In this work the Asymptotic Numerical Method (ANM) with a Moving Least Squares method (MLS) for the simulation of a compressible fluid flow is presented. The strong formulation of compressible viscous isothermal Navier-Stocks equations is the starting point.

This proposed high order implicit algorithm is based on the implicit Euler scheme, a homotopy technique, a Taylor representation, the MLS method and a continuation method. The MLS is a meshless collocation method and has attracted considerable attention in recent years. Thanks to Taylor series development, the nonlinear differential equations, expressing the strong formulation of a compressible fluid, are transformed into a succession of linear differential equations with the same operator.

This algorithm makes it possible to obtain the solution during a very long time interval with a less expensive CPU time. The results obtained using the proposed algorithm will be compared with those obtained using an explicit Runge-Kutta scheme and the Finite Difference Method (FDM) and those calculated using the Newton-Raphson method with MLS method.

The efficiency of this proposed algorithm is tested on a standard benchmark of computational fluid mechanics, the lid-driven cavity problem.