HIGH PRECISION RAY TRACING IN CYLINDRICALLY SYMMETRIC ELECTROSTATICS

David Edwards Jr. ,a)

Ijl research center, Newark, Vermont 05871,USA

With the recent availability of a high order FDM solution to the curved boundary value problem, it is now

possible to determine potentials in such geometries with considerably greater accuracy than had been

available with the FDM method. In order for the algorithms used in the accurate potential calculations to be

useful in ray tracing, an integration of those algorithms needs to be placed into the ray trace process itself.

The object of this paper is to incorporate these algorithms into a solution of the equations of motion of the ray and, having done this, to demonstrate its efficacy. The algorithm incorporation has been accomplished

by using power series techniques and the solution constructed has been tested by tracing the medial ray

through concentric sphere geometries. The testing has indicated that precisions of ray calculations of 10-20

are now possible. This solution offers a considerable extension to the ray tracing accuracy over the current

state of art.

I. INTRODUCTION

A. Elementary considerations.

A particle moving within an electrostatic geometry experiences a force at each point of its movement proportional to

the electric field at that point. Its path is governed by equations of motion which must include an accurate representation of

this field. In the process described below the solution to the equations of motion will be found within the context of the finite

difference method (FDM) and incorporate representations of the field from the potential algorithms themselves.

B. Finite Difference Method.

Descriptions of the finite difference method (FDM) can be found in many references, the most succinct being that of

Heddle [1]. Briefly it consists in placing a rectangular mesh over the geometry and then relaxing this mesh using an algorithmic

process. A long standing problem with FDM has been its inability to incorporate curved boundaries in any but the lowest order

manner. This difficulty has been recently been overcome and a solution has recently been found [2, 3], with the result that

accurate potentials can now be obtained for these curved boundary geometries. The determination of the accurate potential

distributions has necessitated the creation of high order algorithms. In order to improve the accuracy of the ray trace solution,

these algorithms are incorporated into it by the multiple use of power series techniques.

_____________________________

a) Electronic mail: dej122842@gmail.com

C. The solution for the coefficients cj in the power series expansion of v(r, z)

The expansion of v(r,z) as a truncated power series in r, z may be written:

���, �� = + � ∗ � + � ∗ � + � ∗ �� + � ∗ � ∗ � + � ∗ �� +⋯+ ����� (1)

2

Where …���are coefficients of the expansion and n the expansion order, order being defined as the degree of the

highest degree term in the expansion. . It is noted that the expansion is made about a mesh point in the FDM net which in our

representation of FDM occurs at integral values, while the point (r, z) is any point in the vicinity of this mesh point relative to

that mesh point.

The determination of the coefficients cj has been reported [4, 5] and hence is very briefly sketched below. It is noted that

the solution for the coefficients of any given order is called the algorithm for that order. As the underlying problem is

cylindrically symmetric electrostatics, v(r ,z) must satisfy Laplace’s equation at any point r, z within the geometry by:

��������

��� + ��� +

���������� � ���, �� = 0 (2)

where�, � are the coordinates of any point in the geometry with reference to its closest mesh point and the distance of that

mesh point from the axis. This yields an equation in �, � which must be true for any �, � in the neighborhood of this meshpoint.

The latter condition requires that in the resultant equation the coefficient of the term containing �! , �"must be zero, hence

generating k equations from this single equation. As there are s coefficients to be determined an additional s-k equations must

be found in order to have a complete set of s equations and s unknowns. These additional s-k equations are produced by

evaluating ���, �� at a selection of s-k meshpoints surrounding the meshpoint about which the expansion is made. The number

of the additional meshpoints are a strong function of the order of the power series expansion. The solutions for order 2, 4, 6,

8, and 10 has been given in [2] and are used in the ray trace calculation below.

Figure 1 shows the mesh points required for the various order algorithms. In this figure the points of any algorithm are

given by the red discs together with one additional triangle point. The algorithm for that order is formed by averaging all

possible algorithms of that order. It is further noted for order 6 and 10 the base algorithm consists of two types (a, b) but again

the algorithm for those orders is the average of all possible (a and b) algorithms. For a further discussion see reference [2].

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Figure 1. The meshpoints used in the formulation of the various algorithms are depicted. The algorithm for a given order are

formed by the average of all possible algorithms for that order (see text). It is noted that the very symmetric array of

meshpoints used in the average algorithm for orders 6, 8, and 10 are the same as those of the order 8 algorithm, evincing

considerable symmetry.

II. THE RAY TRACING PROBLEM

A. The equations of motion and their solution

Consider a particle of charge q and mass m at a point r, z within a cylindrical geometry over which a rectangular mesh has

been overlaid. It is assumed that the particle is at a point of its trajectory and both its position and velocity are known. The

point itself is not necessarily at a mesh point location but is in the vicinity of the mesh point nearest to it about which the

potential expansions described above are made. The solution for the particles subsequent motion is as follows. The equation

of motion of this particle is determined by Newton’s Law,

# = $ ∗ %/' (3)

where% = −)���, ��, a its acceleration, and ���, �� the potential at �, �.

The equations of motion of the particle follow immediately and after a straightforward but lengthy calculation are put into

dimensionless form resulting in the following 2 equations.

4

*���+�*+� + ,-. ∗ �/���+�,��+��

�� = 0 (4)

*���+�*+� + ,-. ∗ �/���+�,��+��

�� = 0 (5)

where �, �, ., �, and,-. are dimensionless while ,-. is given by:

,-. = $ ∗ /3"+! /�*4"+��*4"+�+�

� �6�deltar being the distance (in cm) between physical meshpoints and deltat the time (in seconds) for a particle with an initial

kinetic energy E0(eV) to traverse a distance of one mesh spacing. To proceed ��.� and ��.�are expanded as power series in t

in which terms higher than degree n are neglected. The order of the expansion is defined as n, consistent with its use above.

��.� = 60 + 61. + 62.� +⋯+ 69.� (7)

��.� = -0 + -1. + -2.� +⋯+ -9.� (8)

60…69, -0…-9 being constants of the expansion. As previously stated, at the start of the time step (i.e. t=0) the position and

velocity of the particle are known and hence also60,61,-0, and-1. To proceed further the solution for the coefficients will be

found for much simplified example of the process by using the order of the time series of 4 and that of the space series 3. This

will allow the output at various phases of the calculation to be displayed. Performing the indicated operations of (4) and (5)

(using ���, �� from equation (1)) the expansion of ��.� and ��.� are inserted into the resulting equations for � and�. This

produces two equations in . alone. Upon collecting similar terms for .: in each equation it is found:

(2d2+f2+e0 f4+2 d0 f5+e02 f7 + 2 d0 e0 f8 + 3 d02 f9) t0

+(6d3+e1 f4+2 d1 f5+2 e0 e1 f7+2 d0 e1 f8+2 d1 e0 f8+6 d0 d1 f9) t1

+(12d4+e2 f4+2 d2 f5+2 e2 e0 f7+e12 f7+2 d0 e2 f8+2 d1 e1 f8+2 d2 e0 f8+6 d2 d0 f9 +3 d12 f9) t2

+ O (t3) = 0 (9)

(2e2+f1+2 e0 f3+d0 f4+3 e02f6+2 d0 e0 f7+ d02f8)) t0

+ (6e3 + 2 e1 f3 + d1 f4 + 6 e0 e1 f6 + 2 d0 e1 f7 + 2 d1 e0 f7 + 2 d0 d1 f8) t1

+ (12e4 + 2 e2 f3 + d2 f4 + 6 e2 e0 f6 + 3 e12f6 + 2 d0 e2 f7 + 2 d1 e1 f7 + 2 d2 e0 f7+2 d2 d0 f8 + d12f8) t2+ O (t3) = 0 (10)

where fj = beta*cj. As these equations must be true for arbitrary t, all coefficients of tj must be 0. This requirement produces

6 (nonlinear) equations.

2d2 + f2 + e0 f4 + 2 d0 f5 + e02 f7 + 2 d0 e0 f8 + 3 d02 f9 = 0 (11)

2e2 + f1 + 2 e0 f3 + d0 f4 + 3 e02 f6 + 2 d0 e0 f7 + d02 f8 = 0 (12)

6d3 + e1 f4 + 2 d1 f5 + 2 e0 e1 f7 + 2 d0 e1 f8 + 2 d1 e0 f8 + 6 d0 d1 f9 = 0 (13)

6e3 + 2 e1 f3 + d1 f4 + 6 e0 e1 f6 + 2 d0 e1 f7 + 2 d1 e0 f7 + 2 d0 d1 f8 = 0 (14)

12d4+e2 f4+2 d2 f5+2 e2 e0 f7+e12 f7+2 d0 e2 f8+2 d1 e1 f8+2 d2 e0 f8+6 d2 d0 f9+3 d12 f9 = 0 (15)

12e4+2 e2 f3+d2 f4+6 e2 e0 f6+3e12 f6+2 d0 e2 f7+2 d1 e1 f7+2 d2 e0 f7+2 d2 d0 f8 + d12 f8 = 0 (16)

from which the coefficients dj and ej may be found. Although the equations are nonlinear they are easily recognized to be

solvable in a sequential manner. Thus knowing d0, e0, d1, e1, one may find d2, e2 followed by d3, e3 and finally by d4, e4. It

is remarked that this situation is quite similar to that of the fourth order Runge Kutta method [1] in which the 4 nonlinear

equations which are obtained are also solved in a sequential manner.

III. TESTING THE SOLUTION

5

A. The test geometries

The geometries used for testing the precision of trajectories will be taken from the class of concentric spheres. There are two

reasons for this choice: 1 the geometries have known theoretical potential distributions, 2 the path of certain trajectories are

known and are circular. A typical geometry is shown in figure 2, together with the path of the circular ray passing along the

median plane.

Figure 2. The basic concentric sphere geometry with the outer sphere at 10 volts, the inner sphere at 0 volts. Also shown is

the path of a ray traversing along the median plane and exiting the geometry at z=55.

It can be shown that if the particle enters the device normal to the entrance plane and with a kinetic energy E0 equal to

the pass energy its trajectory will be one of constant radius hence exit the device at a known point on the exit plane. The pass

6

energy can be shown to be:

; <<-9-�=> = 1/2��?@. + 6-A. � ∗ �B9/��?@. − �B9� − �-� �17�

Where �?@.is the potential on the outer sphere, 6-A. � is the potential difference between the two spheres, �-the potential at

the entrance point of the ray, and D?@., DB9 the radii of the outer, inner sphere respectively. For such a ray the deviation of the

exiting point from that of the circular trajectory provides an immediate and direct measure of the accuracy of the ray tracing

process for the entire ray. The exiting point (the crossing of the exiting plane at r = 0, z ) is called the zero or z-intercept point

and its error will be referred to as the z-intercept error.

B. Properties of ray trace solution

Background details

A ray is traced through the geometry not in a single step but in a number of small steps each of time parameter deltat which as

mentioned above is also the length of the step in units of mesh point spacing. The complete ray is formed by the following: at

the end of any time step the starting position and velocity of the next step is set to the ending position and velocity of the

previous step and the next step commenced. The process continued until either the ray intersects an element or crosses the exit

plane. It is of course clear that the larger the deltat the less precise will be both the time and space interpolations but the more

rapid will be the traversal of the ray through the geometry.

It is remarked that one of two arithmetic units were available for the ray calculation. The first was the one using the native

“doubles” representation provided by the compiler, the other being an external arithmetic unit in which the precision of the

data could be arbitrarily set to a user specified number of bits (the number of bits was taken to be 120 corresponding to ~30

decimal places). Although the double representation has an inherent precision of ~10-17 it was degraded by cumulative effect

of roundoff errors to ~10-11. And so for precisions greater than 10-11, the multiple precision unit was used in order that the

position of z-intercept would not be affected by the calculations themselves.

As previously mentioned a square mesh was overlaid on the geometry. The value of the potential was set on these meshpoints

to its theoretical value which allowed the precision of the ray to be found independent of the precision of the mesh point

potentials themselves. Having done this it is emphasized that the value of the electric field was not taken to be its theoretical

value but determined by the coefficients of potential expansion described above thus enabling the influence of the interpolation

algorithms to be measured apart from any errors in the underlying potential. In a special case considered below the potential

was taken to be that produced by a relaxation process itself so that the influence of the accuracy of the potential on the precision

of z-intercept might be established.

As described above there are two series which have been created, one a time series for r(t) and z(t) and the other a space series

for v(r, z). Seen is that there are two independent parameters which on which the properties of the process are based, i.e. the

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order of the time and space series respectively. As noted above space algorithms were created having order of 2, 4, 6, 8, and

10. For each of these, time algorithms were created having order 4, 6, 8, and 10. Order t and order s are defined as the order of

the time and space algorithms resp.

For each selection of order s, order t one may also select a value of deltat, the time step, where, as mentioned above, for deltat=1

the particle will traverse a distance in each of its steps of ~1 mesh point spacing. The implication is that the smaller the time

step the more steps the particle will make before exiting the geometry and the longer will be the elapsed time for the completion

of the trajectory. The small but finite step distance gives rise to a problem in accurately determining the point at which the ray

crosses the exit plane, since only rarely will the end of the final step land exactly on the axis. To improve the accuracy of this

determination, the time step for the final ~1 mesh point distance from the exit plane was made quite small (.001) and in addition

a fifth order polynomial fit was made using the 4 points immediately preceding the axis crossing thus finding with improved

accurately the position of the axis crossing itself (z-intercept).

The properties of the ray on order t, the order of the time algorithm

To obtain a qualitative description of the intercept error on deltat the highest density geometry of the study was used (Rin,

Rout) = (320,640) for a ray calculated using order t = order s = 10, our highest order algorithms. For this situation the intercept

error was found as a function of deltat and is shown in figure 3.

Figure 3. A plot of the log (intercept error) vs log deltat. Seen is that for deltat < 1 the intercept error approaches a convergent

value of 10-22.

8

In figure 3 two distinct regions are seen. One in which the error changes rapidly as deltat is increased from 1 to 40, and the

other in which the change of the error is somewhat insensitive to a change of deltat. The boundary between the two is seen to

occur at deltat ~= 1 It will be seen in a later study that in the region deltat > 1 the rapid change of the error with deltat follows

an approximately exponential relation which is governed by the order of the time algorithm used. In contrast to this for deltat

< 1 the approach to its convergent value is considerably less rapid and may be interpreted as resulting from a slight improvement

in the precision of the space interpolation algorithm as the interval is shortened (deltat decreased) over which the interpolation

is made.

In order to examine the rapid change in precision with a change of deltat a sequence of data was created for a spatial order of

10 using various time orders and each are plotted as a function of deltat. Points were included in the sequence which were in

the rapidly changing portion of the curve. The result is shown in figure 4 where each data set was for a separate order t.

Figure 4. The log (intercept error) is shown plotted vs the log (deltat) for time order t of 4, 6, 8, and 10, all made with the

space order of 10. Also inserted in the figure are the results of a linear fit to the curves.

9

Clearly evident is that the larger the order of the time algorithm the more rapid the change of intercept error and hence the

faster the curve will decay to the more slowly changing portion of the curve, see figure 3. In order to have a more quantitative

description of the decay, the slope of each curve is plotted vs order t itself and given in the next figure.

Figure 5. A plot the slope of the lines in the previous figure vs order t. The linear fit to the data is shown.

From this figure it is seen that the slope of the intercept error for a given order t is approximated by: slope ~= the order of the

time algorithm – 2. As specified above the order of the time algorithm has been defined as the order of the expansion of r (t) or

z (t) (see equations 7 or 8). However as the equation of motion involves the second time derivative of these quantities the time

order of the process is reduced by 2 which is reflected in the observation that the time order of the process seen in the above

figure is ~2 less than the time order of the basic expansions.

The implication of the above is that the dependence on the intercept error on algorithm time order and deltat can be written:

B9.-�-;.-��?� = �6-A. ./<��"E3�F+G!+F!43�*4��� �18�

Where s is a constant dependent on the algorithm order. Thus the intercept error is seen to be proportional to

6-A. .�"E3�F+G!+F!43�*4��� which is a very rapidly varying function of deltat for the high order algorithms.

10

The properties of the ray on order s, the order of the space algorithm

It is clear from the discussion in the introduction that although the z-intercept may reasonably rapidly converge to its final value

the accuracy of the value to which it converges will be dependent both upon the order of the spatial algorithm together with the

density of the mesh overlay.

The accuracy of the zero intercept is here investigated as a function of the order of the potential or space algorithm used in the

ray trace process. Again using our highest density geometry (Rin, Rout) = (320, 640) and a time order of 10 ensuring the most

rapid time convergence as possible, the error of the z-intercept for a given order s is found by determining the convergent value

of the log (intercept error) vs deltat for small deltat (see figure 3). In this manner the z-intercept error can be determined as a

function of order s, the results plotted in the next figure.

Figure 6. This figure displays the intercept error vs the order of the spatial algorithm used for the ray using the highest

density geometry.

Seen in this figure, depicted for the highest density geometry available, is that there is ~2 order s of magnitude precision increase

for every unit increase in algorithm order. For example, the precision of the order 4 algorithm is seen to be ~ 4 orders of

magnitude more precise than the order 2 algorithm, etc. and displays the precision gain resulting in the use of high order s

algorithms in the ray trace process. In order to determine the dependence of the precision on mesh density, the intercept error

vs order s was determined (using order t = 10) for the 4 test geometries studied: (Rin Rout), (320 640), (160 320), (80 160),

11

and (40 80). In the next figure the results are plotted where 1/Rout was taken as a measure of the mesh density for a particular

geometry.

Figure 7. The log intercept error is plotted vs 1/Rout for the various geometries studied. Data for each geometry was taken

with the indicated order s algorithm.

Seen in the above figure is that the larger the algorithm order the more rapid the increase in precision as the density of the

geometry is increased. The difficulty with the low order algorithms, order 2 in particular, is apparent in the figure, namely that

its precision is relatively insensitive to the mesh density implying that it would be difficult to reach the precisions of the higher

order algorithms simply by an increase of mesh density. To obtain quantitative information from the above graph, linear fits to

the data sets were made and the slopes of the resultant lines are plotted vs the algorithm order in the next figure.

12

Figure 8. The slope of the data of figure 7 is plotted vs order s, the order of the potential algorithm incorporated into the ray

trace process.

From this figure it is seen that the slope of the intercept error vs 1/Rout is approximately order s – 1. This likely follows from

the fact that that the space order enters the calculation via the electric field which due to its definition as the gradient of the

potential has a space order one less than that of the potential series from which it is derived. This observation is consistent with

the interpretation of the dependence of the slope of the intercept error on the time order algorithm which was given in figure 5

and subsequent discussion.

IV ADDITIONAL TOPICS

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A. Overview

It is perhaps useful to provide an overview of the theoretical description of what has been done particularly in the context of

other methods in current use to create high precision potential and field calculations; for example, the Boundary Element

Method and the Finite Element Method. From the point of view of field determinations, the only quantities used in the current

derivation are the spatial expansion coefficients, cj given in equation (1). While the explicit implementation for finding the set

of cj’s has made use of the methods described in references [] this in no way precludes the calculation of these quantities by

the BEM and FEM methods. And so the ray trace process proposed here is in fact implementation independent.

B. Comparison between Runge Kutta and present power series method

The Runge Kutta method is widely recognized as a fast diffeq solver. (A historical reference to this immense and complex

subject is given in Butcher[9] in which is also found further references to orders higher than 4). It thus seems useful to attempt

to make a comparison of it with the present method. More specifically the 4th order Runge Kutta (RK) method, as described

by Heddle [1], will be compared with the above power series method (PS) using the same 4th order space interpolation

algorithm in both. (It should be noted that the 4th order Rung Kutta method used by Heddle and others is here assumed to be

equivalent to the 4th order method presented above provided that a matching of the spatial order s is made.) to the ms. In order

to estimate the time for a single step, the number of elementary operations +, -, *, and / will be found for each since the time

required will be proportional to the number of these operations.

One of the 6 Runge Kutta equations given by Heddle [1] can be represented by:

IJ = ℎ�@� + AJ���,AJ =−ℎ ∗ L���� + AJ��, �� + IJ��� �19�

Where@� ,��, AJ��,IJ�� are constants. The only significant term for the purposes of counting elementary operations is the field

interpolation term L���� + AJ��, �� + IJ��� where L���, �� is ∝ O���, ��/O� and may be evaluated using equation (1). In

order for each method to use the interpolation process a certain number of initialization operations must be performed and the

number will be the same for both methods. The comparison of the two methods is given in the following table:

TABLE I. The number of elementary operations required for each processes.

4th order Runge Kutta (RK) 4th order power series (PS) notes

tosetL���, �� To set 6: , -: 184 450 The ops used in setting each

initialization operations 2202 2202 initialization ops required

2386 2652 Totalops

From this table it is clear that the RK method has a ~3 fold time advantage in setting the field as compared with setting the

coefficients6:, -: for the PS. However in order to use the 4th order potential algorithm a certain amount of initialization must

be done and this initialization will be the same for both methods. It is noted that this initialization will be required for each

time step of the ray and is not simply done once before the start of the raytrace. This initialization is seen to dominate the ops

14

required for each process implying that both process will take an equivalent time for the calculation of a ray step. Thus the

time duration for the RK and PS methods (for an order s = 4) seem to be equivalent.

C. Using a relaxed potential rather than a theoretical potential.

In the above precision tests the theoretical values of the potentials were used. To determine the effect of using relaxed potential,

the various geometries were relaxed with an order 10 algorithm as described in [] and this relaxed potential was used in the ray

trace process again incorporating the most precise algorithms available: order s = 10 and order t = 10. Thus the dependence of

the zero intercept for a particular Rout could be determined for both situations. It was found (not shown) that for Rout of 40

to 320 the zero intercept found using the relaxed potential was ~ 2 orders of magnitude less precise than that using the theoretical

values. This implies that care should be taken when assuming that the accuracy of the rays using the theoretical values reflect

those when using the relaxed potentials, even when using the highest order algorithms possible.

Of more significance is the comparison of the precision using the relaxed mesh to that of the average potential error along the

path of the ray. Such a comparison is given in the following figure.

Figure 9. The ratio of the normalized intercept error to the normalized potential error plotted as a function of log 1/Rout.

In this figure only is the error given as a normalized value. Elsewhere the error is reported as simply the difference between the

determined value its theoretical value. However as Rout depends on the geometry considered whereas the average potential

along the ray does not, it seemed judicious for the sake of comparison to normalize both by their average values. Thus seen in

figure 9 is that the intercept error is within an order of magnitude of the potential error and for the highest density mesh studied

15

in the comparison are the two essentially equivalent. Thus the potential error along a ray path is in fact a lower bound of the

intercept error, but more importantly is within a factor of 10 of the actual intercept error. It should be noted that figure 9 has

been created using very high order spatial and temporal algorithms. No generalization to lower spatial order s is implied.

D. A comparison between low and high order processes

The last figure which is presented is a comparison between the order s = 10 algorithm and the order s = 2 or 4 algorithms on

the precisions of the zero intercept. For this study the order t algorithm was taken to be 10, the theoretical values of the potential

were used, as well as the highest density geometry (Rin, Rout) = (320, 640). Also the unnormalized values of errors are

reported. The following displays the result.

Figure 10. The log of the intercept error is plotted vs the log of the time step deltat.

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Seen is that in the limit of small deltat all 3 curves approach their convergent value. Although as has been remarked above high

order time order processes (Runge Kutta) have been used, few have used any spatial order other than 2. An example of the

precision of the order 2 process can be found in reference [6]. The exceptions are Kursheed [7] and Chmelfk [8] who use an

order 4 potential algorithm which they form from the product of two order 2 polynomials one in r and the other in z. Although

their use of this field interpolation has shown an enhancement of precision of ~10 for the case of Chmelfk, perhaps more for

Kursheed, the power series itself has included all order 2 terms but did lack some order 3 and order 4 terms. It is suspected that

the application of this interpolation give results closer to an order 2 or perhaps order 3 algorithm than to an order 4 algorithm.

It should be noted however that the problem they were solving was one from the finite element method (FEM) where the

overlaid mesh was not regular and their achievement of improvements over the order 2 algorithm is in itself quite remarkable.

V CONCLUSION

With the availability of a high order FDM solution to the curved boundary value problem, it has been possible to determine

potentials with considerably accuracy. As the primary motivation for an accurate potential calculation is the precise tracing of

rays, the use of the high order potential algorithms in a framework for the ray tracing process has been lacking. This paper has

provided such a framework, combining the expansion coefficients of v(r, z) with those of r (t), and z (t), thus forming an

accurate solution to the equations of motion. The conclusion of this study is in fact demonstrated in figure 10 in which an

enhancement in the precision of the ray calculation is seen to be ~ fourteen order s of magnitude over the current order 2 or 4

technologies.

REFERENCES

(1) D.W.O.Heddle, Electrostatic Lens Systems, 2nd ed., 2000, Institute of

Physics Publishing, ISBN 0-7503-0697-1, pp 32 to60.

(2) David Edwards, Jr., “FDM for curved geometries in electrostatics II: the minimal algorithm." Proceedings of the

International Multi Conference of Engineers and Computer Scientists 2014, Vol I, IMECS 2014,March 12 - 14, 2014, Hong

Kong

(3) David Edwards, Jr., "Finite Difference Method for Boundary Value Problems. Application: High Precision Electrostatics"

Proceedings of the International Multi Conference of Engineers and Computer Scientists 2015, Vol I, IMECS 2015,March 18

- 20, 2015, Hong Kong

(4) David Edwards, Jr. “Accurate calculations of electrostatic potentials for cylindrically symmetric lenses”, Rev. Sci. Instr.,

54, 1, 729 (1983)

(5) David Edwards, Jr., “High precision electrostatic potential calculations for cylindrically symmetric lenses”, Rev. Sci.

Instr., 78, 1-10 (2007)

(6) T.J.M.Zouros, F.M.Spiegelhalder, David J.Manura, "Modelling SIMION's accuracy for hemispherical deflector analyzers using a spherical capacitor", Proceedings of the Seventh International Conference on Charged Particle Optics, Physics

Procedia 1 (2008) 461–466

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(7) Anjam Khursheed, The finite element method in charged particle optics, Kluwer Academic Publishers, ISBN 0-7923-

8611-6, 1999.

(8) J Chmelik, Jim E. Barth, "Interpolation method for ray tracing in electrostatic fields calculated by the finite element

method", Proc. SPIE 2014, Charged-Particle Optics, 133 (September 3, 1993); doi:10.1117/12.155693

(9) J.C. Butcher, “A history of Runge-Kutta methods”, Applied Numerical Mathematics 20 (1996) 247-260./