An improved SPH method: Towards higher order convergence G. Oger * , M. Doring, B. Alessandrini, P. Ferrant Fluid Mechanics Laboratory (CNRS UMR6598), Ecole Centrale de Nantes, 44321 Nantes, France Received 6 February 2006; received in revised form 25 January 2007; accepted 30 January 2007 Available online 19 March 2007 Abstract This paper evaluates various formulations of the SPH method for solving the Euler equations. Convergence and sta- bility aspects are discussed and tested, taking into account subtleties induced by the presence of a free surface. The coher- ence 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 gra- dient approximations. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Renormalization; SPH convergence; Free surface considerations; New SPH formulation 1. Introduction Smoothed particle hydrodynamics (SPH) is a promising tool for hydrodynamic applications as it is simple, efficient and robust [1,2]. This easy-to-code meshless method easily handles complex free surface flow prob- lems displaying strong non-linearities, including breaking waves. SPH has a wide range of applications in mechanical engineering topics [3–5] where strong non-linearities occur, as well as coupled fluid/structure inter- action problems [6]. This method is based on the convolution of variables through a kernel (or its gradient) to estimate the velocity and pressure gradients. This convolution theoretically ensures second order accuracy for the continuous approach. However, the discretized convolution approximations are insufficiently accurate for obtaining pressure and velocity fields regular enough to exploit fully. Integrals are approximated using a tra- peze-like quadrature formula. In order to limit the computational costs, only a few points are used, thus com- promising the accuracy. In this paper, particular attention is first paid to the quantification of errors in the gradient approximation procedures using kernel gradients on a set of unstructured interpolation points. Then 0021-9991/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2007.01.039 * Corresponding author. E-mail address: guillaume.oger@ec-nantes.fr (G. Oger). Journal of Computational Physics 225 (2007) 1472–1492 www.elsevier.com/locate/jcp