Finite Difference Method for Solving the Two-Dimensional Laplace Equation in Curvilinear Coordinates

Abstract

Abstract This paper presents a numerical method for solving the two-dimensional Laplace equation in curvilinear coordinates using a finite difference scheme. Laplace equation is a differential equation used to describe various physical phenomena studied within different branches of engineering. The two-dimensional Laplace equation has analytic solutions only for simple shaped domains and simple boundary conditions (first and second kind). Even so, the results are usually complicated to interpret and use. The present paper proposes a method to solve the equation for domains bounded by complex inner and outer curves. The boundary curves are defined by a set of points from which the equations of curves are approximated by using Fourier series. The domain is generated by blending the two curves using a blending function. The results are a set of continuous parametric equations which are defining the domain in angle and radius variables. Following a method presented in the literature, the Laplace equation is transformed from Cartesian coordinates to curvilinear coordinates, as the Fourier series is easily differentiable. Next, the new Laplace equation is approximated using the finite difference method and a MATLAB code is developed to solve the system of linear equations using successive over-relaxation method (SOR). The presented method is used to simulate steady two-dimensional heat conduction in a polygonal shape domain, with uniform and non-uniform Dirichlet boundary conditions. Results are analyzed in order to determine the model’s precision, speed and convergence criteria.