130 IEEE TRANSACTIONS ON EDUCATION, VOL. E-30, NO. 3, AUGUST 1987
Spreadsheet Solution of Partial Differential EquationsMARION HAGLER, FELLOW, IEEE
Abstract-This paper points out that a spreadsheet program on a To be specific [5], suppose we represent the rectangularhome or personal computer permits the solution of partial differential region as a 17 x 9 array of cells. On the spreadsheet, weequations in two independent variables with considerably less effort represent the array in columns A-I and rows 1-17 Thusthan conventional programming languages. The computational mole- rpea ay i nrcolumnta- and rows 1-7 Thus,cule approach is identified as a natural approach for using spread- rray is thesheets to solve elliptic, parabolic, and hyperbolic partial differential and 117 on the spreadsheet, as shown in Fig. 1. Supposeequations in rectangular, skew, and curvilinear coordinate systems. the value of the dependent variable is 400 on the top sideThe technique is relatively independent of the type of spreadsheet or of the rectangle, 300 on the right side of the rectangle,host computer employed, although the spreadsheet must allow forward 200 on the bottom, and 100 on the left side. To minimizeor circular references for iteration. Each of the examples in the paper .'
was~~~~~~~ruon ahmcoptrwh48RA. typlng, we enter the value 400 in cell Al and COPY thisvalue into cells Bl-H 1. We then enter the value 300 intocell I1 and COPY it into cells I2- 16. We next enter 200
I. INTRODUCTION into cell I 17 and COPY it into cells B17-H 17. Finally,we enter 100 into A17 and COPY it into cells A2-A16.
PREADSHEET programs have been used in engi- We have thus specified the value of the dependent varia-Lneering primarily for formula evaluation [1]-[4], al- ble at each boundary cell. To specify the values in the
though the possibilities of analyzing logic networks [2] interior cells, we enter into cell B2 the expression (B1 +and of solving ordinary differential equations have been A 2 + B3 + C2)/4 and COPY it into the other interiorrecognized [3]. This note points out that the spreadsheet cells using relative (as opposed to absolute) addressing.environment is especially convenient for solving partial We then repetitively calculate the spreadsheet until thedifferential equations (PDE's) in two independent vari- values in all cells converge. Iteration is necessary be-ables. The spreadsheet COPY (REPLICATE for some cause, for example, the value computed for cell B2 de-spreadsheets) command with relative addressing permits pends upon the value in cell B3 which in turn dependsthe numerical solutions to be obtained with dramatically upon the value in cell B2. This circumstance is usuallyless programming effort than is required by writing pro- termed a circular or forward reference and must be al-grams in a language such as Fortran. Moreover, some lowed by the spreadsheet program if the program is to bespreadsheet programs include built-in graphics capabili- useful in the present application.ties that permit graphs of the solutions to be viewed on In this particular example, approximately 170 iterationsthe screen or printed with little additional effort. The con- were required to converge to seven figures with 0 as thevenience of solving partial differential equations with initial value in the internal cells. The number of iterationsspreadsheets may therefore permit students to experiment can be reduced by inserting more realistic values in thewith numerical solutions to a larger number and variety intemal cells. With Syncalc on an Atari 800, each itera-of equations than would normally be possible. tion required slightly more than 2 s. About 4K of memory
was required [6]. With Lotus 1-2-3 on a Texas Instru-ments Professional Computer with an 8087 coprocessor,
As an especially simple example of solving a PDE with each iteration required slightly more than a second. Thea spreadsheet, consider finding the solution of the two- values obtained are shown in Table I. Fig. 2 shows graphsdimensional Laplace' s equation in Cartesian coordinates of columns C, E, and G (corresponding to x = 0. 25, 0.50,within a rectangular region for which the value of the de- and 0.75, respectively) prepared and printed with spread-pendent variable is specified on the boundary. If we imag- sheet commands.ine the x-y plane to be subdivided into a grid of squarecells, then the value of the dependent variable in each cell III. COn MpLECULESnot on the boundary is given simply by the average of the Laplace's equation is an example of an elliptic partialvalues in the four cells immediately above, below, right, differential equation. The finite difference equations forand left of the cell. such PDE's are easily formulated in terms of computa-
tional molecules [7] . Computational molecules for the La-placian in rectangular [8] and polar coordinates [9] are
Manuscript received February 21, 1986; revised September 2, 1986. shown in Fig. 3.The author is with the Department of Electrical Engineering/Computer Asa aplcto ofomutinloeuesn
Science, Texas Tech University, Lubbock, TX79409.Asa plcto ofomuton oeuesiIEEE Log Number 8714265. spreadsheet solution of PDE's, consider the solution of
Fig. 4. Grid pattern for solving Poisson's equation over the cross sectionof a keyed hollow circular shaft. (Adapted from [10].)
3 I I I I I I I I I I I 1 and K = 1) the expression (1.67 * C2 + 2.16 * B3 +ci (.5 1 1.5 1 *A2 + 2.16 * BI - 1/12)77. This expression should
2 be copied into cell Bl and into cells B3-B16. Fig. 4 showsF ig.5 t X = 0.5 xA.75 that in the expression within cell Bi it is necessary to re-
Fig. 2. Graphs of solutions to Laplace's equation verses y, for the bound- n th expreary conditions of Fig. 1, at x = 0.25 (column C ), x = 0.50 (column E) place BO by B16, and in cell B16 that it is similarly nec-andx = 0.75 (column G). essary to replace B17 by Bi. From Fig. 5(b), we see that
in cell C3 we should write (again assuming R = 1 and KPoisson's equation (also elliptic) in the irregular cylindri- = 1) the expression (1.62 * D3 + 1 * C4 + 1.16 * B3cal geometry of Fig. 4 [10]. Notice that R is the major + 1 * C2 - 1/11.5)/4.78. This expression should beradius of the keyed hollow circular shaft, and that K is the copied into cells C4-C 15. Because of the boundary con-constant on the right-hand side of Poisson's equation. ditions shown in Fig. 4, zeros should be entered into allSpreadsheet cell Al corresponds to the first node on ring cells on rings A and D and into cells Cl, C2, and C 16.A (r = R/4), and so forth. The rings are separated in Table II shows the solution to Poisson's equation in theradius by R/4 and the nodes in angle by 22.5 degrees. geometry of Fig. 4. The solution converged after aboutThe computational molecule of Fig. 3(b) is specialized to 15 iterations of approximately 1.5 s each on the Atari 800ring B (r = R/2) in Fig. 5(a) and to ring C (r = 3R/4) and required 3K RAM. On the TI PC, each iteration re-in Fig. 5(b) [9]. From Fig. 5(a), we see that in cell B2 quired less than one second. Fig. 6 shows full circlewe should write (for simplicity in notation assuming R = graphs of the dependent variable on rings B ( r-=R/2 )
132 IEEE TRANSACTIONS ON EDUCATION, VOL. E-30, NO. 3, AUGUST 1987
RING B: and C (r = 3R/4). The graphs were made with LOTUS1-2-3 graphics commands on the TI PC.
W167J [7] gives additional computational molecules for skewcoordinate systems and for cells on irregular boundaries.
-7.0 It also shows how to impose boundary conditions on de-1V 2 rivatives of the dependent variable in ways that fit easilyR ~ into the spreadsheet environment [1 1].
( I.0 )IV. THE WAVE EQUATION
(a) The preceding examples have applied what is usually
RING C: termed the explicit formulation of the finite differenceequations to the solution of elliptic PDE's. This approach
1.62 can be applied to parabolic PDE's (such as the diffusionequation) or to hyperbolic PDE's (such as the wave equa-
4.78 tion) as well, although numerical instabilities can arise in2 11.5 these two cases (as opposed to the elliptic case) unless the
R2 grid size is sufficiently small [12].1.16 As an example, consider the one-dimensional wave
equation in x, t with phase velocity, c. The computationalmolecule for this equation is shown in Fig. 7(a). This
(b) molecule is slightly different from the previous ones inFig. 5. The computational molecule of Fig. 3(b) specialized to: (a) ring B that it represents the entire one-dimensional wave equa-
of Fig. 4., and (b) ring C of Fig. 4. (Adapted from [10].) tion rather than just the Laplacian operator. In any case,it allows us to find the values of the dependent variable at
TABLE II a particular x and t in terms of its values at earlier times.This molecule therefore corresponds to a step-by-step or
r 0.25 0.50 0.75 1.00 marching method of finding the solution as time pro-
(degrees) gresses [13]. If the computational grid is square in the0.00 0.0000 -0.0363 0.0000 0.0000 coordinates x and ct, then the computation is (barely) nu-22.50 0.0000 -0.0395 0.0000 0.000045.00 0.0000 -0.0531 -0.0424 0.0000 merically stable and the computational molecule simpli-67.50 0.0000 -0.0613 -0.0540 0.0000 fies to that of Fig. 7(b). We suppose the boundary con-90.00 0.0000 -0.0653 -0.0577 0.0000
112.50 0.0000 -0.0670 -0.0590 0.0000 ditions are those in Table III and, for simplicity, that the135.00 0.0000 -0.0678 -0.0594 0.0000157.50 0.0000 -0.0681 -0.0596 0.0000 phase velocity is unity [14]. These boundary conditions180.00 0° 00°00 -0.0°66882 _0.0597 ° °°°° correspond to, for example, a taut string, fixed at both202.50 0.0000 -0.0681 -0.0596 0.0000225.00 0.0000 -0.0678 -0.0594 0.0000 ends, initially at rest, stretched in the form of a parabola,247.50 0.0000 -0.0670 -0.0590 0.0000270.00 0.0000 -0.0653 -0.0577 0.0000 and released at time t = 0.292.50 0.0000 -0.0613 -0.0540 0.0000 From Fig. 7(b) notice that the solution at a given time315.00 0.0000 -0.0531 -0.0424 0.0000337.50 0.0000 -0.0395 0.0000 0.0000 step depends not only on the values of the dependent var-
iable for the preceding time step, but also on the valuesfor the time step before that. This circumstance leads to
Poissoi's Equation a problem in beginning the calculation, since the depen-Li dent variable is specified at the single time t = 0. The
/ X problem is easily overcome, however, by using the thirde 1.1 \ / boundary condition in Table III to obtain a modified com-
-ILa] l l putational molecule, shown in Fig. 8, to begin the cal-culation [14].
-0. We choose an increment of 0.2 over the range 0 < x< 1 and 0 < t < 2.2. We let cell A I correspond tox =0-0, B1 to x = 0. 2, * * *, and F1 to x = 1.0. According
[email protected] \a Xs jag ,!,m,! to the boundary conditions of Table III, we write 0.00 in_______________-\ A1, 0.64 in BI, 0.96 in Cl, 0.96 inDI, 0.64 inE1, and-I.E6 _ |0.00 in F 1. The boundary conditions also require us to1 ~~~_______ write 0 in cells A2-A12 and in cells F2-F12. From Fig.
117 8, we write (Al + C1)/2 in cell B2 and COPY it withrelative addressing into cells C2, D2, and E2. From Fig.
theta IdeQrees)or.8.51 t r - .75 7(b), we write (A2 + C2 -B1) in cell B3 and COPY itFig. 6. Full circle graphs of the solution of Poisson's equation on rings B with relative addressing into the rectangular region
(r = R/2) and C(r = 3R/4) of Fig. 4. bounded by cells B3 and E12.
HAGLER: SPREADSHEET SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 133
Fig. 7. Computational molecules for the one-dimensional wave equation t- v_~
with a: (a) rectangular grid, and (b) square grid. (Adapted from [141.) a,]8
BOUNDARY CONDITIONS FOR 020 aX2 -22/at2 0 ~ 1> s'
LR 0.I A.2h,0,I,b, I.7 0.8 M,(Oft) 0:
0 t: h,0 t El .21 t:= 0. t: 0.6 1% t: 08. 7 t = .0
(1 0 t)0 ~~~~~~~Fig. 9. Graph of the spatial dependence, with time as a parameter, of the¢ ( 1 . 0 tt)= O ~~~solution to the one-dimensional wave equation with the boundary con-
ditions of Table III.
f(XI 0) =
Wave Equatio'A
- -e 1 {C~-
----------.- .- -- --
Fig. 8. Computational molecule of Fig. 7(b) modified for beginning the -11 nlweencalcualtio nin row 2 . (Adapted from [14].) , .1,'.;0 1eeh, 5 0.6 0 7 M 0 1i I.l l i 1.5I 6 1. l8 I92
O x 0 0.1 t X 0.2 Oxh 0.3 AX:0.4 XX:0.5
The results, presented in Table IV, show the solution Fig. 10. Graph of the time dependence, with position as a parameter, ofhas progressed through one period in time. The solution the solution to the one-dimensional wave equation with the boundarycnnditionsvqof Table-IonI
each and required lessthan 2K RAM. A spreadsheet with~~~~~~~~~~~~~War Eqato
134 IEEE TRANSACTIONS ON EDUCATION, VOL. E-30, NO. 3, AUGUST 1987
say 10 x 10, are very modest, less than 5K RAM beyond pendent variable, and hence for each column) for the present valuesthat requiredfor the spreadsheet program itself. of the dependent variables in terms of the time and of the previousthat requiredl for the spreadsheet program itself. values of the dependent variables. Special expressions may be re-
quired as cell entries in the first few rows if the solution algorithm isACKNOWLEDGMENT not self-starting. (Runge-Kutta algorithms are popular because they
The author wishes to acknowledge that the Texas In- are self-starting, and hence easier to program.) After the first fewrows, however, all cell entries in a given column are the same andstruments Personal Computer and the LOTUS 1-2-3 can therefore be replicated using the COPY command with relativespreadsheet program used in the work presented here are addressing. The approach outlined here is a slight extension of that
part ofadonaiontothe Colege ofEnginering a Texas described in [31, which considers a single ordinary differential equa-part of a donation to the College of Engineering at Texas drtionand uses the spreadsheet columns only to keep track of interme-Tech University by Texas Instruments. diate variables calculated in the course of the solution. Chapter 10 of
[7] offers good treatments of various algorithms, including Runge-REFERENCES Kutta, for numerical solution of coupled sets of first-order ordinary
differential equations, and of the conversion of higher order ordinary[1] S. R. Trost and C. Pomernacki, VisiCalc for Science and Engineer- differential equations into coupled sets of first-order ordinary differ-
ing. Berkeley: Sybex, 1983. ential equations for solution.[2] L. P. Huelsman, "Electrical engineering applications of microcom- [14] See [7], pp. 418-420.
puter spreadsheet analysis programs," IEEE Trans. Educ., vol. E- [151 In principle, the approach described here can be extended to PDE's27, pp. 86-92, 1984. in more than two independent variables by constructing a super
[3] N. D. Rao, "Typical applications of microcomputer spreadsheets to spreadsheet that consists of an array of smaller spreadsheets, each oneelectrical engineering problems," IEEE Trans. Educ.. vol. E-27, pp. of which might correspond, if for example, the independent variables237-242, 1984. were two spatial variables and time, to the spatial values of the de-
[4] A. Hills, "Use of a spreadsheet program in an antenna design appli- pendent variable at a particular instant of time. See [7], pp. 385, 392-cation," IEEE Trans. Antennas Propagat., vol. AP-14, pp. 585-587, 395 for a computational molecule for such a case.1986.
[5] R. W. Hornbeck, Numerical Methods. New York: Quantum, 1975,pp. 286-287.
161 If the array of nodes is increased to 17 x 33, then 19K of RAM Marion Hagler (S'61-M'72-SM'79-F'80) re-beyond that required for the spreadsheet program itself is needed, each ceived the B.A. and B.S.E.E. degrees from Riceiteration requires about 9 s on the Atari, and about 700 iterations are University, Houston, TX, 1962 and 1963, respec-needed to converge to 8 figures. Changes in the values of the depen- tively. He received the M.S.E.E. and Ph.D. de-dent variable with the reduced grid size were a few tenths of a percent. grees from the University of Texas, Austin, in
[7] R. L. Ketter and S. P. Prawel, Jr., Modern Methods of Engineering 1964 and 1967, respectively.Computation. New York: McGraw-Hill, 1969, ch. 11. He joined the Department of Electrical Engi-
[8] See [7], p. 325. neering, Texas Tech University, Lubbock, in 1967[9] See [7], pp. 337-338. and is now P. W. Horn Professor and Chairman.
[10] See 171, pp. 349-352. He has taught most of the required courses in the[IlI See [71, pp. 386-390. undergraduate curriculum and a variety of gradu-[12] See 171, pp. 381-385, 418. ate courses as well. His research interests include coherent optical systems[13] A very similar step-by-step method can be used to solve coupled sets and plasmas.
of ordinary differential equations. With this approach, column A con- Dr. Hagler is a Fellow of the Optical Society of America and a membertains the sequence of time (independent variable) values and columns of the American Physical Society, the American Society for EngineeringB, C, * * *, contain corresponding values ofthe several dependent var- Education, Sigma Xi, Tau Beta Pi, and Eta Kappa Nu. He presently servesiables. Row I contains the initial values of the dependent variables. as Chairman of the IEEE Technical Activities Board Awards and Recog-The interior cells contain expressions (generally different for each de- nition Committee and, thereby, on the IEEE Awards Board.
