Evolution of CFD numerical methods and physical models towards a full discrete approach

The physical models and numerical methodologies of Computational Fluid Dynamics (CFD) are historically linked to the concept of continuous medium and to analysis where continuity, derivation and integration are defined as limits at a point. The first consequence is the need to extend these notions in a multidimensional space by establishing a global inertial frame of reference in order to project the variables there. In recent decades, the emergence of methodologies based on differential geometry or exterior calculus has changed the point of view by starting with the creation of entangled polygonal and polyhedral structures where the variables are located. Mimetic methods and Discrete Exterior Calculus, notably, have intrinsic conservation properties which make them very efficient for solving fluid dynamics equations. The natural extension of this discrete vision relates to the derivation of the equations of mechanics by abandoning the notion of continuous medium. The Galilean frame of reference is replaced by a local frame of reference composed of an oriented segment where the acceleration of the material medium or of a particle is defined. The extension to a higher dimensional space is carried from cause to effect, from one local structure to another. The conservation of acceleration over a segment and the Helmholtz–Hodge decomposition are two essential principles adopted for the derivation of a discrete law of motion. As the fields covered by CFD are increasingly broad, it is natural to return to the deeper meaning of physical phenomena to try a new research or new path which would preserve the properties of current formulations.