On meshless collocation approximations of conservation laws: Preliminary investigations on positive schemes and dissipation models

We consider meshless collocation methods for the numerical solution of tran sport processes described by hyperbolic conservation laws. The future goal is the construction of a robust and reliable meshfree discretization method for the equations of gas dynamics in complex geometries. In this paper we start with the simplest scalar model problems and analyze basic problems oc curring in the grid-free approach. A topological condition on clouds of poi nts is derived and several possible versions of a generalized Lax-Friedrich s scheme are discussed with respect to their numerical dissipation. A movin g least-squares approach is followed to construct a positive discretization for solutions with shocks which is thoroughly analyzed and applied to test problems.