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IM A Journal of Numerical Analysis (1985) 5, 201-214

Implicit Finite-difference Solutions of the EnthalpyFormulation of Stefan ProblemsV . R. VOLLER

Centre for N umerical M odelling a nd Process Analysis,School of Mathem atics, Statistics and Computing,Tham es Polytechnic, Wellington Street, Londo n SE\i 6PF

[Received 29 February 1984 and in revised form 26 September 1984]When related to a phase-change problem, an implicitfinite-differencediscretizationof the enthalpy formulation results in a system of non-linear equations at each timestep. In this paper, various numerical enthalpy methods based on suchdiscretizations are outlined and examined. An alternative discretization for anenthalpy formulation is developed on separating the sensible and latent heat terms.This approach also results in a non-linear system of equations but with the non-linearity isolated as a source term of nodal latent heat. This offers an advantageover the previous techniques in that only one variable (i.e. temperature) is solved forin the resulting iterative scheme. Comparison with simple one- and two-dimensionaltest problems indicate that the computing requirements, with the alternativediscretization, are reduced by between 20 and 50%.

1. IntroductionTH E SO-CALLED ENTHALPY METHODS are a common choice in the numerical solutionof moving phase-change problems (Stefan problems). An impo rtant reason for this isthe fact that the continuously moving phase front does not have to be tracked over adiscrete numerical grid. Investigations have shown, however (Bonacina, Comini,Fasano & Primicerio, 1973; Voller, Cross & Walton, 1979), that the accuracy ofenthalpy solutions is influenced by the position of the phase front Stemming fromthese investigations methods have been developed which overcome the problems ofinaccuracy (Voller & Cross, 1981, 1983a, b; Voller, 1983). Another problem withenthalpy solutions arises when an implicit finite difference discretization is used.Such an approach results in a set of non-linear equations to be solved at each timestep. Not only are these equations relatively difficult to solve (especially incomparison with explicit enthalpy methods) but they are also computationallydemanding. A number of schemes for solution of implicit finite difference enthalpymethods have been proposed (Meyer, 1973; Shamsundar & Sparrow, 1975;Longworth, 1975; Furzeland, 1980, White, 1983). The aim of the current work is toinvestigate these methods using simple one and two dimensional Stefan p roblems. Inaddition an alternative implicit scheme will be developed and tested against theexisting schemes.2. Test Problems

The enthalpy H is defined as the sum of sensible and latent heats. The standardformulation for a m ultidimensional melting/freezing phase change problem takes the201. 0272-4929/85/0202 01 + 14J03.0 0/0 _ _ .1985 Academic Pros Inc. (London) Limited.

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2 0 2 V. R. VOLLER

formp ^ = V.(K(VT )), (1)where T is temperature, K conductivity and p density. The enthalpy andtemperature may be related via{ H/C, -Cs>H,

e(H - L/2)/(Ce+L/2\ -Ce^H^Ce + L, (2a)(H-L)/C, H>C e + L,

or alternatively byCT, - e > 7 \

-eTe, (2b)T>e,where C is the specific heat, L is the latent heat of the phase change and e > 0 is atemperature half range over which the phase change occurs. Note that equations(2a) and (2b) have been derived assuming that the phase change occurs abouttemperature Tm = 0 and that the thermal conditions are constant throughoutTwo test problems are introduced: (1) a one-dimensional problem of freezing inthe semi-infinite half space with fixed surface temperature T < 0 at the surfacex = 0; (2) a two-dimensional problem of freezing in a square duct (side 1 m) withfixed surface tempera ture T < 0. The thermal, boundary and initial conditions forthese problems are given in Table 1. It may be noted that in both these problems thethermal conditions (i.e. conductivity etc.) are constant throughout By suitablemodification of the discretization equations [see equation (4a)], all the methodsexamined in this paper can be applied to problems with varying thermal conditions.Such considerations in the current work, however, were thought to introduceunnecessary complications.3. Solution Methods

On discretizing the area of interest and using Taylor Series approximations(Smith, 1965) or "control volume" conservation (Patankar, 1980) the followingTABLE 1

Con ditions for test problem s (S I un its)

KCLTm^Inittal5x

One dimension22-5 x 10 610 '0- 1 020-125

Two dimensions22-5 x 106100- 1 020-1 (=

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ENTHALPY FORMULATION OF STEFAN PROBLEMS 2 0 3

weightedfinite-differencereplacement for equation (1) may be derived.H ' + 1 = H J+(5(1 9)F(T*) + 8t 0F(T'+1 ) (3)where

H* = (tf5,fl2l# $ , . . # ; ) and 1* = (7* 7 } , . . , 1* )are vectors of nodal enthalpies and temperatures respectively at time t = k 5t. Th eparameter 9 is a weighting factor and can take values between 0 and 1. The vectorfunction F results from the space discretization of the right hand side of equation (1).The form that F takes depends on the geometry and conditions of the problem inquestion. For a one-dimensional problem with constant thermal properties an ithcomponent of F is

_ Kp 5x 2 ~1

If there is a step change in the conductivity at the phase change interface thenequation (4) is modified asL 7 3 ] . (4a)

whereK K K if T T1 "-2 -^liquid U li > J rIn a two dimensional problem in a square region nh x nh covered with a squaremesh of side h, with node numbering from bottom left to top right, an ithcomponent of F is _ KNote that both (4) and (5) give components of F at internal nodes. For nodes on oradjacent to boundaries the form of a component of F will need modification.The relationship between the respective nodal enthalpies and temperature inequation (3) is given by equation (2). Since the three possible forms of thisrelationship are not known a priori, in cases when B > 0 (i.e. implicit solutions),equation (3) becomes a non-linear set of equations. A common approach to solvingsuch equations is to use a modified Gauss-Seidel iterative scheme as suggested byOrtega & Rheinbolt (1970). Meyer (1973) uses equation (2b) writing (3) in terms of T

If at time t = j 5t the vector of nodal temperatures T1 is known the correspondingvector at time t = (j+l) 5t, i.e. T ; + 1 may be calculated from (6). The components ofan initial estimate TJ0+i are calculated fromtF{], (7)

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2 0 4 V. R. VOLLERi.e. the explicit scheme (6 = 0) is used to approximate H J + 1 and the initial estimatefor T J + 1 follows from equation (2a). The initial estimate for TJ+ 1 is improved uponvia the Gauss-Seidel iteration in which the k+ lth estimate is given by

where f 0, -e>Thl-L, Tt>erc+xe, -e>T,,

R*m = e

an dThe value of the parameter X and the form of the vector function G depends on theproblem geometry and conditions. In a one-dimensional problem with F defined byequation (4)

(9)For a two-dimensional problem with F defined by equation (5)

a n d GT) = -^iJi_l + T i-n + T i+a +T i+ l). (10)In all cases the most current iterative values of temperature are used to calculate

Equation (8) is applied until a convergence criterion is satisfied with the values ofa* and p* calculated at the beginning of each iteration. On convergence theprocedure is repeated to find T1+ 1 and so on.The major drawback in using equation (2b) is that as e -> 0 the iterative schemedefined by equation (8) will not converge due to the rapid change in H(T\ reducingto a jump discontinuity when = 0, near T = 0. In practice as E becomes small thevalue of a,* associated with the node which is centred on the control volumeunderg oing th e phase cha nge, jum ps between values close to 0 and L in successiveiterations making the convergence of equation (8) impossible.One way of overcoming the above problem is to use equation (2a) and writeequation (3) in terms of the enthalpy H. This step will involve employing thefunction (H). This function is piecewise continuous for all values of e (in particular = 0). In this m anner equation (3) becomes

X (11)where

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ENTHALPY FORMULATION OF STEFAN PROBLEMS 2 0 5

Shamsundar & Sparrow (1975) have solved equation (11) via a Gauss-Seidel-typeiterative scheme, a method that has been numerically investigated by White(1983a, b ) . With known nodal enthalpies H J at time t = j 5t initial approximationsare obtained from the explicit scheme,

(12)These initial approximations are improved upon via the Gauss-Seidel iterationsdefined by

where F and G are now functions of O and0, -C

X8EL

, H,>Ce + L ,1 -C >HC' ^

XBThe value of X an d X form of G , are defined above, equations (9) and (10), for simpleo n e - and two-dimensional examples.An alternative way of solving equation (11) is to use a Newton method. Such amethod has been presented by L ongw orth (1975). With the noda l enthalpy fieldknown at time t = j dt an initial approximation for the nodal enthalpies at timet = (J+l)5t is made via equation (12). This is improved upon by a sequence ofcorrections

H t t i = H + 1 + t C., (14)where the ith component of Cm is

C C ( ] - " l -and

In the three methods outlined above, for solving the implicit discretization of theenthalpy method, the form of the iterative scheme, equations (8), (13) and (15), at

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2 0 6 V. R. VOLLER

each time step depend on the nodal temperature or enthalpy field at that time step.Since only approximations exist (viz. equations (7) and (12)) for these values at thestart of an iterative loop the form of the scheme may be expected to change duringthe iterations. This makes it necessary to check the nodal enthalpy or temperaturefields at the end of each iterative sweep and modify the vectors a* and P* or at, j) orJm as appropriate. This restriction introduces an element of inefficiency into thevarious schemes.An approach proposed by Furzeland and Elliott (see Furzeland, 1980) results inan iterative scheme which does not require a detailed check of the solution fields.The enthalpyfinite-differencc-scheme equation (3) is written as

H{+1+WT+1 =b, (17)where

bi = H{+5t(l-0)F{+5t6Gl+i. (18)Equation (17) may be written in a point iterative form. If at each node point i, whencalculating thefcth terative values, bt is considered to be a constan t (i.e. independentof [7IK +1 ) a Newton linearization leads to an "inner" iterative scheme for [73^ + 1which on dropping the k subscript is

b ' [ ^ ^ ^ [ 7 J i + 1 , (19)

where the subscript p indicates the position of the inner iteration.The Furzeland/Elliott iterative scheme may be implemented as follows. Initialestimates [T] + 1 are generated from equation (7). During the k + lth iteration ateach node point in turn bt is calculated using the most current available values.From equations (2) if-e(C + X6)

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ENTHALPY FORMULATION OF STEFAN PROBLEMS 2 0 7

4. A New Implicit Solution MethodThe aim of this section is to present a new implicit solution method. Like theFurzeland/Elliott scheme the form of the iterative equations derived are independentof the nodal solution fields. In addition the method performs efficiently in its rawstate without the need of additional enhancements.The basic principle in the new implicit enthalpy method is to separate out thelatent and sensible heat components. First equation (1) is written as

! ( C r + A#) V.(KVT), (20)orwhere on assuming the relationship between enthalpy and temperature as given byequation (2b)CO , - e ^ T,

AH = eis the latent heat component of the enthalpy. On re-arrangement equation(20) becomes

pC = V.(KVT) + S (22)atwhere S is a latent heat source given by

The essential feature of equation (22) is that the latent heat contribution has beenincluded into the formulation via a source term. On seeking an implicit finite-difference solution this fact ensures that the non-linearity associated with the latentheat may be isolated and dealt with efficiently. On defining AH to be the vector ofnodal la tent heats given by equatio n (21) a g eneral finite-d ifferen ce scheme forequation (22) isCT> +1 = CT i + 5t(l-ff)F(J') + 5teF(T J+ 1) + S'+\ (24)

where

The vector S in equation (24) can be regarded as a vector of heat sources, eachcomponent representing the change in the latent heat content of the control volumessurrounding the nodes. The physical significance of S can be understood onconsidering a freezing problem. In a freezing problem over each time step there is anet heat loss from each control volume. For the control volumes in the (/+ l)th timeinterval (i.e. \J 5t, (j+1) +1 = 5t(l-ff)F(TJ)+5t8F(JJ+ l). (25)

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2 0 8 v- R- VOLLERIn control volumes in which no phase change occurs in the (j+ l)th time interval allthe heat loss is accounted for by a change in the sensible heat (i.e. CT). If a phasechange occurs within a control volume in the time interval then only a fraction ofthe heat loss is accounted for by a change in the sensible heat The remainder of theheat loss is accounted for by a latent heat change. The value of Sj*1 will give thislatent heat change in the ith control volume. Hence in each control volume duringthe (J+ l)th time interval the latent heat loss can be calculated as

_r0; " > eS{ - | _ 4 + l ; otherwise (26)where the condition

T/ eimplies that no phase change occurs in the ith control value during the (J+ l)th timeinterval. The value of A{+1 in equation (26) will depend on the fraction of the timeinterval in question over which the phase change occurs in the ith control volume.There are three possibilities. The phase change in control volume i (1) commences ata point in the time interval, (2) occurs over the entire interval or (3) is completed at apoint in the time interval. With these possibilities in mind A{+1 can be calculated as

- i ? [ e i + 1 - C ( ^ - ) ] , T/>e,RQ{+\ (27)

The value R represents the fraction of heat loss accounted for by the latent heatchange during the phase change. If the temperature enthalpy relationship is of theform given in equation (2) then

For a melting problem similar equations may be employed for calculating S{+1 withappropriate sign changes in the conditions.An efficient solution methodology for equation (24) is as follows. In each time stepa. Gauss-Seidel iteration is defined as

(28)

where F and G are functions of T as defined above and

If the ith control volume is identified as changing state (i.e. 0 < [Aff j j(+ 1 < L) thenthe nodal temperature is set as

via equation (2). On the completion of each iterative sweep the values of [S])(ti and

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ENTHALPY FORMULATION OF STEFAN PROBLEMS 209Initial values T + 'Generated from explicitscheme equation (30)

Initial values S + 'Generated from equation (26)

fcth estimate [TJi+ 'Generated from equation (28)

kth estimate [S]i + IGenerated from equation (26)

Convergence check

If convergence go tonext time stepFIG. 1. Major steps in solution over one time itep.

hence [A tfJ^ +i are updated via a su itable iterative form of equation (26). Thisprocedure is continued to convergence.To initialize the procedure on each new time step, values [T]o +1 a r e generated viathe use of a fully explicit scheme with S^ + * set t o zero, i.e.(30)

(31)In the control volumes where [SJ 0 + 1 ^ 0 0 e - t n e control volumes changing phase)

Initial values [S yo+ i are then estimated via equation (26) with

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2 1 0 v - R- VOLLERthe initial estimates for temperature are modified as

. (32)Figure 1 is a flowsheet which illustrates the major steps in the solution for one timestep.5. Comparison of Methods

In comparing the techniques outlined in this paper for computational efficiencycare has to be taken that none of the methods is biased by the chosenimplementation. To safeguard against this, where possible, the basic structure of thesoftware for each technique was similar. In addition the convergence criterion, (33,

where y is a prescribed tolerance, was used in all techniques.The performance of all the techniques presented in the paper may be enhanced byrelaxation techniques. The major aim of the current work, however, is to test therelative performance of the available techniques in their basic form. Hence in thecomparison runs relaxation was not employedIn comparing the results four methods will be examined. The method based onequation (13) will be called "White", the method based on equation (15) will bereferred to as "Longworth", the method based on equation (19) will be referred to as"Furzeland" and the method described by equations ((26) and (28)) will be referredto as "New". Each of these methods was applied in a "Crank-Nicolson" mode, i.e.9 was fixed at i. In all five runs were carried out on both the one- and two-dimensional test problems. The specifications for each of these runs are recorded inTable 2.Table 3 shows numerically predicted nodal enthalpies at x = 05 m in dayintervals up to 12 days for Run 1. These results are typical of all runs in thatpredictions from each of the methods are in agreement up to the third decimal place.In fact, the Furzeland, White and New methods are in agreement up to the fifthdecimal place. In addition the number of iterations per time step required in theFurzeland, White and New methods were identical at all time steps in all runs.These results clearly indicate that each of the above methods is solving the samediscrete problem.

TABLE 2Specification of runs

Run no12345

DimensionsOn eOn eOn eOn eTw o

5t (hours)4124124

Range e000-50-50

Convergence y10"210"2l O ' 210"210"*

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ENTHALPY FORMULATION OF STEFAN PROBLEMS 211In presenting results associated with the efficiency of the methods the Whitemethod is chosen as a benchmark for CPU comparisons. In this way the CPU

requirement for the White method is given the value 1 in each run with the CPUrequirements of the other methods normalized appropriately.Table 4 shows a run-by-run comparison for the CPU usage on a Dec PDP 11/34for each method to reach the termination point (i.e. 12 days in the one-dimensionalproblem, complete phase change in the two-dimensional problem). The majorconclusion that may be drawn from these results is that the New method onlyrequires between 049 and 0-81 the CPU time of the next best method.In addition to the above conclusion the following points are made.(1) Introduction of a mushy zone range e favours the new method.(2) The improvement in CPU usage for the new method when applied to a two-dimensional problem is not as marked. This is due to the fact that inmultidimensional problems m ore than one control volume is changing state atthe same time. Hence the procedure for updating the latent heat source termhas to be used more than once in each iterative sweep. This fact suggests thatin complex problems in which many control volumes are changing state

TABLE 3Enthalpy Predictions at x = 0-5 m, Ru n 1

Time (days)123456789101112

Run no12345

Longworth103 5 x 10102110171O091O0710060-9210-6850-4350182- 0 O 3 0- 0 O 4 7

White11111

White1035 x 10910221017100910071O060922068704370184- 0 O 3 3- 0 0 4 7

TABLE 4CPU usage

Relative CPUFurzeland

07260728076207830738

Furzeland1035 x 1010221017100910071O060922068704370184- O 0 3 3- O 0 4 7

requirementNew05380521042303860606

New1035 x 109102210171O091O071O060922068704370184- 0 0 3 3- 0 0 4 7

Longworth1-3401-7541-3481-760No run

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2 1 2 v - R- VOLLERsimultaneously the CPU usage of the White or Furzeland and New methodsmight be compatable.

(3) The programming requirements for the Furzeland method are less, e.g. in onedimension the F urzeland method required abou t 60 lines and the new methodabout 80 lines.(4) As noted above, SOR techniques may enhance the results given in Table 4.Investigations suggest that due to the small number of iterations per time step(on average 23) in the one dimensional runs, SOR would have no noticeablebenefit. In two dimensions, employing the simple SOR strategy of a globalvalue of 2 > a> > 1 for control volumes not changing state and a value ofco = 1 for regions which are changing state (Elliott, 1981), does reduce theiterations. The additional complexity in the programming, however, offsetsany improvement in efficiency. If SOR is to improve the efficiency then itappears that sophisticated methods for choice of an optimum co will need tobe developed. One such candidate, proposed by Elliott (as reported inFurzeland, 1980) is where co is varied automatically according to the relativesizes of the two-phase regions.

6. ConclusionsThe aim of this paper has been to investigate and develop implicit numericalsolution methods for the enthalpy formulation of phase change problems. Theapproach adopted has been pragmatic in nature and the major interest has been inthe development of an implicit enthalpy method which is computationally moregeneral and efficient than existing methods. A rigorous analysis of the numericaltechniques presented is left to another time.The basic principle in the development of the new implicit method is theseparation of sensible and latent heat terms in a discretized enthalpy formulation,thereby introducing a latent heat source in the equations which acts as alinearization term in subsequent iterative solutions. This represents an advantageover previous schemes in that the form of the iterative scheme does not depend onthe iterated values. In addition, only one variable (i.e. temperature) is explicitlysolved for at each time step. In application to simple one- and two-dimensionalsolidification test problems the New method is 20-50% faster than previousmethods.Having an efficient means of solving the enthalpy formulation does not create a"magical" means of solving all phase-change problems. The inherent drawbacksfound in use of the enthalpy method, Voller et al. (1979), Bonacina et al. (1973), willstill make themselves noticed. Hence there is still a need for remedial schemes suchas those proposed by Voller & Cross (1981, 1983a, b) . With an efficient implicitenthalpy solution, however, such schemes may be applied more effectively (Voller &Cross, 1983b).Application of the methods developed in this work to more complex problems,e.g. geometric complex regions or variations in thermal properties, although"messy", should be possible. The reason for this is that the development of themethod is compatible with the control volume conservation numerical technique

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ENTHALPY FORMULATION OF STEFAN PROBLEMS 213

(Patanker, 1980; Schneider, Strong & Yaranovich, 1975). This approach in thedevelopment of a numerical scheme copes well with such situations. Essentially allthat is required is the correct formulation of the discrete equations describing heatflow in and out of control volume [e.g. Equation (4a)].

Enthalpy methods have proved a useful means of solving relatively simple phase-change problems. Current interest, however, is directed towards applications ofenthalpy formulations in more complex problems, e.g. convective/conduction phasechange (Morgan, 1981; Gartling, 1980). To move into such areas, accurate, efficient,flexible and robust enthalpy schemes are required. The enthalpy scheme presented inthis paper, it is hoped, goes some way towards meeting these requirements.The author would like to thank Dr C. M. Elliott of Imperial College, London, DrM. Cross of Thames Polytechnic and the referees of the IMA Journal of NumericalAnalysis for many helpful comments.

REFERENCESBONACINA, C, COMINI, G., FASANO, A. & PRIMICERIO, M. 1973 Numerical solutions of phasechange problems. Int. J. Heat Mass Transfer 16, 1825-1832.ELLIOTT, C. M. 1981. On thefiniteelement approximation of an elliptic variational inequalityarising from an implicit time discretization of the Stefan problem. IMA J. num. Analysis1, 115-125.FURZELAND, R. M. 1980 A comparative study of numerical methods for moving boundaryproblems. / . Inst. Maths Applies 26, 411-429.GARTLING, D. K. 1980 Finite element analysis of convective heat transfer problems withchange of phase. In Computer Methods in Fluids (Morgan, K., Taylor, C. & Brebbia,C. A., Eds). London: Pentech.LONGWORTH, D. 1975 A numerical m ethod to determine the temperature distribution around'a Moving Weld Pool. In Moving Boundary Problems in Heat Flow and Diffusion(Ockendon, J. R. & Hodgkins, W. R., Eds). Oxford: OU P.MEYER, G. H. 1973 Multi-dimensional Stefan problem s. SIAM J. num. Analysis 10, 522-538.MORGAN, K. 1981 Numerical analysis of freezing and melting with convection. Comput.

Meth. app. Mech. Engng 28, 275-284.ORTEGA, J. & RHEINBOLT, W. 1970 Iterative Solutions of Non-linear Equations in SeveralVariables. New York: Academic Press.PATANKAR, S. V. 1980 Numerical Heat Transfer an d Fluid Flow. New York: Hemisphere.SCHNEIDER, G. E., STRONG, A. B. & YARANOVICH, M. M. 1975 A physical approach to thefinite difference solution of the conduction equation in orthogonal curvilinear co-ordinates. ASME Annual Meeting, Houston, Texas, 1975. New York: ASMEpublications.SHAMSUNDAR, N. & SPARROW, E. M. 1975 Analysis of multi-dimensional conduction phasechange via the enthalpy model. J. Heat Transfer ASME 97, 33-340.VOLLER, V. R. 1983 Interpretation of the enthalpy in a discretized multi-dimensional re gionundergoing a phase change. Int. Commun. Heat Mass Transfer 10, 323-328.VOLLER, V. R. & GROSS, M. 1981 Accurate solutions of moving boundary problems using theenthalpy method. Int. J. Heat Mass Transfer 24, 545-556.VOLLER, V. R. & CROSS, M. 1983a An explicit numerical mediod to track a moving phasechange front. Int. J. Heat Mass Transfer 26, 147-150.VOLLER, V. R. & CROSS, M. 19836 Use of the enthalpy method in the solution of Stefanproblems. In Numerical Methods in Thermal Problems, Vol. 3 (Lewis, R. W., Johnson,J. A. &Smith, W. R., Eds). Swansea: Pineridge Press.VOLLER, V. R., CROSS, M. & WALTON, P. 1979 Assessment of the weak solution technique forsolving Stefan problems. In Numerical Methods in Thermal Problems (Lewis, R. W. &Morgan, K., Eds). Swansea: Pineridge Press.

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21 4 v - R- VOLLERWHITE, R. E. 1982a An enthalpy formulation of the Stefan Problem. SIAM J. num. Analysis19, 1129-1157.WHITE, R. E. 1982* A numerical solution of the enthalpy formulation of the Stefan Problem.SIAM J. num. Analysis 19, 1158-1172.