Enforcing conservation laws and dissipation inequalities numerically via auxiliary variables

We propose a general strategy for enforcing multiple conservation laws and dissipation inequalities in the numerical solution of initial value problems. 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 Navier-Stokes equations. We generalize to arbitrary order the energy-dissipating and helicity-tracking scheme of Rebholz for the incompressible Navier-Stokes equations, and devise a time discretization of the compressible equations that conserves mass, momentum, and energy, and provably dissipates entropy.