Numerical solution of Laplace’s equation through the relaxation method

Laplace’s equation is a differential equation that can describe different physical phenomena. In this work we discuss how it can be numerically solved using the relaxation method. The Laplacian operator is written in terms of finite differences, establishing a relationship between the value of the function at one point and the arithmetic mean of the function at neighboring points. The obtained relationship is applied iteratively and numerical solutions are obtained. We apply the algorithm for the solution of the heat equation considering steady states and for the determination of electrostatic potential between two coaxial conducting cylinders. The agreement between analytical and numerical solutions is very good, the algorithm is computationally cheap and can be implemented with basic knowledge of algorithms in a few lines of code. We discuss how this didactic tool can be applied in different teaching contexts.

Keywords:
Laplace’s equation; Computational Simulations; Python; Differential Equations