VOL. 9, NO. 3 WATER. R.ESOURCES RESEARCH JUNE 1973

Rayleigh-Ritz and Galerkin Finite Elements for Diffusion-Convection Problems

I. M. SMITH, R. V. FARRADAY, AND B. A. O'CONNOR Simon Engineering Laboratories, University o,• Manchester

Manchester M13 9PL, England

Finite element methods are presented for the solution of certain two-dimensional partial differential equations of interest in water resource problems. Earlier work using Galerkin's method for one-dimensional problems is shown to be a prototype finite element technique. It is suggested that previous variational (Rayleigh-Ritz) formulations of finite elements for some problems are misleading and are of limited application when compared with Galerkin's method. The accuracy and stability of the techniques presented are discussed in relation to the Well-known 'numerical diffusion and dispersion' phenomena prevalent in popular finite difference methods.

In recent years, considerable attention has been given to the numerical solution o.f linear, or piecewise linear, second-order partial dif- ferential equations of the form

04• • xO ( 04• u•4•) • s (1) Ot --•=_• •Xi Dii Ox i where

% variable, e.g., salt concentration in a river; t, time;

x•, Cartesian coordinate (x, y, or z); u•, convective velocity (u•, u•, u•);

Dii, second-order tensor, e.g., governing the dif- fusion of salt in an estuary;

s, general source and/or sink term, e.g., -k4• is used to specify the decay of a particular pollutant species in lakes.

Equation 1 is often written with reference to the principal axes of the tensor D,• to yield, in two dimensions,

o,/ot = O/Ox o,/Ox -

+ Olay (D• O4•/Oy -- u•4•) + s (2) This is believed to have wide application and has been used with suitable source and sink terms to describe separation in chromatographic processes, dispersion in porous media, sus- pended sediment transport in rivers and estuaries, the attenuation of flood waves in open channels, salt intrusion in estuaries, beach changes due to groynes, the dispersal of pollu-

Copyright ̧ 1973 by the American Geophysical Union.

tants in the sea, and heat transfer problems in rivers and lakes.

Various finite difference algorithms or 'schemes' have been presented for the solution of (2) or its simpler derivatives, such as the classical diffusion equation. It is well known that many of these schemes are partially un- satisfactory due to the formation of oscillations and numerical diffusion within the solutions. The occurrence of such errors has been demon- strated [Stone and Brian, 1963] by examining a Fourier series solution to (2). Oscillation errors were shown to occur due to the inability o.f the finite difference analog to convect in- dividual harmonics at the true flow velocity. Numerical diffusion resulted from a selective

damping of the high frequency harmonics. It is possible to minimize schematic errors

of the type discussed either by the inclusion of higher order terms. in the difference analog [Ro, berts and Weiss, 1966] or by cyclic applica- tion of the difference analog with weighting coefficients in small time steps [Stone and Brian, 1963; Shamir and Harleman, 1967].

These corrections are desirable in many prob- lems such as salinity intrusion in estuaries. However, the increased computational difficulty and time required to achieve a given accuracy makes the authors follow Price et at. [1968], Guymo• [1970], and Bredehoe]t [1971], who consider other methods of solution to be more advantageous.

593

594 SMITH ET AL.: DIFFUSION-CONVECTION

The present paper is thus concerned with pendent diffusion. The writers have found that finite element approaches to the solution of (2). some of the two-dimensional finite elements A va.riational method based on a new functional described in the following sections reduce to is considered initially, following on from the the 'chapeau' and 'smooth cubic' solutions in different functional given by Guymon [1970] one dimension of Price et al. However, the and Guymon et al. [1970]. However, this ap- finite element concept proves to be the ve- proach, which is dependent on an exponential hicle by which the extension of Galerkin tech- transform of •, is found to be limited in its niques to two- and three-dimensional situations application to real estuary problems, for ex- becomes straightforward and systematic in con- ample (see below), and therefore a finite ele- trast to the rather laborious classical applica- ment method based on a. Galerkin approach tions previously published. Two papers describ- is derived. The Galerkin method is found to ing finite elements generated by variational have much wider applicability, and, by adopt- techniques for the solution of one- and two- ing it, the same algorithm can be employed dimensional diffusion-convection problems have for the solution of (2), classical diffusion, or already been published by Guymon [1970] and simple convection, an •ttractive practical fea- Guymon et al. [1970]. They make use of the ture. fa.ct that although (2) is not self-adjoint, any

The methods of solution described involve second-order partial differential equation can be the storage of influence coefficients for • values rendered self-adjoint by the use of a trans- at fixed points in space as banded matrices and formation. In the case of (2), the transforma- the reduction of these matrices. These methods tion is simply are in common use in many fields of con- tinuum analysis. h = • exp (uxx/2Dxx) exp (uyy/2Dyy) (5)

PREVIOUS WORK which renders (2) in the form Many algorithms for finite difference solu- Dxx O•'h/Ox•'-] - Dy• O•'h/Oy •'-- [(u•'/4D•)

tions of (2) or its one-dimensional equivalent have been presented in the literature, and an -]- (u•'/4D•)]h = Oh/Or (6) exhaustive review need not be given again. The Guymon et al. [1970] then state that minimiza- work of Peaceman and Rach/ord [1955, 1962], tion of the functional Stone and Brian [1963], Shamir and Harleman [1967], Oster e• al. [1970], Lawson [1971], P = (D•/2)(Oh/Ox)•'-• - (Dy•/2)(Oh/Oy) •' O'Connor [1971], and Hobbs and Fawcett [1972] gives an idea of the scope of the meth- -]- [(u•/2) Oh/Ox + (u•/2) Oh/Oy]h ods. The results of some finite difference compu- -]- [(u•'/8Dx•) -]- (u•'/8D•y)]h •' (7) tations are compared in a later section with integrated over the element area is equivalent those obtained by finite element techniques. to solving the steady state form of (6). How- Of more immediate concern are other con-

ever, (7) represents only one such functional, tributions that a.nticipate the content of the and the writers suggest the alternative func- present paper. Price et al. [1968] solved the tional one-dimensional diffusion-convection equation by several techniques including conventional F = (D•/2)(Oh/Ox)•'-• - (D•/2)(Oh/Oy) •' finite differences, the method of characteristics, and a Galerkin type of method called 'varia- -]- (ux•'/8Dx•) -]- (u•'/8D•y)h •' (8) tional,' although it does not appeal to the which is nonnegative quadratic. In fact, the calculus of variations. The extension of this solution obtained by Guymon [1970] of a one- latter method to two dimensions was men- dimensional problem is correct only because rioned but no solutions were given. Cavendish equally sized elements were used and h (i.e., et al. [1969] did, however, give some solutions •) was prescribed at both boundaries. How- to two-dimensional non-time dependent diffusion ever, the writers believe that if different sized problems, and Culham and Varga [1971] con- elements are contiguous or if boundary con- sidered nonlinear one-dimensional time de- ditions of the type

SMITI-I ET AL.: DIFFUSION-CoNvECTION 595

(•)/(•n = C1•) (9) or

Oq•/On = C2 (10) where C• and C.o are constants are encountered, as is often the case in practical problems, that solutions based on (8) will differ from solu-

:_ la) (b)

tions based on (7). Further, Guymon et al. Fig. 1. Coordinate system. (a) Global coordi- [1970] claimed that solutions of practical in- nates. (b)Local coordinates.' terest can be obtained by these methods to estuary problems where the ratios of length times velocity to dispersion coefiqcient are in within such an element can be achieved by general higher than the ratios for porous media using the shape functions in local coordinates problems, although they did not verify this t/J, 7)as was suggested by Taig: belief with actual field tests. (Since the sub- mission of the present paper, Guymon [1972] Si - •(1 --•j)(1 -- v) has qualified the circumstances under which S• =-•(1- •j)(1-•-7) his formulation is valid.) The writers take an (11) opposing view because of the exponential $k = •(1 -•- •j) (1 -•- 7) nature of the transformation in (5). In long estuaries the exponent u•x/2D• commonly at- St = •(1 -•-•j)(1 -- 7) tains magnitudes of the order of several hun- To simplify the equations that follow, it is dreds, whereas a computer with 48 bit registers assumed that functions SF•, SFj, SFk, and SF•, will overflow at something like exp (80). In in global coordinates (x, y), which are equiva- contrast, Price et al. [1968] were able to solve lent to (11), are known, although in practice problems by using a Galerkin method when it is best to work in t/j, 7), to transform to u•x/D• was 100,000. In the following sections, (x, y) by using the Jacobian transformation finite elements are generated both by the [Taig, 1962], and to carry out the integrations Rayleigh-Ritz (variational) method based on numerically [Irons, 1966]. (8) and by the Galerkin method, operat- ing directly on the untransformed equation (2). Numerical results are then presented that allow a comparison of the two types of element. Of Guymon's [1970] five objections to Galerkin's method, only one, namely that it involves the solution of unsymmetrical co- efiqcient matrices, is upheld.

LINEAR QUADRILATERAL ELEMENTS None of the formal techniques described in

or

Thus for any element

ß

q• = [SFiSF•SFkSFi] •• l•,J

(12)

the present paper is new, and an excellent cov- and similarly if h is the variable. Since the erage of the scope of finite element methods time dependence in (1) and (6) is simulated is given by Zienkiewicz [1971]. The central by a series of instantaneous steady state solu- idea., common to all varieties of the method, tions, as is described subsequently, the varia- is the description of the variation of unknowns tional approach to the solution of (6) when (• or h in the present instance) in an element the right-hand side is 0, is to minimize F in by shape functions relating to the nodal values (8) and to integrate over the area of the finite of the unknowns. For example, a useful ele- element. Zienkiewicz [1971] shows that this ment is the quadrilateral described by Taig can be achieved by substituting the equivalent [1962] that has four corner nodes as is shown of (12) in terms of h into the right-hand side in Figure 1. A linear approximation of • or h of (8) and by equating the derivatives of I

596

with respect to the nodal h values in turn to 0. That is,

I - f,• FdA (13) where A is the element area, and four minimiz- ing equations can be written for elements not on a boundary,

OI/Oh, - 0 a = i, j, k, 1 which leads to the four equations

SMITI-I ET AL.'. DIFFUSION-CONVECTION cedures only exists for self-adjoint equations [Duncan, 1938].

Boundary conditions in which • or h is prescribed are straightforward, but other types commonly encountered are given in (9) and (10). Figure 2 shows a boundary element sub- ject to both types of boundary conditions. These conditions are handled in the same way both in Rayleigh-Ritz and in Galerkin methods.

(14) In the former methods, extra terms are added to the functional, whereas in the latter methods similar terms are generated when the second- order derivatives are integrated by parts. The additional terms are of the form

where [KR] is a 4 X 4 matrix obtained by integrating the following over the element area:

Ox Ox • D• Oy Oy'

( u: u: + k4O• + 4• a,• = i,j,k, 1

The symmetry of [KR] is evident from this expre•ion.

To implement Galer•n's method one may fohow Z&nkiewicz [1971] and use the shap e functions as weighting functions. Thus the fight-hand side of (1) is set to 0, • is sub. stituted from (12), and multiplication by the f•our'shape functions in turn is carried out, the result being integrated over the element area. The second-order derivatives are r•uced to first-order derivatives by integration by par•s. This reduction leads to the four equations

[KG]{•} ' = 0 (17)

s D.,,SF• a6/an •. dS (19) where n is the direction normal to surface S on which the boundary condition applies, and 1. is its direction cosine. T,hus the boundary condi- tion of the type given by (9) is

f•'• D,,,,SF,• C•ekl,, dS (20) whereas the condition of the type given by (lO) is

•• DxxSF,• C•.lx dS (21) For the boundary element shown in Figure

2, (20) leads to an addition to [KG] in (17) of

-- C, D,,,,(xk -- xi)

0 0 0 0

2 1

I 2

0 0

(22)

where [KG] is the integral over the element area of

Ox Ox + D,,,, Oy Oy 3

oy a, • = i, j, k, 1

Note in this case the lack of symmetry in the convective terms, but had Galerkin's method b•n applied to (6), the symmetrical form of (16) would have been regained. This equiva- lence between Rayleigh-Ritz and Galerkin pro-

½2

Fig. 2. Boundary conditions where the co.ncen- tration gradients are specified.

SMITH ET AL..' DIFFUSION-CONVECTION 597

whereas (21) leads to an addition to the right- methods [Mitchell, 1969] that approximate hand side of (17) of

C2 Dxx(Yk- Y•) (23)

METHODS OF SOLUTION

(24) by

ß + = 0 where 0 is a scalar parameter. When 0 -- •, the Crank-Nicolson method is obtained and

forward and backward differencing in time When the elemental [KR] or [KG] matrices yields

have been computed in any real problem, in- cluding boundary effects if necessary, they are {04•/0t}o q- {04•/0t}1 built together to form a large matrix [HI that has a.s many rows as there are unknowns • (2/At)({qb}l -- {qb}o) (26) but is often banded. For example, in an Combining (25)and (26)yields estuary problem there might be thousands of unknowns but the bandwidth might only be [[HI q-(2/At)[C]]{r)}i 20, the estuary being shallow in comparison to its length. Thus the problem of storing the • [(2/At)[C]- [H]]{qb}o- 2{F} (27) [HI matrix is not as serious as might at first appear. The [HI matrix is symmetrical in self- adjoint problems but generally unsymmetrical, although still banded, in non-self-adjoint prob- lems. In the former case the writers use Choleski reduction and in the latter case Gaussian elimination to solve the sets of linear equations. Thus Galerkin solutions take longer {0qb/0t}l • (1/A/)({qb}l -- {qb}o) (28) on the computer, and use about twice as much Combining (25) and (28) yields storage as solutions in the transformed un- known h. [[H]-[- (1/At)[C]]{qb}x

The time dependent nature of (1) and (6) is introduced in an approximate manner by • (1/At)[C]{c)}o- {F} (29) assuming that the time derivative is fixed at a recurrence relation guaranteeing uncondi- a certain instant in time and therefore that tional stability. In the next section, numerical time dependent problems can be solved as a examples are given of solutions obtained by series of steady state ones, connected by a the two methods given above. It should be recurrence relation of some kind. Each 'steady noticed that the matrices on the left-hand state' solution takes the form sides of (27) and (29) are constant in linear

[H]{4•} q-[C]{Oqb/Ot} q- {F} = 0 (24) problems with fixed time steps. Thus the re- duction of these matrices, the most time-con- At t = 0, {•b)o is some known starting con- suming part of computation in large problems,

dition. Matrix [C] is built up from elemental need only be done once and the solution of [TB] matrices in exactly the same way as [HI subsequent time steps involves only forward is built up from elemental [KR] or [KG] and backward substitutions in the Choleski matrices, where [TB] is found by integrating and the Gaussian elimination methods. The [SF,, SF•] for a, fi = i, j, k, l, over the ele- Gaussian method, which the writers use in the ment area. Various methods could be used to Galerkin solutions, does require considerably construct relationships between (•b)o and (•b)•, more storage in the computer, since an extra the solution at time At, but the writers have matrix is required for working space, compared considered the class of methods called implicit with only a vector in the Choleski method.

as the recurrence relation. This. method is be- lieved to be the most accurate, but has a rather smaller margin of stability than has the case of 0 -- 1, which is the fully implicit method. Here backward differencing in time yields

598 SMITH ET AL.: DIFFUSIoN-CoNVECTION

NUMERICAL EXAMPLES one mall subroutine is substituted for another and the only other alterations are in data The objective in this section is to compare

Rayleigh-Ritz solutions and Galerkin finite specifications. element solutions with one another and with ONE-DIMENSIONAL DIFFUSION-CONVECTION other published solutions for linear one- and (MISCIBLE DISPLACEMENT) two-dimensional situations. Roberts •nd' Weiss [1966] have indicated attributes that good This problem enables a comparison between numerical solutions should possess: Rayleigh-Ritz and Galerkin finite element solu-

(1) Conservation' material must not be lost tions to be made. A theoretical solution was due to numerical approximation. obtained by Ogata ancl Banks [1961], and

for the boundary and initial conditions (2) Stability: exponential growth of errors

must be absent, as must unacceptable oscilla- •(0, t) = •o t > 0 tions. --

(3) No dissipation: short wavelength com- •(x, 0) = 0 x > 0 (32) ponents must not be damped out (the nu- merical diffusion phenomenon). •(eo, t) = 0 t •_ 0

(4) No dispersion: different wavelength corn- is given by

ponents must not have different velocities. To achieve better accuracy in specific cases, • = •o exp u,x erf e •D,, . t•7•.j many authors have developed optimization

techniques that suit the particular problem [•(_u, tl with which they are dealing. The finite element -{- « erfc D•t)•/• (33) solutions presented in this section are not optimized in this way, although of course they This solution is exact for 0 _• x _• L provided could be. The writers' purpose is to show the that the concentration profile has not ad- adaptability of their methods to a variety of vanced more than approximately 2L/3 from situations, and in the time domain, for ex- x- 0. ample, only the Crank-Nicolson and fully im- A measure of the accuracy of a numerical plicit discretizations are discussed. However, solution may be defined by a few results obtained by using higher order finite elements in space are included. In par- ticular, calculations have been performed by using quadratic or cubic quadrilaterals in place of the linear quadrilaterals previously discussed. The quadratic quadrilaterals have midside nodes, as well as corner nodes, and thus 8 de- grees of freedom per element. A typical shape function is

E(t) = max I(t) - l (a4) O_<x_<2L/3

This error (subject to At << LAx/D• to en- sure that the time truncation error is neg- ligible) may be shown to have the form [Price eta/., 1968]

E(t) = G(t)(1/n)" (35) where n equals the number of elements used

«(1 --•)(1 -- V)(1 -{- V) (30) and G(t) equals some function of t. Hence The cubic quadrilaterals have corner nodes but in general 4 degrees of freedom per node, making 16 per element. A typical shape func- tion is

Sk = •(•fa _ 3•-t- 2)

log E(t) = log G(t) -- ot log n (36) and a plot of log E(t) against log n should result in a straight line of slope a.

Before the comparisons between the Ray- leigh-Ritz solution and the Galerkin solution

2 ß (va -- v -- v + 1) (31) are made, the limitations of the former method, as previously mentioned, need to be amplified. It should be emphasized that the incorporation When the exponent used in the transform be- of these different elements in a finite element comes large, any errors in h due to such factors computer program does not mean a large re- as roundoff are magnified. For example, for programing effort. In the writers' programs an exponent of 5 and an error in h of 0.001,

SMITH ET AL.: DIFFUSION-CONVECTION

the corresponding error in • is 0.3. It is clear that for larger values of the exponent, the same error in h will result in the • solution being completely masked.

For a valid comparison to be made, values of ux, Dx•, and x must be chosen so. that ex- cessive errors due to the transformation are avoided. Figures 3 and 4 show, for two values of the exponent, how the errors produced by the Rayleigh-Ritz and the Galerkin methods compare for linear elements. The graphs are straight lines of slope 2 and confirm the pres- ence of second-order spatial discretization error terms. The larger errors in the Rayleigh-Ritz solution are a consequence of a combination

599

I10 '

I00 •0

80 Rayleigh Ritz c 70 • 60

; 50

•, 40

30

0-001 0'005 Error E: 0.01 (>05

Fig. 4. Decrease of error with increasing num- of (1) error magnification due to the ex- ber of linear elements. The lines have slope ponential transform, and (2) the h solution for an exponent value of 1.25 and U, -- 1, D,x being more difficult to model mathematically 10, and L- 100. since Oh/Ox > O(k/Ox. Figure 5 shows a com- parison of the h and • solutions for an ex- ponent value of 3.125.

ONE-DIMENSIONAL DIFFUSION-CoNVECTION (SEDIMENT TRANStORT)

where • is the sediment concentration, D,• is the sediment diffusion coefficient, and u, is the sediment fall velocity (negative in (37) and (38)). The boundary conditions of the prob-

A second diffusion-convection problem of lem are interest is concerned with a description of sedi- ment transport in channels. An analytical solu- tion has been published by Dobbins [1944] and a numerical (finite difference) solution by O'Connor [1971].

The equation solved has the form

c3ek/Ot ---- D,, O2ek/Oy 2 - u• &k/c3y (37)

80

70

60

50

E 20 •1• •,Galerkin •

io 0'001 0'005 Error F_ 0-01 O-OS

Fig. 3. Decrease of error with increasing num- ber of linear elements. The lines have slope a for

(c3ek/Oy),_-•t = (u,/ D,,)ek (38) = (u,/ ©

where H is the water depth and •' is the steady state (t = o•) sediment concentration aty = 0.

I'O

(>4

0•3

(>2

(>1

0 0 0'1 0•2 0•3 0.4 O'S 0-6 (>7 O•

=/L

Fig. 5. Equivalent of • and h solutions for an exponent value of 0.5 and U, -- 1, Dx, -- 4, linear elements. The exponent value is 3.125 and and L_-- 100. U• -- 1, D,• -- 4, L -- 100, and t -- 25.

600 SMITYI ET AL.' DIFFUSION-CoNVECTION

0-024

O'O20

O.Ol 6

O-OI2

'•- -o-oo4 I

'"' ø0.008

• -o.012

-0'016

-O.OI2

- o. 024

IOO 200 300

NO OF TIME LABEL ELEMENT ELEMENTS At DISCRETIZATION O LINEAR 40 20 CRANK NICOLSON

• LINEAR 40 20 FULLY IMPLICIT LINEAR 40 5 CRANK NICOL50N

(• CUBIC 20 I0 FULLY IMPUClT

400• 700 •oo •K)O--._1000 • ,

Fig. 6. Maximum error bounds at y -- 0 in the P•ayleigh-Ritz finite element solution œor various types of elements, where Us -- --0.0527 ft/sec, D,, -- 0.49 ft¾sec, and H -- 56 feet.

Figures 6 and 7 show comparisons between in the main, as accurate as the number of the errors involved in the Rayleigh-Ritz and degrees of freedom allowed, whatever the types the Galerkin solutions, respectively, when com- of element used. pared with Dobbins's solution. Finite element In this problem, for physically reasonable results for the Crank-Nicolson method and the values of (u,y/2D,,), the discrepancies between fully implicit method show that for small time the Rayleigh-Ritz and the Galerkin solutions steps the Crank-Nicolson solutions are the more are not as great as was the case before. This accurate. However, at larger time steps they is because the gradients in the transformed oscillate about the true results as indicated solution are this time less than those in the by the error bounds in Figures 6 and 7, whereas untransformed solution, and in fact for the the fully implicit solutions proceed monoton- physical parameters chosen, the Rayleigh-Ritz ically to the steady state, despite an inferior solutions are the more efficient. It is evident accuracy at large time. Also, for large At, that in such diffusion-convection problems, it significant errors occur at y = 0 where the is worth solving problems by both methods 'shock' input cannot be accommodated. Figures before deciding on one or the other. However, 6 and 7 show the errors involved. These errors in two-dimensional situations, the presence of are less at y > 0 and, for example, at y = %H the double exponential transform is bound to and y = H, errors were less than 0.001 in all limit the applicability of the variational ap- cases tested. In fact the parameter values used proach. impose a severe shock condition and also cause In many real problems, D,, is a function of y a rapid convergence to the steady state, two [O'Connor, 1971] and this may readily be in- factors which limit the size of At. For smaller corporated in finite element solutions. values of u, and larger values of D,, the size of At may be appreciably increased. ONE-DIMENSIONAL CONVECTION

The improvement in accuracy with higher Stone and Brian [1963] considered a purely order elements is not particularly marked, and convective process for which conservative these results are reinforced by the writers' variational methods such as the Rayleigh-Ritz experience in other fields of analysis where method break down. Their solution is an ex- it. is found that finite element solutions are, ample of an optimized finite difference ap-

SMITH ET AL.' DZrrVSZoN-CoNvr•cTION 601

0-024

0.020

0'016

0012

8 0'008

• 0.004 ._

,• o

ß .e. -o.oo4

• - 0-CX38

• -o.•2

-0.016

-0-020

-0'024

200

ILABEL IELEMENT IEL%g ' TIME TSIzttlDISCRETIZ-A ,, '40 I 2 IOUid3RAnC I 2O I 3 IOU•K:I 20 1201FULLY ,lytPI•.ICIT ',• I 4 IQUADRA11C ! 20 I'S I cumc I 2o

300 400 SO0 600 700 800 )0 I000

Fig. 7. Maximum error bounds at y -- 0 in the 'Galerkin finite element solution for various types of elements, where U, -- --0.0527 ft/sec, D• -- 0.49 ft•/sec, and H -- 56 feet.

proach, weighting coefficients chosen to op- timize solutions to a particular finite difference analog being used. By contrast the same prob- lem can be solved by Peaceman and Rach[ord's [1962] scheme, which uses 'standard' finite dif- ferences. It is of interest to note that the equations produced by using Galerkin's method

with linear elements and a Crank-Nicolson time discretization can be reduced to Stone and Brian's optimized equations before cycling. Further, the spacewise discretization is the same as in Peaceman and Rachford's scheme.

Figures 8 and 9 show comparisons of results obtained by using finite elements generated by

06

Linear element solution '"-•...A F-•,•. -"Lineør element solution /•x= 0-04 . 7 ¾ •- AX = 0-02 I ß ,'-. /•t 00004168 •'-• f• [ •t- O'OOO416B

/ /

/ / • •.o.• /

i i i i i i

Cubic element solution

_

/ / /

i I i i I

I.I -

1.0

0'9

0-7

O6

0.$ 0'4

0'3 0.2

0'1

0

-01

-0.2 0 0.1 0'2 x 0'3 0-4 0'5 0 6

Fig. 9. Comparison of • cubic finite element solution with the Stone and Brian solution, where Ux -- 1.0 ft/sec and t -- 0.4 sec.

0 0'1 0'2 x 0'3 0-4 0.S 0-6

Fig. 8. Comparison of linear finite element solutions with the Peaceman-Rachford solution, where Ux -- 1.0 ft/sec and t -- 0.4 sec.

602 SMITI-I ET AL.' DIFFUSION-CoNvECTION

Galerkin's method and those obtained by Stone Mild oscillations near x -- 0 appeared (Figure and Brian [1963] in the boundary value prob- 11) in the solution when the former boundary lem' condition was imposed, but were removed by

forcing the nodes at x -- 0'•through 0. The •(x, t) - 0 0 •_ x _• 1 results in this case were very similar to those •(0, t) - 1 0 < t < 0.2 (39) obtained by Siemons.

-- -- The nine-point spread of the time derivative •(0, t)- 0 t> 0.2 in the finite element solution contrasts with

For the same spacing in x the linear finite the three-point finite difference spread, allied element solutions appea.r to be superior to with optimized weighting coefi%ients, as used the Peaceman and Rachœord solution but in- by Siemons. The complexity of the problem ferior to Stone and BriaWs optimized solution makes it difiqcult to ascertain which of the after cycling within time steps. However, for solutions is the more accurate; however, the the same Ax, cubic elements lead to an efiqcient finite element solution with • -- 0 at x - 0 solution without the need for cycling. was found to. be as conservative as Siemons'

solution in which the loss in mass was of the Two-DIMENSIONAL DIFFUSION-CONVECTION order of 1%. Siemons [1970] has solved the two-dimen- TWO-DIMENSIONAL CONVECTION

sional diffusion-convection equation with u, -- 0 by an iterative alternating direction implicit Roberts and Weiss [1966] have considered (ADI) finite difference method employing a finite difference method applied to the prob- Stone and Brian optimized weighting co- lem of convecting a Gaussian hump distribution efiqcients. A direct comparison can thus be (Figure 13) diagonally across a large square made between the ADI and Galerkin finite region. The following tabulation shows the con- element solutions for the convection and dif- centrations to be associated with the five con- fusion of the concentration distribution shown tours in Figures 13-16' in Figure 10. The following parameters were used in the comparison'

Ax = Ay = 200 meters

DI• = D• = 2 m2/sec u• = 0.25 m/sec

Con'tour Concentration

1 0.85 2 0.7 3 0.5 4 0.3 5 0.1

Finite element solutions using linear rec- Again the writers have used Galerkin finite tangular elements were computed for boundary elements (linear rectangles) to solve the prob- conditions O•/Ox -- 0 or • -- 0 at x -- 0. lem for a square mesh size of side 1/14, and The results are presented in Figures 11 and 12. a Crank-Nicolson approximation in the time

o o o o o o o o o

o o o o •o 0 o o o

+0 0 0 250500250 0 0 0 o 0 2•0•s002•0 o o ß + + .+ .+ +

0 250500750 I000••• 0 ,i- + .4. + .+ .4. .4. +

0 0 2%•35(:x:)7+.•0 5002•50 ? 0 0 0 0 250•250 0 0 0 4 + + + + + .* 4.

250 0 0 0 0 + •. .+ +

0 0 0 0 0 0 0 0 0 + + + + 4. ., + +. +

Fig. •0. Initial concentration distribution.

domain. Figures 14 and 15 show the computed distribution after convection over a distance of 0.3 for time steps of 0.07143 and 0.0102, respectively. For the larger time step, distor- tion of the contours is significant, and use of a fully implicit solution to alleviate this dis- tortion causes unacceptable dispersion and de- pression of the peak of the distribution. For the convected distance at which the result of Roberts and Weiss's fourth-order finite dif- ference solution has been plotted it appears that the. finite element solution (Figure 16) has somewhat less distortion of the contours for about the same diminution in height (13%).

SMITH ET AL.' DIFFUSION-CoNvECTION

0 0 0 0 0 0 0 0 -I -II 0 I I I 0 0 0 0 0 0 0 0

0 0 0 0 •

, • -S 7 -9 9 -7 I S -8 S 38 •• 531 274 • S -3 0 0 0 •

ß •x

Fig. 11. Comparison of the Galerkin •ite element solution (linear elements) with Siemon's solution (at x -- 0, 0•./0x -- 0) t -- 2• sec. The asterisks indicate the center of the concen- tration dist•bution at t -- 0.

603

In summary, for the range of problems solved described should have application to many by finite element methods, the solutions ap- branches of water resource research and de- pear to exhibit conservation and stability and velopment. It has been shown that the Galerkin to suffer from dissipation and dispersion to elements can be used with equal facility about the same degree as higher order finite whether diffusion or convection be the dom- difference solutions, many of which have been inant physical process being modelled. If, how- optimized in some way. ever, the scale of the problem is such that

formally equivalent Rayleigh-Ritz and Galerkin DISC•SSSIO• A•D CO•CL•Ss•o•s elements can be used, in terms of a trans-

Finite elements for use in diffusion-convec- formed variable, these are more efficient com- tion problems have been derived by the Ray- putationally from both the storage and the leigh-Ritz and the Galerkin procedures. To speed aspects than the equivalent solutions in judge from the volume of recent publications terms of the untransformed variable. All of on the solution of such problems, the methods the methods described involve the solution

x-O 0.' 0 0 0 0 0 0 0 -I -II 0 I I It 0 0 0 0 0 0 0 0

: Z I -2 2 -3 3 -3 I I -4 S -I -6 II II 8 0 -I 0 0 0 0 0 • 3 -S 7 9 • -9 S 3 -14 21 -II -21 98 169 • 13 -3 -I 0 0 0 0 •

Z • -S 7 -9 9 •7 I S -8 S 38 2• •6• S• 274 79 S -3 0 0 0 0

I -3 S -8 II -II 5 9 -23 43 165 3• • 843 7• • 217 53 I -2 0 0 • ,

0 -3 6 -9 II -I0 4 • -23 32 I• 372 6• 821 •0 •+ I• 41 -2 0 0 0 • • .• ,• .• .• .• + .. + + • -. -,- + 1- .l- +

C• -6 -7 -B 8 -6 2 4 -B II 49 221 512 664 .543287 91 II -2 0 0 0 • -4 4 -S 7 -7 S 0 -9 0 -lB 32 270 400 272 8B 6 -4 0 0 0 0 0 -6 7 -9 I0 -I0 6 4 -IS 22 -II -22 9S 152 82 12 -3 0 0 0 0 0 ...................... õ•.

, ,-, ,-,-, ,, ,,, o o o o o o o ,? + + + + + ß . + .i. + + .• + .i. + 0 o o o o o o o o o -I o I -I 2 o o o o o o o

Fig. 12. Comparison of the Galerkin finite element solution (linear elements) with Siemon's solution (at x -- 0, • -- 0) t -- 2500 sec. The asterisks indicate the center of the concentration distribution at t -- 0.

604 SMITH ET AL.' DIFFUSION-CoNvECTION

Fig. 13. Initial concentration distribution.

of sets of linear algebraic equations. When large problems are being solved, the storage required for the equation coefficients is substantial. However, many authors, when discussing ADI finite difference procedures, refer to the avail- ability of large capacity, high-speed computers. These were not available in the early 1950's when methods putting a premium on minimum storage and computation were evolved, and the writers do not believe that a good case has ever been made for not storing large matrices, when this can be done. Most modern scientific computers, even if they are incapable of treat- ing all of the store on one level as has been the case for 10 years on Atlas machines, have large, fast access backing stores. Therefore large matrices can be processed by partitioning or wave front techniques at the cost of some

Fig. 15. Concentration distribution at convected distance of 0.3 (.At -- 0.010204).

extra programing effort. To judge from the volume of debate in the literature about the accuracy and stability of such methods as ADI, this would be a small price to pay for guar- anteed, dependable results.

Further arguments in favor of ADI schemes have been stated by Smith [1965]. He fol- lowed Peaceman and Rachford in hypothesizing that stable implicit procedures of the type described in this paper, but with the equations solved iteratively by successive over-relation, would be some seven times slower than Peace- man and Rachford's ADI scheme. The writers cannot accept this figure in the light of the banded nature of the equation coefficients and of the need to reduce these coefficients only

Fig. 14. Concentration distribution at convected distance of 0.3 (At -- 0.071428).

Fig. 16. Concentration distribution at convected distance of 1.5 (2) TM (At --0.010204).

SMITH ET AL.: DIFFUSIoN-CoNvECTION 605

once in linear problems. Indeed an adjacent statement of Peaceman and Rachford's that 'solutions of these equations by an elimination procedure is out of the question' is certainly untrue in many hundreds of computing lab- oratories o•t•l•ywIn nonilinear problems, how- ever, such methods as ADI may retain their attractiveness.

The writers believe that finite element meth- ods have more than numerical advantages over finite differences. Bredehoeft [1969] has raised the question of how, in a finite difference approximation, the situation can be modelled in which the principal axes of the transmissi- bility tensor change in orientation, as they un- doubtedly do in an aquifer, for example. Since in finite elements, integrations are carried out element by element, it is possible to allow for these changes in orientation by simple co- ordinate transformation. The interested reader is referred to Zienkiewicz [1971] where such problems are considered. Other advantages lie in the interchangeability of programs. Although the writers' Galerkin programs use four pro- cedures or subroutines that are different from the Rayleigh-Ritz procedures, owing to the need to store a full band matrix rather than a symmetrical half, the actual programs differ only marginally. Similarly, programs to solve radially symmetric problems differ marginally from their plane two-dimensional counterparts. Three-dimensional elements, elements with curved boundaries, elements in time, and so forth are all available so that if some agree- ment can be reached on the solution of equa- tions such as (1), then more attention can be directed toward the essential engineering aspect of the physical basis of such equations.

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606 SMITH ET AL.: DIFFUSION-CoNvECTION

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(Received July 17, 1972; revised October 5, 1972.)