Convergence of SPH method for scalar nonlinear conservation laws

This paper is devoted to the study of the convergence of weighted particle approximation of nonlinear multidimensional conservation laws. For Euler equations the method is closely related to the smooth particle hydrodynamics (SPH) method. Extension of the original algorithm is proposed. We use approximate Riemann solvers instead of artificial viscosity in order to stabilize the scheme. The mathematical analysis is performed by connecting this new approach with the finite volume scheme. Convergence of the approximate solution in L-loc(p) (p < infinity) towards the unique weak entropy solution of the Cauchy problem is obtained in the scalar nonlinear case by using uniqueness of measure valued solutions.