Aspectos computacionales del método de diferencias finitas para la ecuación de calor dependiente del tiempo

Abstract

En este artículo se describe en detalle un algoritmo para la eficiente implementación computacional del método de diferencias finitas (MDF) en la ecuación de calor dependiente del tiempo, con condiciones de frontera de Dirichlet no homogéneas, en dos dimensiones. Para validar el método presentado aquí se utiliza el paquete computacional MATLAB®, sin embargo, los procesos se exponen independientes al lenguaje de programación. Finalmente se presentan resultados numéricos que validan el algoritmo propuesto.

Convergency and Stability of Explicit and Implicit Schemes in the Simulation of the Heat Equation

“…The numerical solution of the heat equation is discussed in many textbooks. References [1][2][3][4][5][6][7][8] provide a more mathematical description of the development of finite difference methods. See Cooper [4] for a modern introduction to the theory of partial differential equations along with brief coverage of numerical methods [9,13,16] that takes a more applied approach and introduces implementation problems.…”

“…The term on the right-hand side of Equation ( 6) is called the truncation of the finite difference approximation. It is the error that results from truncating the series in Equation (5). In general, ξ is not known and is determined from the simulation.…”

“…Although the expression does not give the exact magnitude of the error, it tells us how fast that term approaches zero as ∆x becomes smaller. Using the notation O large, one can write Equation (5) as…”

“…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 increasing use of multiprocessor-based computers, either using parallel programming or multiple processors, has recently encouraged the use of explicit methods, which are much more apt to be parallelized due to their local information processing condition [4,5]. It could be said that an explicit method, although more computationally expensive than an implicit method, is much more likely to be more efficient in parallel computation, and may even exceed it, depending on the number of parallel processors used [6][7][8].…”