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.
Highlights
Heat conduction problems with suitable boundary conditions exist in many areas of engineering applications [1,2,3,4,5,6,7]
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
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
Summary
Heat conduction problems with suitable boundary conditions exist in many areas of engineering applications [1,2,3,4,5,6,7]. Highly accurate compact finite difference schemes are developed in the work by Lele [8]. These higher-order compact finite difference schemes only offer good accuracy at the interior nodes or for periodic boundary conditions. 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. Dai et al [11, 13,14,15,16] proposed a new idea to achieve higher-order accuracy with unconditional stability. Authors introduced a new parameter θ that adjusts the location of nodes near the boundaries in symmetric way
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