Contribution to Lagrangian and Arbitrary-Lagrangian-Eulerian numerical schemes. (Contribution au domaine des méthodes numériques Lagrangiennes et Arbitrary-Lagrangian-Eulerian)

This thesis presents our work related to (i) Lagrangian schemes and (ii) Arbitrary- Lagrangian-Eulerian numerical methods (ALE). Both types of methods have in commun to solve the multidimension compressible Euler equations on a moving grid. The grid moves with either the fluid velocity (Lagrangian) or an arbitrary velocity (ALE). More specifically we deal with some problems related to artifical viscosity, internal consistency, stability, accuracy, exceptional points and slide line treatments. In the ALE chapter we study the remap and rezone phases but also the mesh reconnection to build a ReconnectionALE scheme and further some interface reconstruction techniques. In a third chapter (iii) other resarch topics are presented like interface reconstruction techniques on a fixed grid finite volume scheme, ultra fast kinetic scheme, and very high-order finite volume schemes.