The SPH method (Smooth Particle Hydrodynamics) is our starting point. It has been “discovered” in 1977 by Lucy ([15]) a British astrophysicist. At this time his computation only requires 100 particles. The effective development of the method is due to J. Monaghan, an Australian applied mathematician ([7],[8],[16]), who also developed most of the extensions of the original techniques to multifluid equations, MHD, etc.

Until 1985, the SPH method was specialized in Astrophysics applications, W. Benz (yet an astrophysicist) is among the first which use SPH methods for complex applications such as high velocity impacts problems with damage models ([4],[5]). Actually, a lot of research center use SPH methods ([19],[11]) as an efficient alternative to Finite Element Lagrangian codes in the field of high velocity impacts. Industrial codes using SPH methods have been available only recently. This is partly due to some difficulties in the theoretical and numerical basis for handling with boundary conditions. In this paper, I first give an overview of classical recipes for designing SPH methods. I then introduce a weak discrete formulation, which provides an efficient tool for understanding and solving problems related with the global conservation property. I show how Renormalization ([12],[20]), a new efficient tools in the field of SPH methods can be used in this context. I also introduce new hybrid SPH-Godunov type schemes. Combining this new approach with renormalization I overcome usual restrictions on the ratio of the smoothing length to the size of the mesh, which needs to be large enough (or equal to some specific value depending on the kernel function). I have originally presented this approach, which really mixes Finite Difference type Riemann solver and SPH in [24]. At this time I only use standard SPH (ie without renormalization), note that this approach leads to a com-