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 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 structure appears to offer considerable qualitative advantages over other structure-preserving schemes (e.g. preserving symplecticity).