BORIS ANDREWS
Postdoctoral Research Associate, University of Oxford (he / him)
ENFORCING CONSERVATION LAWS AND DISSIPATION INEQUALITIES NUMERICALLY VIA AUXILIARY VARIABLES
Boris Andrews | Patrick Farrell
29.APR.2025 (arXiv) | 31.DEC.2025 (SISC)
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 a time discretization of the compressible equations that conserves mass, momentum, and energy, and provably dissipates entropy.
No matter whether you’re modelling a fluid, a plasma, any mechanical system, whatever, your partial differential equations (PDEs) almost always possess some fundamental physical structures. These might be include conserved quantities like mass, momentum, or energy, or dissipated quantites such as (as famously dictated by the Second Law of Thermodynamics) entropy.
In the continuous world, these laws are mathematical certainties. Exact solutions to our PDE models must satisfy these laws.
In the discrete world of numerical simulations, however, they’re often the first casualties. Numerical simulations give approximate solutions; approximate solutions only necessarily satisfy physical laws approximately.
When a solver fails to respect these invariants, simulations can drift. They can become unstable. They can produce physically impossible results, like the coffee in your cup magically generating its own energy and jumping onto your desk.
That’s not something you want to just “accept” when you’re simulating the air flowing past the wings of an aircraft…
The challenge
Conserving simple, quadratic quantities (e.g. the (L^2) energy in Navier–Stokes) is pretty well-understood (in particular in finite element discretisations). The challenge spikes dramatically when dealing with non-quadratic quantities of interest, in particular when there are multiple such structures.
Standard time-stepping schemes generally force an unhappy compromise:
- Either accept some drift in your conserved quantities,
- or introduce some “artificial dissipation” (a fictious damping force to keep the simulation stable at the cost of accuracy).
Conservative or accurately dissipative PDE integrators have then generally historically been constructed on an ad-hoc system-by-system basis.
The literature to date
If you’re familiar with the field, you may know of some of these integrators. In particular, I’ll highlight two disparate bodies of work:
- Auxiliary variables (i.e. mixed finite element methods): These have been effective in preserving the behaviour of multiple quantities of interest, but have struggled with non-quadratic properties.
- Finite elements in time: These have been effective in preserving the behaviour of arbitrary, potentially non-quadratic, quantities of interest, but have generally only been applicable for a single structure to be preserved.
Unlike classical Runge–Kutta (RK) methods which treat time as a sequence of discrete points, CPG treats time as a continuous dimension (similar to space in traditional finite elements). This works for arbitrary order in time, and in theory is no more computationally difficult than a classical implicit RK method: The \(S\)-stage CPG method has no more degrees of freedom than an \(S\)-stage implicit RK method.
To see why CPG is so vital for non-quadratic quantities of interest, you'll just have to read the paper!
Our work unifies these distinct ideas.
Yet, this belief is not entirely accurate...
Symplecticity enhances the collective behaviour of a group of simulations.
Don't get me wrong, this is great! If I'm e.g. simulating the motion of a host of example asteroids passing through our solar system, it'd nice to have an accurate prediction about what percent of these might hit Earth.
However, this does not guarantee energy conservation.
In fact, as noted by Ge & Marsden in 1988:
Symplectic integrators cannot (in general) conserve energy.
Our proposed framework
We introduce a general, arbitrary-order framework that doesn’t just give you one integrator; we provide a sequence of procedural steps that allows you to turn pretty much any specific PDE model of your choosing into a structure-preserving scheme. The core innovation is the systematic joint use of auxiliary variables and finite elements in time.
Here’s the key idea:
- Quantify the physics: For each conservation or dissipation laws you want to preserve, identify a certain “associated test function” (explained in more detail in the paper).
- Project: Make a mixed finite element discretisation by introducing these associated test functions as auxiliary variables (projections of these associated test functions onto the discrete test set).
- Discretise in time: Discretise in time using finite elements in time.
It’s as simple and general as that! All those existing mixed finite element methods and finite-element-in-time schemes mentioned above for example just turn out actually to be special cases of our wider framework.
A neat little demo: compressible Navier–Stokes
To prove the power of this framework, we applied it to one of the most challenging systems in computational physics: the compressible Navier–Stokes equations.
This is a nasty system with a lot of physically important structures:
- Mass conservation
- Momentum conservation
- Energy conservation
- Entropy dissipation/generation (i.e. the Second Law of Thermodynamics)
We use our framework to preserve all of these properties simultaneously. To our knowledge, this is the first scheme to achieve this in the general setting we consider.
Just to get you excited, here’s a fun video of some numerical results from the paper, illustrating a numerical shockwave simulation using our discretisation:
Ready to upgrade your integrator?
If you’re looking to build simulations that are both simple and physically meaningful, we invite you to explore the full details of the framework, proofs, and implementation strategies in the manuscript.
We would both gladly discuss it further!
VIDEOS
Check out Patrick’s Langtangen Seminar (22.APR.2025) at Simula below:
His earlier ACM Colloquium (13.NOV.2024) at the University of Edinburgh and Heriot-Watt University can be found here.
RELATED WORKS
In a recently submitted manuscript, Patrick Farrell and I apply these ideas to two very general classes of problems:
- Arbitrary ODEs with multiple invariants: We construct a general integrator that’s able to preserve as many invariants as desired
- Arbitrary ODEs and PDEs from the GENERIC formalism: We’re similarly able to construct a general integrator that preserves both energy conservation and entropy generation. Notably, this includes the Boltzmann equation!
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.
RELATED OPEN PROBLEMS
CO-AUTHORS
Patrick Farrell
- Affiliation
|University of Oxford - Joint publications
|Helicity-preserving finite element discretization for magnetic relaxation / Enforcing conservation laws and dissipation inequalities numerically via auxiliary variables / Conservative and dissipative discretisations of multi-conservative ODEs and GENERIC systems / An augmented Lagrangian preconditioner for natural convection at high Reynolds number* / Automated Galerkin time stepping in Irksome*
TALKS
2026
- SciCADE, University of Edinburgh
- ECCOMAS WCCM, Munich #
- Invited Seminar, Chinese Academy of Sciences
2025
- Biennial Numerical Analysis Conference, University of Strathclyde
- 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