A description of new Riemann solver that is incorporated in the Least squares Finite Difference Scheme which is a strong form of meshless method. Three types of Schemes are presented; the first Riemann Least Squares Finite Difference Scheme (RLSFD), the conservative form of the RLSFD sheme and the second order Riemann Least Squares finite difference sheme. It is then proved that the first oder RLSFD is consistent with the linear advection equation and that is conditionally stable. The existence of a weak solution for the first order RLSFD. Finally, we present our numerical results when the RLSFD is applied to the 1-D linear advection equation, 1-D Burgers equation, the shock tube problem and to some the two phase flow problems.