Absorption Measurements of Certain Changes in the Average by Allison S. K., Duane W.

By Allison S. K., Duane W.

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1 The limitations of the five point method. The accuracy of the five point method may be estimated by noting that the potential at a given point differs from the mean of that at the nearest neighbours by an amount proportional to h4 or N −2 where N is the number of mesh points. However this is not the absolute accuracy, because these latter points do not have their ‘correct’values either so there is a progressive change which will depend on N , leaving the absolute accuracy proportional to N −1 .

It is possible to increase the accuracy of the calculation by considering not just the average of the potentials at four nearest neighbours, but by taking account of four further points. This leads to relaxation methods known as ‘nine point relaxation’in distinction to the ‘five point relaxation’ we have discussed so far. 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.

11 mentioned above correspond to large negative values of V . The reader may care to examine the dependence on g of the number of iteration cycles required to reach the end point. A graph of log n against g emphasizes the importance of the correct choice of value, while a graph of n against log(2 − g) is also interesting. Both graphs show the effects of incipient oscillation for values above the optimum. 4 Precision and accuracy The precision of the final result of a relaxation calculation depends on the size of the array.

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