CONSERVATIVE-DISSIPATIVE INTEGRATORS FOR REVERSIBLE-IRREVERSIBLE SYSTEMS

Boris Andrews

In preparation (Draft available on request)

The GENERIC formalism extends Hamiltonian systems to include both:

a conserved energy
a non-decreasing entropy

FULL DETAILS

The general GENERIC ODE in \(\mathbf{x} : \mathbb{R}_+ \to \mathbb{R}^d\) is \[ \dot{\mathbf{x}} = L(\mathbf{x})\nabla E(\mathbf{x}) + M(\mathbf{x})\nabla S(\mathbf{x}). \] Here, \(E, S : \mathbb{R}^d \to \mathbb{R}\) are the (conserved) energy and (non-decreasing) entropy, and \(L, M : \mathbb{R}^d \to \mathbb{R}^{d\times d}\) are the skew-symmetric (Poisson) matrix and positive-semidefinite (friction) matrix. With the following orthogonality conditions, \[ \nabla S(\mathbf{x})^\top L(\mathbf{x}) = 0, \qquad \nabla H(\mathbf{x})^\top M(\mathbf{x}) = 0, \] the conservation of \(E\) and non-dissipation of \(S\) can be identified by testing against \(\nabla E\) and \(\nabla S\) respectively. Extending to PDEs is fiddly (for the introduction of Fréchet derivatives) but similar.

As the name suggests, this is extremely general. Examples of such systems include:

the compressible Navier–Stokes equations
the Boltzmann equation
pretty much any irreversible thermodynamic system

We can apply the framework from mine and Patrick Farrell’s preprint to preserve both the conservative and non-dissipation structures. As such, we have a general way to construct structure-preserving finite element methods for any of the above systems, with arbitrary finite elements and at arbitrary order in space and time.

These properties are crucial for accurately capturing the dynamics of these systems.

(Further details available soon!)

This scheme can be viewed as a special case of my earlier work with Patrick Farrell, on general constructions for conservative and dissipative finite element integrators.

Problem	Reward
Conservative integrators for PDEs with arbitrarily many invariants	★★★★★
Dissipative integrators for ODEs with arbitrarily many dissipated quantities	★★★★★
Application of the auxiliary variable framework to a viscoelastic fluid system	★★☆☆☆
Structure-preserving integrators for compressible MHD	★☆☆☆☆

TALKS

2025

Invited talk, Brown Unversity

⬆️ UPCOMING | PAST ⬇️

2024

External ("tiny desk") seminar, Rice University