ENFORCING CONSERVATION LAWS AND DISSIPATION INEQUALITIES NUMERICALLY VIA AUXILIARY VARIABLES

Boris Andrews `|` Patrick Farrell

29.APR.2025 (arXiv)

    THIS REPRESENTS PART 1 OF AN EARLIER MANUSCRIPT, CURRENTLY BEING PARTITIONED INTO MULTIPLE SUBMISSIONS 

    CLICK HERE TO CHECK IT OUT!

We propose a general strategy for enforcing multiple conservation laws and dissipation inequalities in the numerical solution of initial value problems. […] We demonstrate these ideas by their application to the Navier-Stokes equations. We generalize […] the energy-dissipating and helicity-tracking scheme of Rebholz for the incompressible […] equations, and devise the first time discretization of the compressible equations that conserves mass, momentum, and energy, and provably dissipates entropy.

FULL 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 the first time discretization of the compressible equations that conserves mass, momentum, and energy, and provably dissipates entropy.

In my earlier work with Patrick Farrell, we proposed a framework for the construction of finite-element integrators that preserve multiple conservation laws and dissipation inequalities, alongside various applications to different PDE systems. This preprint serves as part 1 of a partition of this work.

We re-establish the framework, including alongside the discussions of its applications to the Navier–Stokes equations, deriving (to arbitrary order) integrators that:

for the incompressible equations, dissipate energy and, in the ideal case, conserve helicity;
for the compressible equations, conserve mass, momentum and energy and generate entropy.

For further details, check out my earlier manuscript here.

As stated above, this work represents part 1 of a resubmission of an earlier manuscript with Patrick Farrell, partitioned into multiple parts.

For a neat and related application of these ideas to a problem in magnetic relaxation that really highlights their importance, check out my subsequent work with Mingdong He, Patrick Farrell & Kaibo Hu, on structure-preserving integrators for the magneto-frictional equations.

Problem	Reward
Application of the auxiliary variable framework to a viscoelastic fluid system	★★☆☆☆
Structure-preserving integrators for compressible MHD	★☆☆☆☆

CO-AUTHORS

Patrick Farrell

Affiliation | University of Oxford
Joint publications | High-order conservative and accurately dissipative numerical integrators via auxiliary variables / Topology-preserving discretization for the magneto-frictional equations arising in the Parker conjecture / Enforcing conservation laws and dissipation inequalities numerically via auxiliary variables / An augmented Lagrangian preconditioner for natural convection at high Reynolds number*

TALKS

2025

Biennial Numerical Analysis Conference, University of Strathclyde
Numerical Analysis Group internal seminar, University of Oxford

⬆️ UPCOMING | PAST ⬇️

Numerical Mathematics & Scientific Computing seminar, Rice University
SIAM CSE, Fort Worth, Texas
Scientific Computing seminar, Brown Unversity

2024

External ("tiny desk") seminar, Rice University
Computing Division technical meeting, UKAEA
Firedrake User Meeting, University of Oxford
PDEsoft, University of Cambridge
Finite Element Fair, University College London (UCL)
Exploiting Algebraic and Geometric Structure in Time-integration Methods workshop, University of Pisa
UKAEA PhD student engagement day, UKAEA
Junior Applied Mathematics Seminar, University of Warwick

2023

ICIAM, Waseda University
Numerical Analysis Group internal seminar, University of Oxford
Junior Applied Mathematics Seminar, University of Oxford
Met Office presentation, University of Oxford