Numerical simulation. Diffusion equation

Abstract

Numerical simulation is an effective tool in applied science, for determining and predicting solutions without experimentation. It allows that the physical processes, that frequently occur at the engineering area, would be shown by mathematical algorithms. Partial differential equations, particularly the diffusion equation, usually need a careful study of numerical methods. It is also important to be careful at the algorithms and the fundamental processes included in the simulation. Numerical simulation and mathematical model differ. While the first is a simulated representation in every moment, the second is a mathematical abstraction of the fundamental equations necessary to analyze a phenomenon. Using a numerical simulation to study a given problem usually requires careful planning to use the mathematical model and algorithms needed to solve it. We can determine the numerical solution of problems which may or may not know its analytical solution. In this article we develop and exemplify model partial differential equations, particularly we study the diffusion equation. This equation was chosen because it has multiple applications in mechanical problems. Using a scientific software, we can develop computing material to compare different methods. We analyzed both, the advantages and disadvantages of its implementation. Conclusions are obtained through out the application of solving methods.