluid models using their metriplectic 4-bracket structure

Journal Article · Mon Feb 24 00:00:00 EST 2025 · Communications in Nonlinear Science and Numerical Simulation

DOI:https://doi.org/10.1016/j.cnsns.2025.108683· OSTI ID:2997714

^[1]; Morrison, Philip J. ^[1]; Zaidni, Azeddine ^[2]

The University of Texas at Austin, TX (United States)
Mohammed VI Polytechnic University, Marrakech-Safi (Morocco)

Thermodynamically consistent models in continuum physics, i.e. models which satisfy the first and second laws of thermodynamics, may be expressed using the metriplectic formalism. In this work, we leverage the structures underlying this modeling formalism to preserve thermodynamic consistency in discretizations of a fluid model. The procedure relies (1) on ensuring that the spatial semi-discretization retains certain symmetries and degeneracies of the Poisson and metriplectic 4-brackets, and (2) on the use of an appropriate energy conserving time-stepping method. Here, the minimally simple yet nontrivial example of a one-dimensional thermal-fluid model is treated. It is found that preservation of the requisite symmetries and degeneracies of the 4-bracket is relatively simple to ensure in Galerkin spatial discretizations, suggesting a path forward for thermodynamically consistent discretizations of more complex fluid models using more specialized Galerkin methods.

View Accepted Manuscript (DOE)

Research Organization:: The University of Texas at Austin, TX (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Fusion Energy Sciences (FES)

Grant/Contract Number:: FG02-04ER54742

OSTI ID:: 2997714

Alternate ID(s):: OSTI ID: 2539973

Journal Information:: Communications in Nonlinear Science and Numerical Simulation, Journal Name: Communications in Nonlinear Science and Numerical Simulation Vol. 145; ISSN 1007-5704

Publisher:: Elsevier BVCopyright Statement

Country of Publication:: United States

Language:: English

References (19)

How is local material entropy production represented in a numerical model?: Local Material Entropy Production Gassmann, Almut; Herzog, Hans-Joachim Quarterly Journal of the Royal Meteorological Society, Vol. 141, Issue 688 https://doi.org/10.1002/qj.2404	journal	July 2014
Fully Conservative Higher Order Finite Difference Schemes for Incompressible Flow Morinishi, Y.; Lund, T. S.; Vasilyev, O. V. Journal of Computational Physics, Vol. 143, Issue 1 https://doi.org/10.1006/jcph.1998.5962	journal	June 1998
Linear energy-preserving integrators for Poisson systems Cohen, David; Hairer, Ernst BIT Numerical Mathematics, Vol. 51, Issue 1 https://doi.org/10.1007/s10543-011-0310-z	journal	January 2011
A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations Shakib, Farzin; Hughes, Thomas J. R.; Johan, Zdeněk Computer Methods in Applied Mechanics and Engineering, Vol. 89, Issue 1-3 https://doi.org/10.1016/0045-7825(91)90041-4	journal	August 1991
A paradigm for joined Hamiltonian and dissipative systems Morrison, Philip J. Physica D: Nonlinear Phenomena, Vol. 18, Issue 1-3 https://doi.org/10.1016/0167-2789(86)90209-5	journal	January 1986
Energetically consistent model reduction for metriplectic systems Gruber, Anthony; Gunzburger, Max; Ju, Lili Computer Methods in Applied Mechanics and Engineering, Vol. 404 https://doi.org/10.1016/j.cma.2022.115709	journal	February 2023
A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier–Stokes equations Palha, A.; Gerritsma, M. Journal of Computational Physics, Vol. 328 https://doi.org/10.1016/j.jcp.2016.10.009	journal	January 2017
Thermodynamically consistent Cahn–Hilliard–Navier–Stokes equations using the metriplectic dynamics formalism Zaidni, Azeddine; Morrison, Philip J.; Benjelloun, Saad Physica D: Nonlinear Phenomena, Vol. 468 https://doi.org/10.1016/j.physd.2024.134303	journal	November 2024
Compatible finite element methods for geophysical fluid dynamics Cotter, Colin J. Acta Numerica, Vol. 32 https://doi.org/10.1017/S0962492923000028	journal	May 2023
Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems Cockburn, Bernardo; Shu, Chi-Wang Journal of Scientific Computing, Vol. 16, Issue 3, p. 173-261 https://doi.org/10.1023/A:1012873910884	journal	September 2001
Metriplectic integrators for the Landau collision operator Kraus, Michael; Hirvijoki, Eero Physics of Plasmas, Vol. 24, Issue 10 https://doi.org/10.1063/1.4998610	journal	October 2017
A new class of energy-preserving numerical integration methods Quispel, G. R. W.; McLaren, D. I. Journal of Physics A: Mathematical and Theoretical, Vol. 41, Issue 4 https://doi.org/10.1088/1751-8113/41/4/045206	journal	January 2008
Geometric integration using discrete gradients McLachlan, Robert I.; Quispel, G. R. W.; Robidoux, Nicolas Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Vol. 357, Issue 1754 https://doi.org/10.1098/rsta.1999.0363	journal	April 1999
Inclusive curvaturelike framework for describing dissipation: Metriplectic 4-bracket dynamics Morrison, Philip J.; Updike, Michael H. Physical Review E, Vol. 109, Issue 4 https://doi.org/10.1103/PhysRevE.109.045202	journal	April 2024
Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics Morrison, Philip J.; Greene, John M. Physical Review Letters, Vol. 45, Issue 10 https://doi.org/10.1103/PhysRevLett.45.790	journal	September 1980
Hamiltonian description of the ideal fluid Morrison, P. J. Reviews of Modern Physics, Vol. 70, Issue 2 https://doi.org/10.1103/RevModPhys.70.467	journal	April 1998
A Skew-Symmetric Discontinuous Galerkin Spectral Element Discretization and Its Relation to SBP-SAT Finite Difference Methods Gassner, Gregor J. SIAM Journal on Scientific Computing, Vol. 35, Issue 3 https://doi.org/10.1137/120890144	journal	January 2013
Variational and thermodynamically consistent finite element discretization for heat conducting viscous fluids Gawlik, Evan S.; Gay-Balmaz, François Mathematical Models and Methods in Applied Sciences, Vol. 34, Issue 02 https://doi.org/10.1142/S0218202524500027	journal	January 2024
Irksome: Automating Runge–Kutta Time-stepping for Finite Element Methods Farrell, Patrick E.; Kirby, Robert C.; Marchena-Menéndez, Jorge ACM Transactions on Mathematical Software, Vol. 47, Issue 4 https://doi.org/10.1145/3466168	journal	December 2021

Similar Records

Inclusive curvaturelike framework for describing dissipation: Metriplectic 4-bracket dynamics

Journal Article · Thu Apr 04 20:00:00 EDT 2024 · Physical Review E · OSTI ID:2997874

Thermodynamically consistent Cahn–Hilliard–Navier–Stokes equations using the metriplectic dynamics formalism

Journal Article · Tue Jul 23 20:00:00 EDT 2024 · Physica D: Nonlinear Phenomena · OSTI ID:2998044

Metriplectic four-bracket algorithm for constructing thermodynamically consistent dynamical systems

Journal Article · Thu Jul 31 20:00:00 EDT 2025 · Physical Review E · OSTI ID:2998046

Related Subjects

Hamiltonian structure
Metriplectic dynamics
Navier–Stokes–Fourier
Structure-preserving discretization
Thermodynamic consistency

A thermodynamically consistent discretization of 1D thermal-fluid models using their metriplectic 4-bracket structure

Citation Formats

References (19)

Similar Records

Related Subjects