An improved SPH method: Towards higher order convergence

This paper evaluates various formulations of the SPH method for solving the Euler equations. Convergence and stability aspects are discussed and tested, taking into account subtleties induced by the presence of a free surface. The coherence between continuity and momentum equations is considered using a variational study. The use of renormalization to improve the accuracy of the simulations is investigated and discussed. A new renormalization-based formulation involving wide accuracy improvements of the scheme is introduced. The classical SPH and renormalized approaches are compared to the new method using simple test cases, thus outlining the efficiency of this new improved SPH method. Finally, the so-called “tensile instability” is shown to be prevented by this enhanced SPH method, through accuracy increases in the gradient approximations.