Conservation Laws in Fluid Dynamics and the Enforcement of their Preservation in Numerical Discretizations

Abstract

The existence of conservation laws for fluid dynamics problems has been recognized to be of vital importance to the understanding of basic physical and mathematical properties of problems of interest. Almost all the equations describing fluid dynamics problems can be interpreted as the laws of conservation of mass, momentum and energy. In this short lecture note we will adopt two different approaches which essentially characterize different lines of thought as far as conservation laws are concerned. The first is the ‘theoretical’ approach whereby one considers what is a conservation law and the problem of obtaining conservation laws in the sense of physics from a system of equations as an integrability problem on a manifold (Eiseman and Stone 1980 [1]).KeywordsConservation PropertyNonlinear InstabilityBarotropic Vorticity EquationPotential EnstrophyGauss Divergence TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.