# Convergence and Stability of Explicit and Implicit Schemes in the Simulation of the Heat Equation
## 1 Introduction  
Most physical phenomena are represented by partial differential equations. Steady-state physical problems or time evolution problems are modeled through elliptic and parabolic partial differential equations, respectively; these equations are difficult to solve by analytical methods when the initial and boundary conditions are not simple. An example of this is shown in [1, 2, 3] and references therein, where several equations of physics-mathematics describing phenomena related to continuous element dynamics, electrodynamics, quantum mechanics, and mass and heat transfer phenomena are highlighted. For the solution of some of these equations, they use methods such as the Fourier Method, the Riemann Integration, and the use of the Green's Function [4, 5, 6, 7, 8]. However, these procedures do not always guarantee analytical solutions that illustrate the behavior of a mechanical system; this is where numerical methods provide powerful techniques that allow a solution as close as possible to the reality of a specific problem. The numerical methods developed by [PERSON] [9, 10, 11, 12, 13, 14, 15] were used to solve equations and transcended in an important way to engineering in 1943 when [PERSON] developed the finite element method. [PERSON] used the numerical methods of variation proposed by [PERSON] to obtain approximate solutions for mechanical systems [16, 17]. The Courant-Friedrichs-Lewy (CFL) condition is a necessary condition for convergence in the numerical solution of partial differential equations with partial differentials. A case of special importance is when time discretization schemes are used as a numerical solution. As a consequence, inmany simulations, the time step has to be less than a characteristic value; otherwise, the code may be unstable and not yield the correct results. Thus, simulations are an effective tool in applied sciences to find solutions and predict their behavior. Additionally, they allow the mathematical recreation of physical processes that frequently appear in engineering [18; 19; 20]. The use of simulations to study partial differential equations, in particular the diffusion equation, normally requires a careful study of numerical methods, the algorithms to be used, and the fundamental processes to be included in the simulation. A simulation differs from a mathematical model in that the former is a representation at each instant of the process to be simulated, while the model is a mathematical abstraction of the fundamental equations necessary to analyze that phenomenon. Normally, the use of a simulation to study a given problem requires careful planning of the mathematical model (partial differential equations and initial and boundary conditions) to be used and of the algorithms necessary to solve the model. Numerical methods are used to determine the numerical solution of problems for which the analytical solution may or may not be known. These allow the translation of complicated mathematical schemes using algorithms, the results of which can be contrasted with the analytical solutions, in cases where such solutions exist. In this work, the heat conduction equation is simulated to obtain the temperature patterns in a thin bar [21; 22; 23]. The analysis of non-stationary heat transfer is of practical interest, not only because of the importance that cooling and heating processes have in a large number of industrial applications but also because of their similarity with several other equations of physics-mathematics that present the same difficulties to be solved numerically. Among the problems directly related to non-stationary heat transfer problems are the starting and stopping of machines, processes that can lead, on the one hand, to the appearance of important thermal loads on the solid with the consequent deterioration (turbine blades, for example), and on the other hand, to an increased emission of polluting species (steam generators, gas turbines, and alternative internal combustion engines, among others).