A new numerical grid truncation scheme for the finite difference of Laplace's equation

A modified form of Laplace's equation, based on coordinate stretching, is proposed, that lends itself to a new convenient numerical grid truncation methodology for the solution of Laplace's equation in open regions. The proposed method eliminates the need for approximate local boundary conditions the truncation boundary of the numerical grid. Thus, it provides for a simple, robust, computationally efficient and very accurate grid truncation scheme. First, an analytic justification of the proposed grid truncation methodology is presented. Next, its numerical implementation is discussed in conjunction with the numerical solution of Laplace's equation in unbounded two-dimensional regions. Numerical studies are used to illustrate the choice of the parameters used in the numerical implementation of this new truncation scheme, and quantify their impact on solution accuracy. A separable boundary value problem is used to motivate the proposed truncation scheme. The numerical studies and associated numerical results emphasize the calculation of per-unit-length capacitances for two-dimensional conductor configurations of the type used for unshielded planar transmission lines.