The accuracy of numerical solutions for electron gun design

The solution of Poisson's equation and the equation of motion by numerical methods is discussed. By considering the truncation error due to the use of difference equations in the one-dimensional case, it is shown, on the basis of a perturbation theory, that the error in the solution is proportional to the mesh size
h
. This theory is shown to be in good agreement with numerical solutions obtained on a computer. A correction formula is then derived which makes it possible, by obtaining sointions for two different mesh sizes h
a
and h
b
, to form a space-charge-flow solution accurate to second order in
h
. It is also shown that more accurate forms of the difference equation may be obtained to represent the steady space-charge flow. By these various means, it is possible to increase the accuracy of numerical solutions by an order of magnitude or more. Thus, by using a coarse mesh, but better accuracy, the speed of computation may be considerably increased.