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

Numerical solutions to Laplace's equation for an electrostatic potenti al can easily be found in undergraduate physics courses by approximati ng the Laplacian on a mesh and solving the resulting difference equati ons using a spreadsheet program or a simple program written in BASIC o r 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 equati ons, and this raises the question of how stringent to make the criteri on 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 exac t 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 magnitud e larger than the error in the solution of the difference equations. I t was also found that the commonly used nearest-neighbor approximation to the Laplacian gives the most accurate numerical solutions.