Meshfree methods aim to approximate accurate and efficient numerical solutions to partial differential equation problems without the necessity for a mesh, relieving some of the difficulties associated with mesh generation. In this thesis, a hybrid method is presented which alternates between an implicit sequential function approximation scheme and an explicit Runge-Kutta time marching scheme. In the sequential function approximation scheme, a solution is improved incrementally as a summation of Gaussian radial basis functions (RBFs) which have been optimized using a genetic algorithm. Spatial differentiation is performed under both schemes using a meshfree, local differential quadrature method. This thesis focuses on the governing equations for compressible flow problems relevant to aerospace applications, namely aerodynamics. Consequently, numerical methods from finite difference and finite volume methods are adapted to the meshless scheme to account for characteristics particular to these governing equations. The method presented is used to approximate solutions for some common validation cases and the results are compared with those found in the literature.