Numerical solution of Laplace’s equation in polar coordinates by a Fourier expansion and the finite difference scheme

The steady-state heat equation for a circular plate with Dirichlet boundary conditions has been solved using a method that combines Fourier expansion and a second-order finite difference scheme. This method is implemented in a Fortran 90 program called Laplace Solver in Polar Coordinate (LSPC). The numerical solution for a Dirichlet boundary condition with significant mathematical complexity is obtained and compared with its corresponding analytical one, showing good agreement. Furthermore, numerical solutions are obtained for two more physically realistic Dirichlet boundary conditions, where constant values are assigned over two hemispheres and four quadrants. Finally, an error analysis between the numerical and analytical solutions is performed using the error vector norm, along with a self-convergence analysis. From this, it is concluded that the method is self-convergent and exhibits a decreasing error, which depends on the number of points in the discretization for finite difference and the number of terms in Fourier expansion.

Keywords
Steady-state heat equation; Dirichlet boundary conditions; Fourier expansion; finite difference scheme