Soluções analíticas e numéricas usando o método SPH: aplicações à condução de calor e ao tubo de choque

Abstract

In the context of the development and advancement of numerical approximation techniques for scientific and engineering research, the present work proposes to explore traditional numerical methods of finite differences, such as Euler, Lax-Wendroff and Mac-Cormack, and compare with the Smoothed Particle Hydrodynamics (SPH). For this, two physical problems are used: the shock tube and the heat transfer by conduction in a rectangular flat plate. The mathematical models of these problems are presented and analytical solutions are developed for them, so that they are the reference elements for the comparison of the numerical methods. The numerical models of each method are introduced with a brief bibliographic review, and in the case of finite difference methods, a brief development is doing, giving more attention to the SPH method. Given this, numerical approximations are applied to the physical problems, generating results that are compared with the exact solutions. In addition, some aspects of the numerical methods are explored, trying to understand the influence of the same on the results, allowing a more detailed analysis. Finally, the methods are compared among them, seeking to understand the potential of the SPH method.