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Enforcing conservation laws and dissipation inequalities numerically via auxiliary variables

Boris D. Andrews, Patrick E. Farrell

TL;DR

This work introduces a general, order-agnostic framework for enforcing multiple conservation laws and dissipation inequalities in time-discretized PDEs by embedding auxiliary variables that project invariant-related test functions onto a discrete test space. The approach applies to both incompressible and compressible Navier–Stokes equations, enabling simultaneous preservation of mass, momentum, energy, and entropy (or helicity in the incompressible case) without reparametrization of the system. Central to the method is a systematic identification of associated test functions for the quantities of interest and the construction of a modified right-hand side via auxiliary variables, yielding fully discrete schemes that exactly reproduce the desired invariant behavior up to quadrature and solver tolerances. Numerical experiments on Hill vortices, inviscid Euler tests, and supersonic flows illustrate improved fidelity in conserving invariants and dissipating entropy, with code openly available for reproducibility and extension.

Abstract

We propose a general strategy for enforcing multiple conservation laws and dissipation inequalities in the numerical solution of initial value problems. The key idea is to represent each conservation law or dissipation inequality by means of an associated test function; we introduce auxiliary variables representing the projection of these test functions onto a discrete test set, and modify the equation to use these new variables. We demonstrate these ideas by their application to the Navier-Stokes equations. We generalize to arbitrary order the energy-dissipating and helicity-tracking scheme of Rebholz for the incompressible Navier-Stokes equations, and devise a time discretization of the compressible equations that conserves mass, momentum, and energy, and provably dissipates entropy.

Enforcing conservation laws and dissipation inequalities numerically via auxiliary variables

TL;DR

This work introduces a general, order-agnostic framework for enforcing multiple conservation laws and dissipation inequalities in time-discretized PDEs by embedding auxiliary variables that project invariant-related test functions onto a discrete test space. The approach applies to both incompressible and compressible Navier–Stokes equations, enabling simultaneous preservation of mass, momentum, energy, and entropy (or helicity in the incompressible case) without reparametrization of the system. Central to the method is a systematic identification of associated test functions for the quantities of interest and the construction of a modified right-hand side via auxiliary variables, yielding fully discrete schemes that exactly reproduce the desired invariant behavior up to quadrature and solver tolerances. Numerical experiments on Hill vortices, inviscid Euler tests, and supersonic flows illustrate improved fidelity in conserving invariants and dissipating entropy, with code openly available for reproducibility and extension.

Abstract

We propose a general strategy for enforcing multiple conservation laws and dissipation inequalities in the numerical solution of initial value problems. The key idea is to represent each conservation law or dissipation inequality by means of an associated test function; we introduce auxiliary variables representing the projection of these test functions onto a discrete test set, and modify the equation to use these new variables. We demonstrate these ideas by their application to the Navier-Stokes equations. We generalize to arbitrary order the energy-dissipating and helicity-tracking scheme of Rebholz for the incompressible Navier-Stokes equations, and devise a time discretization of the compressible equations that conserves mass, momentum, and energy, and provably dissipates entropy.
Paper Structure (23 sections, 1 theorem, 56 equations, 5 figures)

This paper contains 23 sections, 1 theorem, 56 equations, 5 figures.

Table of Contents

  1. Introduction
  2. General strategy: incompressible Navier--Stokes
  3. Definition of semi-discrete form
  4. Definition of timestepping scheme
  5. Identification of associated test functions
  6. Modification of right-hand side
  7. Construction of structure-preserving scheme
  8. Analysis: structure-preserving properties
  9. Numerical example: Hill spherical vortex
  10. Compressible Navier--Stokes
  11. Definition of semi-discrete form
  12. Definition of timestepping scheme
  13. Identification of associated test functions
  14. Introduction of auxiliary variables
  15. Modification of right-hand side
  16. ...and 8 more sections

Key Result

Theorem 2.4

\newlabelth:avcpg_sp0 Where solutions to eq:generalavcpgweak exist, they preserve the sign of the changes to the functionals $Q_q(u)$, $q=1, \dots, P$, across each timestep. In particular, if $Q_q(u)$ is conserved by the exact solution, then it is also conserved by the discretization, up to quadra

Figures (5)

  • Figure 1: Evolution of the energy $Q_1$ and helicity $Q_2$ in the $(Q_1, Q_2)$-preserving scheme \ref{['eq:incompnsavcpg']} and the $Q_1$-preserving scheme derived from \ref{['eq:incompnsintermediateF']}, with varying $\mathrm{Re} = 2^{2s}$ for $s \in \{0, \dots, 8\}$.
  • Figure 1: Error in the entropy $|Q_4 - Q_4(0)|$ over time within the inviscid test (Section \ref{['sec:inviscid_test']}) for implicit midpoint and our proposed scheme.
  • Figure 2: Cross-sections of streamlines of the velocity $\mathbf{u}$ for the Hill vortex at times $t \in \{0, 3 \cdot 2^{-6}\}$ in the $(Q_1, Q_2)$-preserving scheme \ref{['eq:incompnsavcpg']} and the $Q_1$-preserving scheme derived from \ref{['eq:incompnsintermediateF']} with $\mathrm{Re} = 2^{16}$. Coloring indicates $\|\mathbf{u}\|$.
  • Figure 2: Contours of the velocity magnitude $\|\mathbf{u}\|$, density $\rho$, temperature $\theta$, and specific entropy $s$ at times $t \in \{0, 1 \cdot 2^{-4}, 2 \cdot 2^{-4}\}$ in the supersonic test (Section \ref{['sec:supersonic_test']}).
  • Figure 3: Errors in different invariants over time within the supersonic test (Section \ref{['sec:supersonic_test']}) for implicit midpoint and our proposed scheme.

Theorems & Definitions (5)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3: Final discretization
  • Theorem 2.4: Structure preservation of the framework
  • Proof 1