HIGH-ORDER CONSERVATIVE-DISSIPATIVE INTEGRATORS FOR REVERSIBLE-IRREVERSIBLE SYSTEMS

Boris Andrews

Upcoming, 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!)

Talks

2025

• Invited talk, Brown Unversity

2024

⬆️ UPCOMING | PAST ⬇️
• External ("tiny desk") seminar, Rice University