Advances in Numerical Analysis

Volume 2016, Article ID 8376061, 12 pages

http://dx.doi.org/10.1155/2016/8376061

## Eighth-Order Compact Finite Difference Scheme for 1D Heat Conduction Equation

^{1}Department of Mathematics, University of Engineering and Technology, Lahore 54840, Pakistan^{2}Dipartimento di Scienza e Alta Tecnologia, Universita dell’Insubria, Via Valleggio 11, 22100 Como, Italy^{3}Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Comte d’Urgell 187, 08036 Barcelona, Spain^{4}Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 13 April 2016; Accepted 19 April 2016

Academic Editor: Kenneth Karlsen

Copyright © 2016 Asma Yosaf et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The purpose of this paper is to develop a high-order compact finite difference method for solving one-dimensional (1D) heat conduction equation with Dirichlet and Neumann boundary conditions, respectively. A parameter is used for the direct implementation of Dirichlet and Neumann boundary conditions. The introduced parameter adjusts the position of the neighboring nodes very next to the boundary. In the case of Dirichlet boundary condition, we developed eighth-order compact finite difference method for the entire domain and fourth-order accurate proposal is presented for the Neumann boundary conditions. In the case of Dirichlet boundary conditions, the introduced parameter behaves like a free parameter and could take any value from its defined domain but for the Neumann boundary condition we obtained a particular value of the parameter. In both proposed compact finite difference methods, the order of accuracy is the same for all nodes. The time discretization is performed by using Crank-Nicholson finite difference method. The unconditional convergence of the proposed methods is presented. Finally, a set of 1D heat conduction equations is solved to show the validity and accuracy of our proposed methods.

#### 1. Introduction

Heat conduction problems with suitable boundary conditions exist in many areas of engineering applications [1–7]. Historically, highly accurate compact finite difference schemes are developed in the work by Lele [8]. But these higher-order compact finite difference schemes only offer good accuracy at the interior nodes or for periodic boundary conditions. Usually, compact finite difference schemes have first- or second-order accuracy [9, 10]. The low-order of accuracy near boundary grid points affects the whole numerical results and it reduces the accuracy of overall numerical solution [11]. Some authors offer one-side finite difference approximations for the Dirichlet boundary condition [8, 12] but they cannot offer unconditional stability for the whole finite difference scheme. Recently, Dai et al. [11, 13–16] proposed a new idea to achieve higher-order accuracy with unconditional stability. Actually, authors introduced a new parameter that adjusts the location of nodes near the boundaries in symmetric way.

#### 2. Higher-Order Compact Finite Difference Method

The 1D heat conduction equation can be written as Dirichlet boundary conditions are as follows:Neumann boundary conditions are as follows:

Han and Dai [17] have proposed a compact finite difference method for the spatial discretization of (1a) that has eighth-order accuracy at interior nodes and sixth-order accuracy for boundary nodes. Actually, they use eighth-order accurate approximation for the second-order derivative developed in [8] and given as where double dash always means derivative with respect to spatial variable and , , , and . The implicit compact finite difference scheme (4) without the contribution of boundary nodes in matrix form is given byHere,

The governing equation of implicit compact finite difference approximation of second-order derivative isNote that (4) can be obtained from (7).

We can observe that matrix is strictly diagonally dominant and matrix is diagonally dominant. In [17], authors constructed finite difference scheme at boundary nodes in such a way that they conserve the diagonal dominance of matrixes and to attain higher order of accuracy. The achieved order of accuracy at boundary nodes in [17] is six. The essence of article [17] is hidden in the calculation of a parameter that they use to obtain higher-order approximation near boundary nodes. The parameter makes the point distribution unequal near boundary but offer higher order of accuracy and diagonal dominance for matrices and . The diagonally dominance of and is the key to prove stability of whole compact finite difference scheme when they use Crank-Nicholson method for time integration.

We construct a new finite difference scheme for boundary nodes to achieve eighth order of accuracy that exactly matches the order of accuracy at the interior nodes. The inclusion of two parameters and ’s will be introduced in our newly developed finite difference scheme for boundary nodes. The diagonally dominance will be conserved and the whole eighth-order scheme becomes stable.

The stencil of eighth-order implicit finite difference scheme to approximate the second-order derivative for the interior nodes isIt means that if we are at location , then we need two grid nodes to the left of and two grid points to the right of it. It is noticeable that mutual distance between nodes is equal to and points divide the interval with , , , , and for . Figure 1 shows the location of interior and boundary nodes.