DISCRETIZATION AND TRUNCATION ERRORS IN A NUMERICAL-SOLUTION OF LAPLACE EQUATION

Numerical solutions to Laplace's equation for an electrostatic potential can easily be found in undergraduate physics courses by approximating the Laplacian on a mesh and solving the resulting difference equations using a spreadsheet program or a simple program written in BASIC or PASCAL. The simplest numerical method uses iteration and accelerates the convergence by simultaneous overrelaxation (SOR). Truncating the iteration introduces an error in the solution to the difference equations, and this raises the question of how stringent to make the criterion for convergence. This paper considers the relative magnitude of the errors made in approximating the Laplacian (discretization error) and in truncating the iteration (truncation error). Numerical results are given for an electrostatic cavity problem previously investigated by several authors, and the numerical solutions are compared with an exact solution obtained by conformal mapping. It was found that when even a modest convergence criterion is used to truncate the iteration, the rms error inherent in discretization is more than an order of magnitude larger than the error in the solution of the difference equations. It was also found that the commonly used nearest-neighbor approximation to the Laplacian gives the most accurate numerical solutions.