Numerical Error Analysis and Heat Diffusion Models

In this chapter we present the mathematical equations that model the phenomenon of steady state heat diffusion. In addition, we show the variables of interest in our study that are determined locally or globally. The local or global primaries are obtained by solving the linear system of algebraic equations associated with the discretization of the mathematical model when applying the SPH method. This chapter of the book too is devoted to the presentation of fundamental analyses for determining the specific numerical error sources of SPH approximations. In addition, we show the behavior of iteration and round-off errors, so that the reader can make the relationship between the numerical errors of methods using mesh and meshless methods. These types of analyses are quite important and unknown to the authors. We use as examples, the mathematical models of stable and unstable 1D and stable 2D heat diffusion. However, the analyses can be replicated for any mathematical model, such as the Navier-Stokes equations, for example.