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 . 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 haand hb, to form a space-charge-flow solution accurate to second order in . 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.