Designing Conservative and Accurately Dissipative Numerical Integrators in Time

Explore a seminar on designing conservative and accurately dissipative numerical integrators in time, presented by Patrick Farrell from the University of Oxford. Delve into the world of structure-preserving numerical methods for simulating transient systems, focusing on their enhanced accuracy and physical reliability over extended periods. Examine the challenges in developing higher-order-in-time structure-preserving discretizations for nonlinear problems and conserving non-polynomial invariants. Discover a novel, general framework for constructing structure-preserving time steppers using finite elements in time and the systematic introduction of auxiliary variables. Investigate how this framework extends beyond Gauss methods to generate arbitrary-order structure-preserving schemes for nonlinear problems and allows for the creation of schemes conserving multiple higher-order invariants. Explore practical applications of these concepts through examples such as exactly conserving all known invariants of the Kepler and Kovalevskaya problems, developing arbitrary-order schemes for compressible Navier-Stokes equations that conserve mass, momentum, and energy while dissipating entropy, and creating multi-conservative schemes for the Benjamin-Bona-Mahony equation.