By Ruark A. E.

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This is an unrealistic situation, but it has been used as an aid to the normalization of other expressions. The general form can be approximated by a number of analytic expressions chosen more for their mathematical tractability than their connection with the basic physics of the problem. Such an approach has a long history, but with the amount of computing power readily available nowadays, there is little point in using this sort of method though we give a brief account in the next section. Results of much greater accuracy can be obtained by approximations based on the laws of electrostatics.

V9 (z) − V5 (z) V9 (z) where the notation should be transparent. We know that the five point method does not converge to the correct potential and we shall see later that the nine point method does converge correctly in a certain sense, so the differences shown in this figure indicate the scale of the error in the five point method as applied to a lens geometry. 12. 15 calculated by the five point and nine point methods. The finite difference method is one of the easiest to program, provided that the geometry is simple; curved surfaces need to be approximated by chords.

If we consider the four points a distance ±2h from the target point in the two dimensional array [11] we can write V (x − 2, y) + V (x + 2, y) + V (x, y − 2) + V (x, y + 2) − 4V (x, y) = 16h2 ∂ 2V ∂ 2V + ∂x 2 ∂y 2 and this time the first neglected term is 4 ∂ 4V ∂ 4V + h4 + 3 ∂x 4 ∂y 4 which is 16 times as great as for the four nearest neighbour case. We can eliminate this term, leaving the sixth order term as the first to be neglected, if we give the nearest neighbour sum a weight of 16 and the ‘2h’ sum a weight of −1 60V (x, y) = 16[V (x − h, y) + V (x + h, y) + V (x, y − h) + V (x, y + h)] − [V (x − 2h, y) + V (x + 2h, y) + V (x, y − 2h) + V (x, y + 2h)].