Finite difference method

Abstract: This paper presents an extension to the work presented in Part I of this series of two articles to the transient case. Emphasis is placed on the development of a new model for heat flow in a double U-shape vertical borehole heat exchanger and its thermodynamic interactions with surrounding soil mass. The discretization of the spatial-temporal domain of the heat pipe model is done by the use of the space–time finite element technique in conjunction with the Petrov–Galerkin method and the finite difference method. The paper shows that the proposed model and the choice of the discretization technique, in addition to the utilization of a sequential numerical algorithm for solving the resulting system of non-linear equations, have contributed in reducing significantly the required number of finite elements necessary for describing geothermal heating systems. Details of the mathematical derivations and comparison to experimental data are presented. Copyright © 2006 John Wiley & Sons, Ltd.

Abstract: In this paper, the continuous genetic algorithm is applied for the solution of singular two-point boundary value problems, where smooth solution curves are used throughout the evolution of the algorithm to obtain the required nodal values. The proposed technique might be considered as a variation of the finite difference method in the sense that each of the derivatives is replaced by an appropriate difference quotient approximation. This novel approach possesses main advantages; it can be applied without any limitation on the nature of the problem, the type of singularity, and the number of mesh points. Numerical examples are included to demonstrate the accuracy, applicability, and generality of the presented technique. The results reveal that the algorithm is very effective, straightforward, and simple.