Geometry, Energy, and Entropy Compatible (GEEC) Variational Approaches to Various Numerical Schemes for Fluid Dynamics

Since WW2, computer fluid dynamics has seen a staggering expansion of methods and applications fueled by the development of computer power. Now, of the main numerical approaches that have been explored over these years, only a few have become mainstream and make the vast majority of theoretical investigations in academia and practical usage in applications. These mostly hinge on concepts of finite volume discretization, monotonicity preservation, flux upwinding, and the analysis of the associated numerical dissipation processes-common tools here are the Riemann problem at cell interfaces and the Godunov scheme, more or less adapted from their original versions. However, application to what looks as niche problems shows that these dominant approaches may not be as effective as generally accepted and have unduly benefited from a winner-takes-it-all effect. One of these problems is the simulation of isentropic flows which is actually not-so-niche as it is of high practical interest, especially in multi-fluid systems which involve complex energy transfers. The present contribution aims at providing some perspective on CFD numerical schemes recently designed in order to better capture isentropic flows. The basic principle is that isentropic flow is geometric, i.e. potential (or internal) energy only depends on fluid density which in turn is defined by fluid element trajectories. A numerical scheme can thus be obtained by a variational, least action principle. Corrections must be further added to enforce other properties such as energy conservation and positive dissipation. This Geometry, Energy, and Entropy Compatible approach (GEEC) is illustrated here on the historical von NeumannRichtmyer Lagrangian scheme and on our recently developed multi-fluid Arbitrary Lagrangian-Eulerian scheme.