High-order conservative and accurately dissipative numerical integrators via finite elements in time
Boris Andrews (University of Oxford)
https://www.maths.ox.ac.uk/people/boris.andrews
boris.andrews@maths.ox.ac.uk
BorisAndrews
Patrick Farrell (University of Oxford)
Numerical methods for the simulation of integrable systems with conservative properties are known to exhibit greater accuracy and physical reliability, in particular over long durations. In this talk, we propose a general framework for the construction of conservative schemes via finite elements in time, and the systematic introduction of auxiliary variables.
For problems with quadratic invariants, these scheme are provably equivalent to Gauss methods. However, the alternative framework extends to nonlinear systems with potentially multiple non-quadratic invariants. The framework also allows for the construction of numerical methods that accurately preserve dissipation structures, e.g. energy dissipation in the incompressible Navier–Stokes equations.
We demonstrate the ideas by devising energy–conserving schemes for Hamiltonian PDEs, and demonstrating mass/momentum/energy–conserving schemes for the compressible Navier–Stokes equations, with necessarily increasing entropy.