On the Convergence of SPH Method for Scalar Conservation Laws with Boundary Conditions

Abstract

This paper is the third of a series where the convergence analysis of SPH method for multidimensional conservation laws is analyzed. In this paper, two original numerical models for the treatment of boundary conditions are elaborated. To take into account nonlinear effects in agreement with Bardos, LeRoux and Nedelec boundary conditions, the state at the boundary is computed by solving appropriate Riemann problems. The first numerical model is developed around the idea of boundary forces in surrounding walls, recently initiated in Simulating Gravity Currents with SPH. III : Boundary Forces by Monaghan in his simulation of gravity currents. The second one extends the well-known approach of ghost particles for plane boundaries to the case of general curved boundaries. The convergence analysis in $L_loc^p ( p< \infty)$ is achieved thanks to the uniqueness result of measure-valued solutions recently established in Ben Moussa B., Szepessy A., Scalar Conservation Laws with Boundary Conditions and Rough Data Measure Solutions for $L^\infty$ initial and boundary data.