Application of finite difference method of lines on the heat equation

• Published in: Numerical methods for partial differential equations Vol. 34; no. 2; pp. 626 - 660
• Main Authors: Kazem, Saeed, Dehghan, Mehdi
• Format: Journal Article
• Language: English
• Published: New York Wiley Subscription Services, Inc 01.03.2018
• Subjects: Differential equations; Dirichlet problem; Eigenvalues; Eigenvectors; exponential matrix; Finite difference method; heat equation; implicit and explicit methods; Method of lines; Thermodynamics; tridiagonal matrix
• ISSN: 0749-159X; 1098-2426
• DOI: 10.1002/num.22218

In this article, we apply the method of lines (MOL) for solving the heat equation. The use of MOL yields a system of first–order differential equations with initial value. The solution of this system could be obtained in the form of exponential matrix function. Two approaches could be applied on this problem. The first approach is approximation of the exponential matrix by Taylor expansion, Padé and limit approximations. Using this approach leads to create various explicit and implicit finite difference methods with different stability region and order of accuracy up to six for space and superlinear convergence for time variables. Also, the second approach is a direct method which computes the exponential matrix by applying its eigenvalues and eigenvectors analytically. The direct approach has been applied on one, two and three‐dimensional heat equations with Dirichlet, Neumann, Robin and periodic boundary conditions.