10–13 Jun 2024
Inria Center at the University of Lille
Europe/Paris timezone

Designing conservative and accurately dissipative numerical integrators in time

13 Jun 2024, 09:30
Amphitheater, Building B (Inria Center at the University of Lille)

Amphitheater, Building B

Inria Center at the University of Lille

Parc scientifique de la Haute-Borne, 40 avenue Halley, 59650 Villeneuve d'Ascq – France


Patrick E. Farrell (University of Oxford)


Numerical methods for the simulation of transient systems with
structure-preserving properties are known to exhibit greater accuracy and
physical reliability, in particular over long durations. These schemes are
often built on powerful geometric ideas for broad classes of problems, such as
Hamiltonian or reversible systems. However, there remain difficulties in
devising higher-order- in-time structure-preserving discretizations for
nonlinear problems, and in conserving non-polynomial invariants.

In this work we propose a new, general framework for the construction of
structure-preserving timesteppers via finite elements in time and the systematic
introduction of auxiliary variables. The framework reduces to Gauss methods
where those are structure-preserving, but extends to generate arbitrary-order
structure-preserving schemes for nonlinear problems, and allows for the
construction of schemes that conserve multiple higher-order invariants. We
demonstrate the ideas by devising novel schemes that exactly conserve all known
invariants of the Kepler and Kovalevskaya problems, high-order energy-conserving
and entropy-dissipating schemes for the compressible Navier–Stokes equations,
and multi-conservative schemes for the Benjamin-Bona-Mahony equation.

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