GAFD Seminar: Patrick Farrell (University of Oxford)

We propose a general strategy for enforcing multiple conservation laws and dissipation inequalities in the numerical solution of initial value problems, without Lagrange multipliers or brute-force projection operators. The key idea is to represent each conservation law or dissipation inequality by means of an associated test function; we introduce auxiliary variables representing the projection of these test functions onto a discrete test set, and modify the equation to use these new variables.

We demonstrate these ideas by their application to the several problems, including Hamiltonian and GENERIC ODEs, the Benjamin-Bona-Mahony equation, and the Navier-Stokes equations; we devise the first time discretization of the Eulerian formulation of the compressible Navier-Stokes equations that conserves mass, momentum, and energy, and provably dissipates entropy. In several cases preserving conservation and/or dissipation structures appears to offer substantial qualitative advantages over other structure-preserving schemes (e.g. such as those preserving symplecticity).