A finite point method for three-dimensional compressible flow

Numerical simulation came into the focus of interest of applied sciences and engineering in the last decades. As a result, the development of numerical techniques for solving partial differential equation (PDEs) has been growing continuously, mainly simulated by increasing computational resources and ever-challenging demands for practical and theoretical applications. Basically, these numerical techniques reduce the original governing differential equation, or a mathematically equivalent form, into an algebraic system of equations, which is easily solved with a computer. A numerical technique for solving PDEs is set apart from another by the way in which the unknown function and its derivatives are approximated, and this is intimately related to the discretization of the physical domain. Regarding domain discretization, most numerical techniques for solving PDEs could be roughly classified into mesh-based methods or meshless methods. The domain discretization in mesh-based methods or meshless methods. The domain discretization in mesh-based methods consists of a list of points or nodes properly ordered by the definition of connectivities between them. These connectivities originate the cells or elements which compose the mesh. Mesh-based methods like Finite Differences (FD), finite Volumes (FV) and Finite Elements (FE) are usually employed in practice due to their robustness, efficiency and high confidence gained through years and years of continuous use and enhance. In meshless methods, the domain discretization is based on a list of points but an ordered connectivity between them is not required. This fact turns meshless techniques conceptually attractive but their practical implementations are not likely to be so, which explains the relatively little interest that has been devoted to these methods. However, over the last ten years, some difficulties in mesh-based or conventional methods when performing particular applications, have brought meshless methods into focus.
The first mesh less methods appeared in the mid-seventies and numerous formulations have been proposed since then. A retrospective view of the evolution of the most relevant meshless methods as well as their connections is presented by Belytschko et al. In their work, the main features of some typical meshless methods, their implementation issues and practical applications are offered. Another interesting work by Fries and Matthies classifies and analyzes the most important meshless methods. For each, the authors highlight the main characteristic and implementation details as well as the advantages and disadvantages. Other helpful reviews on meshless methods, also considered here, can be found in the literature; see for instance Li an Liu and Duarte.
The present work deals with a meshless technique call Finite Point Method (FPM) which was introduced by Oñate et al. In this method, the numerical approximation to the unknown function an dits derivates is based on a Weighted Least-Squares (WLSQ) procedure known as Fixed Least Squares (FLS). The strong form of the governing PDEs is sampled at each point by replacing the continuous variables with their approximated counterparts. Finally, the resulting system of algebraic equations is obtained by means of a collocation technique.
Next, a review of the development of the FPM, aimed at providing a framework for the present research work and its motivations, is given. Then, by the end of this section, the objectives of this research and the organization of the contents are presented.