Eighth-order compact finite difference scheme for id heat conduction equation

• Published in: Advances in Numerical Analysis 2016-01
• Main Authors: Yosaf, Asma, Rehman, Shafiq Ur, Ahmad, Fayyaz, Ullah, Malik Zaka, Alshomrani, Ali SAleh
• Format: Article
• Language: English
• Subjects: Conduction; Finite element method; Heat; Heat equation; Mathematical models
• ISSN: 1687-9562; 1687-9562
• DOI: 10.1155/2016/8376061

The purpose of this paper is to develop a high-order compact finite difference method for solving one-dimensional (ID) 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 ID heat conduction equations is solved to show the validity and accuracy of our proposed methods.