the numerical solution of partial differential equations (PDEs) with applications in physics and chemistry. His research
includes adaptive finite element discretizations, Adjoints and PDE-constrained optimization problems, and Bifurcation
analysis of nonlinear PDEs. He has received numerous prestigious awards, including the LMS Whitehead Prize, the
Broyden Prize in Optimization, the Wilkinson Prize for Numerical Software, and the IMA Leslie Fox Prize in Numerical
Analysis. A core developer of the Firedrake projects, he actively contributes to advancing scientific computing.
Title: Designing conservative and accurately dissipative numerical integrators in time
Abstract: 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, arbitrary-order schemes for the compressible Navier–Stokes
equations that conserve mass, momentum, and energy, and provably dissipate entropy, and
multi-conservative schemes for the Benjamin-Bona-Mahony equation.
Date and time: Wednesday, October 16, 2024, 5:00 PM (IST)
