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AN APPROXIMATE-SOLUTION-BASED NUMERICAL SCHEME FORSTEFAN PROBLEM WITH TIME-DEPENDENT BOUNDARYCONDITIONSRizwan-uddin a

a Department of Nuclear Engineering, Computational Science and Engineering Program, University ofIllinois at Urbana-Champaign, 214 NEL, Urbana, IL, USA

To cite this Article Rizwan-uddin(1998) 'AN APPROXIMATE-SOLUTION-BASED NUMERICAL SCHEME FOR STEFANPROBLEM WITH TIME-DEPENDENT BOUNDARY CONDITIONS', Numerical Heat Transfer, Part B: Fundamentals, 33:3, 269 — 285To link to this Article: DOI: 10.1080/10407799808915033URL: http://dx.doi.org/10.1080/10407799808915033

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AN APPROXIMATE-SOLUTION-BASED NUMERICALSCHEME FOR STEFAN PROBLEM WITHTIME-DEPENDENT BOUNDARY CONDITIONS

Rizwan-uddinDepartment ofNuclear EngineeringComputational Science and Engineering Program,University of Illinois at Urbana-Champaign, 214 NEL,103 S. Goodwin Aoe., Urbana.Ll: 61801, USA

An approximate-locally analytic in the time variable and globally analytic in Ihe spacevariable-solution is developed for the one-dimensional Stefan problem with limedependent boundary conditions. Because of local analyticity in time, the solution is accurateover lime intervals thas are much Illrger than the step size permitted by most numericalschemes. The solution is developed by reducing the governing partial differential equauon(POE) 10 two ordinary differential equations (ODEs), and then simultaneously solving Ihemalong wilh the ODE for tbe moving-boundary inter/rue condition. The resulting schemerequires the solution of only a singk transcendental (algebraic) equasion at each limeinterval for the time-step-averaged temperature gradient at the moving boundary. Theposition of the moving boundary and the temperature distribution at Ihe end of the timeinterval are Ihen simply evaluated using approximate analylical expressions. Goodagreement with reference solutions is obtained.

INTRODUCTION

Moving-boundary Stefan problems can be solved analytically for only alimited number of special cases. Special numerical techniques with very small timesteps are often needed for accurate solutions. Numerical techniques are speciallyknown to have difficulties with time-dependent boundary conditions. Reviews ofthe analytical and numerical techniques for the Stefan problem appear regularly inthe literature [1-3]. Several numerical methods classified into fixed and variablegrid methods [1-8] have been reported. Gupta and Kumar extended a variable gridmethod to solve the Stefan problem with mixed boundary conditions [7]. A fronttracking method for the one-dimensional problem was proposed by Marshall [8],who introduced a variable-time-step procedure in conjunction with a predictor-corrector scheme. Comparison of various numerical methods has been made byFurzeland [9].

Menning and OZI§Jk reported an integral method that yields the movingboundary for the time-dependent boundary condition problem [10]. Their methodwas restricted to the determination of the moving boundary, and did not necessi-

Received 27 June 1997; accepted 8 October 1997.Address correspondence to Professor Rizwan-uddin, University of Illinois at Urbana-Champaign,

Department of Nuclear Engineering, 214 Nuclear Engineering Laboratory, 103 S. Goodwin Ave.,Urbana, IL 61801-2984. E-mail: [email protected].

Numerical Heat Transfer, Part B, 33:Z69-Z8S, 1998Copyright © 1998 Taylor & Francis

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270 RIZWAN-UDDIN

NOMENCLATURE

a half-length of thc domain (= t) x dimensionless spaceA, (e 2o'e _ I) coordinate (= x'/R')C R(I)dR(I)/dl x' dimensionalspace coordinateC, liquid specific heat a thermal diffusivityI(tl dcfined by Tl;x = 0, I) = 1(1) aTre r reference temperature changeIIsf latent heat [= (T' - T';')]r characteristic length al time intervalR dimensionless moving T, reference value for time when

boundary (= R' / I') movingboundary reaches rR' dimcnsional moving boundary IjJ time-interval-averaged value of theS pseudo source term spatial derivative at x = 1Stc Stefan number (= c,aTre,/h'i)1 dimensionless time [= 1'/(al' )] Subscripts and SuperscriptsI' dimensional time'in dimensionless initial time (prime) indicates a dimensionalT dimensionless temperature quantity

[= (T' - T';')/IJ.T", liquidor C?" - T';')/T';'l -I (bar I) quantity averaged over a

T' dimcnsionaltempcrature time intervalT' dimensional reference temperature -x (bar x) quantity averaged over theh

t; dimensional melting temperature spatial domain

tate a simultaneous solution of the domain 'interior temperature distribution.Semianalytical solutions to the problem-such as the one in [lO]-are of interestin that they can be used in developing more accurate numerical solutions for thoseproblems that do not admit such solutions. Moreover, numerical schemes based onsuch semianalytical methods usually allow time step sizes that are larger than thoseused with more conventional numerical schemes. Realizing these advantages andthe difficulties encountered in solving the Stefan problem with time-dependentboundary condition exactly, our goal here is to develop approximate analyticalsolutions for the moving boundary and for the temperature distribution, which cansubsequently be used in the development of efficient numerical schemes.

The approximate solution of the one-dimensional problem presented herewas suggested by a nodal integral method [11-13] developed to solve the movingboundary problem with time-dependent boundary conditions numerically [14].Numerical calculations showed that, in many cases, a single node (or computationalelement) stretching over the entire domain was sufficient to describe the temperature variation, and quite accurately predicted the moving-boundary position. Infact, very often a single computational node yielded results that were moreaccurate than those from calculations using more than one node [14]. The possibility of using a single computational element then suggested an approximate analytical solution to the Stefan problem with time-dependent boundary conditions.

The phrase time interval is used in this article to refer to the duration overwhich the locally analytic solution remains accurate. Realizing that the solution ofthe Stefan problem in general might be needed at times much larger than timeintervals over which the locally accurate solution remains valid, the approximate


NUMERICAL SCHEME FOR STEFAN PROBLEM 271

solution-much like a numerical scheme-is used repeatedly to march in time,using the solution at the end of a time interval as the initial condition for the nextinterval.

The nonlinear problem requires simultaneous solution of the moving boundary-which depends on the temperature gradient at the moving edge-and thetemperature distribution, which depends on the position of the moving boundary.Taking advantage of the structure of the mathematical formulation in an invariantdomain, approximations are introduced that permit-after some manipulationsthe solution of the time-interval-averaged temperature gradient at the movingboundary by solving a single transcendental equation. Position of the movingboundary at the end of the time interval is found in terms of the moving-boundaryposition at the beginning of the interval and the time-interval-averaged temperature gradient at the moving boundary. The spatially averaged temperature in thedomain can also be evaluated explicitly at the end of the time interval. Moreover,the development also allows for a reconstruction of the temperature, averaged overa time interval, as a function of space. The time-step-averaged, space-dependenttemperature is given explicitly in terms of the average temperature in the domainat the beginning of the time interval and the temperature or heat flux on the fixedboundary during the time interval. While the development is carried out andresults are presented only for the time-dependent Dirichlet (temperature) boundary condition, extension to the time-dependent Neumann (flux) boundary conditionis straightforward.

The resulting scheme is applied to two problems. In the first problem, after astep jump at t = 0, the surface temperature varies linearly. The second-a moredifficult-problem has an exponentially varying temperature, and a known exactsolution.

MODEL

The dimensionless formulation of the Stefan problem in an invariant domain(0 .. x .. 1) is

a2T(x, t ) dR(t) aT(x, t) eti», t)---'2:-- +xR(t) -d- = R 2(t) ---

ax t ax at

dR(t) aT(x = t.r)R(t) -- = -Ste ----

dt ax(2)

subject to the boundary conditions

for t < 0:

for t > 0:

R=O

T(x = 0, t ) = f(t)

Tt;x = I, t) = 0

where the Stefan number Ste "" C/IiT,eflh,{, X"" x'IR', R "" R'II', where theprime indicates a dimensional quantity and I' is an arbitrary characteristic length[1].


272 RIZWAN·UDDlN

METHOD

Our approach relies on the observation that the above set of equations canbe solved approximately for short time interoals given the initial conditions at anyarbitrary time, t = Ii' This approximate solution is then used to find the position ofthe moving boundary and the temperature distribution at the end of the timeinterval, t = ti + !:it, which is then used as the initial condition for the nexttime interval. Because of the approximate, locally analytical solutions in time, theinterval !:iT can in fact be-as is shown later-relatively much larger than thoseused in traditional numerical schemes. Below we solve for the space-averagedjtime-dependent and time-averagedjspace-dependent temperatures, defined respectively as

- [1P(t)== T(X,t)dxo

- 1 j"+/lo,1j'(X) == ~ J Ti;x, t) dt

t 't

where the time average is over a time interval, !:it, and the subscript j indicates thejth time interval.

Given the position of the boundary at the beginning of the time interval ti ,

R(t = ti) = R(t), the position at a time t > ti can be found by integrating Eq. (2):

(4)

Now, evaluating the above equation at t = ti + !:it, and realizing the definition ofthe time-averaged temperature from Eq. (3), we find

(5)

where '" is defined as the derivative of the time-step-averaged temperature at theright boundary,

a'J:.I(x = 0.p. == ----'1'--__

1 ax(6)

(7)

and subscript j indicates the jth time interval. Note that '0 must be consistentlydetermined using Eq. (0, to which we now tum.

Equation (0, in the spirit of the nodal integral method [11), is reduced to twoordinary differential equations (ODEs). By operating Eq. (1) with (lj!:it) j,';+tll dt,

Jand using the definition of the time-averagedjspace-dependent temperature, weget

d 2'J:'(x) dT'(x)_1 + ex 1 = S'(x)

dx 2 dx 1

where C is the time-averaged value of [R(t}R(t}), given by Eq. (2) as ( - Ste "'). Inwriting Eq. (7), the average of the product has been replaced by the product of the


averages,

NUMERICAL SCHEME FOR STEFAN PROBLEM

1 JII+I1I[ dR(t)][aT(X,t)]- R(t)-- dttJ.t II dt ax

1 JI.+I1I[ dR(t)] 1 JI-+111 aT(x,t)'" - } R(t) --dt - } dttJ.t 'I dt tJ.t 'I ax

273

(8)

An approximation of this kind is often used in nodal integral methods. It has beenshown that, in general, replacing the average of a product by the product of theaverages leads to second-order error [11]. The right-hand side (RHS) of Eq. (7),Sj(x)-usually called the pseudo source tenn-represents the third term in Eq. (1)averaged over the time interval,

- 1 JI-+111 sri», t)S'(X) =- } R 2(t ) dt} tJ.t 'I at

The boundary conditions are also averaged over the time interval to yield

- 1 JI.+ 111 1 J'+ 111 ~r(x = 0) = - J T(x = O,t)dt =- } I(t)dt =//

tJ.t II tJ.t 'I

and

'i?(x = 1) = 0J

(9)

(10)

(11)

The equation for the space-averaged, time-dependent temperature is obtained byoperating Eq. (l) by udx,

where the RHS is given by

_ jl[a2T(x,t)

dR(t) aT(x,t)]s-i» = + xR(t) -- dxo ax 2 dt ax

(12)

(13)

So far the only approximation made in the development is that of equatingthe average of the products with the product of the averages to obtain Eq. (7), anapproximation that is known to lead to second-order error [11]. Equations (7) and(12) are now solved approximately. Equation (7) is solved first. The pseudo sourceterm on the right-hand side, Sj(x), is expanded in, say, Legendre polynomials, andtruncated at the zeroth order, leading to a constant, Sjo. Truncation of thepolynomial at the zeroth order has been shown to lead to second-order error [11] inthe final numerical scheme when such a procedure is used repeatedly over adjacent


274 R1ZWAN·UDDIN

nodes coupled via continuity equations. Though Eq. (7) with this approximationcan be solved, it leads to the error function for the solution of the time-averagedtemperature. Subsequent steps in the development of the solution involve integration of the temperature distribution over space, which in the presence of errorfunctions lead to rather messy algebra. Hence, to simplify, the independentvariable x in the second term on the left-hand side (LHS) of Eq. (7) is replaced byits averaged value (a '" i) over the domain (0 .;;; x .;;; 1). The solution to Eq. (7)with imposed boundary conditions at x = 0 and x = 1, is then

_ [2S10+ clte<- 2a' C) ] [SIO] [(-2SI0- CII

) ]T'(x) = J J + _J_ X + J J e<-Cax) (14)J CAl aC CAl

and the derivative of the time-averaged, space-dependent temperature is given by

a't.I(x) [stO] [(2StO+ Clt)a]J = _J_ + J J e(-Cax)ax aC Al

(15)

where Al '" [e<-2a'C) - 1). Equation (14) will be used later to reconstruct thedomain interior temperature distribution. Note that the entire spatial domain isbeing treated here as a single node, and the solution sought is an approximatesolution valid over the entire space (0 .;;; x .. 1). Hence, the last approximationthat of replacing x in the second term in Eq. (7) by its averaged value over thedomain, a = ~-does not lead to an error that can be associated with the node orgrid size. It is rather an approximation of the kind introduced in obtainingapproximate solution of differential equations-say, via approximate analyticaltechniques or via variational techniques.

Equation (12) is now solved with the initial condition 'j'x(t = tj ) = 'j'X(t j ) . TheRHS is expanded and truncated (replace it with sXO), and the already determinedsolution for R(t) from Eq. (4) is used to obtain

Equation (16) can be evaluated at t = tj + tJ.t to find the space-averaged temperature at the end of the time interval.

At this stage there are three unknowns that must be evaluated to determinethe solution at t = t j + t!t-R(tj + tJ.t), 'j'X(t j + tJ.t) and I/Jj' We have one equationfor the position of the moving boundary, Eq. (5); one equation for the spaceaveraged, time-dependent temperature of the node, Eq. (16); and Eq. (15),which-when evaluated at x = I-becomes an equati~n for I/Jj:. But, with theintroduction of the approximate pseudo source terms, SJo and sxo, we actuallyhave two additional unknowns in these equations. The two additional equations forthese unknowns are obtained by ensuring that the PDE [Eq. (I)] is satisfied overthe space-time domain (0 .;;; x .. 1, t j .;;; t .. tj + tJ.t) in an integral sense, and byrequiring the uniqueness of the space-time-averaged temperature, independent ofthe order of integration [11].



Operating Eq. (1) with JdJ,'i+ 1i1 dxdt, and using the definitions for, 5;0 and- Isxo from Eqs. (9) and (13), we get

(17)

The second equation is obtained by operating Eq. (14) with Jd dx, and equatingthat to Ecj. (16) operated with (l/ilt)J,'i+ li t dt. The resulting equation-which also_ _ J _

has both SjO and sxo-:-is then solved simultaneously with Eq. (17) for st Anexplicit expression for SjO is given in the Appendix. The two pseudo source termsare then simply eliminated from the set of equations [Eqs. (14)-(16)] in terms ofthe other three unknowns. Fortunately, these three equations are not coupled. Thespatial derivative of the time-interval-averaged temperature, If1j , can be determinedby evaluating Eq. (15) at x = 1. Hence,

aT.'(x = 1) [5tO] [(2510

+ cai/) ]1f1. sa 1 = _1_ + 1 1 e(-Ca)

1 ax aC A j

(18)

where C == - Ste If1j , and 5jO is given in the Appendix. Equation (18) is therefore atranscendental equation that can be solved iteratively at each time interval for If1j ,

given the boundary and the initial conditions-time-interval-averaged temperatureat x = 0, ~t, and the domain-averaged temperature at t = Ij' 1'X(I/ The positionof the moving boundary_R(tj + ill), and the domain-averaged temperature at theend of the time interval TX(t j + ill) are then evaluated explicitly using Eqs. (5) and(16), respectively. The domain interior temperature distribution is reconstructedusing Eq. (14). Note that reconstruction using Eq. (14) is not a necessary part ofthe solution scheme. If the temperature distribution is desired, reconstruction canbe carried out at the end of each time interval. However, since the solution at theend of any time interval does not depend on the temperature distribution-only onthe average temperature over the domain-at the beginning of the time interval,the reconstruction step can be skipped entirely.

RESULTS AND DISCUSSION

The scheme developed above is applied to two problems, both with timedependent boundary conditions.

Example 1

This problem was studied by Menning and OZI§lk [10]. In this model problem,the temperature on the left boundary is ramped linearly. The correspondingsurface temperature at x = 0 in our formulation varies as Tt;x = 0, I) == f(1) = (1- 0.2t).

We use the exact temperature distribution, corresponding to the solution forthe step increase in the left boundary surface temperature, at a short time intervaltin after t = 0 as our initial condition for the time-dependent problem. The initial


276 RIZWAN·UDDIN

condition at t = tin is necessitated by the singular nature of the problem for thevery first time step [1]. For this problem we used tin = 0.01, and determine R(t),I/Jj' and P(tj ) for j = 1,2,3, ... , where tj = tin + j 6t. The transcendental equation for I/Jj is solved using the simple Brent method [15], which takes, in most cases,only four to five iterations to converge.

In Table 1, the position of the moving boundary is tabulated as a function oftime for Ste = 0.2, and compared with the fine mesh results reported in [10]. Thereference fine-mesh, finite-difference solution [10] is for the time Tr , r =0.1,0.2, ... , etc., at which the moving boundary reaches r. Reference values fortime Tr are accurate to four places after the decimal point and in general do notmatch with an integer multiple of different time intervals 6t used here. Hence, todetermine R(Tr ) , where tj < Tr < tj + 1, we use R(tj ) and ~ in the locally accuratesolution of R(t) [Eq. (5)] in the time interval tj < t < tj + 1:

(19)

We will henceforth refer to this procedure as the locally accurate reconstruction orlocally accurate interpolation of the moving boundary.

Shown in Table 1 are the computed moving-boundary positions at differenttimes (not all intermediate tj are tabulated), and the locally accurate interpolatedvalues (underlined) at the desired times Tn for five different values of the timeinterval, 6t = 0.01, 0.05, 0.1, 0.25, and 0.5. Also shown are the percent relativeerrors between the computed value and the reference value [10]. The relativelylarge percent errors at t = 0.02618 are due to the approximate initial conditionused at t = tin' The error due to the initial condition, however, quickly drops. Theresults clearly show the accuracy of the semianalytical solution of the movingboundary problem. With 6t as large as 0.5 (Table I), the relative error in themoving-boundary position when it reaches 0.9 is only about 1%. Moreover, usinglarger 6t-up to a limit-does not reduce the accuracy of the results. In fact, theerror actually decreases slightly as the time interval is increased from very smallvalues. This is due to the fact that larger time-interval calculations-while asaccurate as smaller time-interval calculations-require fewer steps to reach thedesired final time.

For comparison with reference values, cases with large values of 6t necessitated the interpolation of several moving-boundary positions within two computedpositions. For example, consider the 6t = 0.5 case in Table 1. The movingboundary is specified at the initial condition, tin = 0.01. It is calculated at the endof the first time interval at t = 0.51. Hence, interpolated values are evaluated fort = 0.02618,0.1068,0.2441, and 0.4425. Even in such a case, percent relative error,which is initially large due to the approximate initial condition, drops quickly asone moves away from the contaminated initial condition.

The previous calculations are repeated for a more difficult case with Ste = 1.0,and the results are compared with reference values in [10] in Table 2. With thelarge Stefan number, the moving boundary moves faster and reaches 1.0 att = 0.6783 [10]. The percent relative error for different values of time interval forthis more difficult case at t = tfinal = 1.0 is of the order of 2-3%.



Table 1. R(t) for numerical example 1 (Ste = 0.2)

Time, t R(t) Percent error Ref. R(t)1O

(a) tit ~ 0.010.0200 0.08690.02618 0.0995 0.477 0.100

0.0300 0.10660.1000 0.19430.1068 0.2007 0.357 0.200

0.1100 0.20370.2400 0.29920.2441 0.3017 0.574 0.300

0.2500 0.30530.4400 0.40160.4425 0.4027 0.670 0.400

0.4500 0.40590.7000 0.50050.7094 0.5037 0.729 0.500

0.7100 0.50391.0500 0.60291.0570 0.6047 0.776 0.600

1.0600 0.60541.5000 0.70451.5080 0.7061 0.867 0.700

1.5100 0.70652.1000 0.80722.1030 0.8076 0.952 0.800

2.1100 0.80862.9500 0.90972.9510 0.9098 1.091 0.900

2.9600 0.9107(b) tit = 0.05

0.0100 0.06130.02618 0.0994 0.572 0.100

0.0600 0.15070.1068 0.2007 0.351 0.200

0.1100 0.20370.2100 0.28030.2441 0.3017 0.564 0.300

0.2600 0.31120.4100 0.38820.4425 0.4027 0.664 0.400

0.4600 0.41020.6600 0.48690.7094 0.5036 0.729 0.500

0.7100 0.50381.0100 0.59241.0570 0.6047 0.776 0.600

1.0600 0.60541.4600 0.69651.5080 0.7061 0.867 0.700


278 RlZWAN·UDDIN

Table I. R(t) for numerical example J (Continued)

Time,l R(t) Percent error Ref. R(t)'O

(b) 6.t = 0.051.5100 0.70652.0600 0.80122.1030 0.8076 0.951 0.800

2.1100 0.80862.9100 0.90582.9510 0.9098 1.090 0.900

2.9600 0.9107(c) 6./ = 0.10

0.0100 0.06130.02618 0.0993 0.7\4 0.100

0.1068 0.2007 0.327 0.200

0.1100 0.20360.2100 0.28030.2441 0.3016 0.532 0.300

0.3100 0.33900.4100 0.38820.4425 0.4026 0.647 0.400

0.5100 0.43100.6100 0.46920.7094 0.5036 0.728 0.500

0.7100 0.50381.0100 0.59241.0570 0.6046 0.764 0.600

1.1100 0.6J801.4JOO 0.68621.5080 0.7061 0.866 0.700

1.5100 0.70652.0100 0.79362.1030 0.8076 0.950 0.800

2.1100 0.80862.9100 0.90582.9510 0.9098 1.085 0.900

3.0100 0.9154(d) 6./ = 0.25

0.0100 0.06130.02618 0.0988 1.152 0.100

0.1068 0.1994 0.319 0.200

0.2441 0.3013 0.439 0.300

0.2600 0.31100.4425 0.4022 0.552 0.400

0.5100 0.43110.7094 0.5032 0.642 0.500

0.7600 0.51991.0100 0.59251.0570 0.6044 0.735 0.600

1.2600 0.65351.5080 0.7061 0.866 0.700

1.5100 0.7065



Table 1. R(t) for numerical example 1 (Continued)

Time,/ R(t) Percent error Ref. R(t)1O

(d)!>.t = 0.252.0100 0.79362.1030 0.8073 0.909 0.800

2.2600 0.82982.7600 0.89042.9510 0.9096 1.064 0.900

3.0100 0.9154(e)!>.t ~ 0.50

0.0100 0.06130.02618 0.0981 1.886 0.100

0.1068 0.1972 1.409 0.200

0.2441 0.2978 0.724 0.300

0.4425 0.4009 0.217 0.400

0.5100 0.43030.7094 0.5015 0.2922 0.500

1.0100 0.59281.0570 0.6043 0.719 0.600

1.5080 0.7056 0.797 0.700

1.5100 0.70602.0100 0.79382.1030 0.8069 0.860 0.800

2.5100 0.86172.9510 0.9093 1.034 0.900

3.0100 0.9155

Example 2

The second test problem we solve has an exact solution [9]. For the specialcase of Ste = 1, the differential equations (1)-(2) are satisfied by

rc«, t) = e'(I-x) - 1

R(t} = t

T(x=O,t)=e'-1

T(x=l,t)=O

T(x,t=O)=O

(20)

With exponentially rising temperature at the fixed boundary, this clearly is a moredifficult problem than the previous example. Taking advantage of the known exactsolution for this problem, the exact initial condition at t = tin is used to avoid anycontamination of numerical results due to the approximate initial condition.

Exact time-averaged and space-averaged temperatures can be easily determined from Eq. (20). Exact R(t) and calculated R(t) for dt = 0.05 are shown inFigure 1, and percent errors for several different time intervals and tin are


280 R1ZWAN·UDDIN

Table 2. R(t) for numerical example 1 (Ste = 1.0)

Time, t R(t) Percent error Ref. R(t)1O

(0) At ~ 0.010.0200 0.17790.02561 0.2017 0.869 0.200

0.0300 0.21860.0500 0.28300.05816 0.3054 1.790 0.300

0.0600 0.31020.1000 0.40070.1040 0.40862 2.154 0.400

0.1100 0.42020.1600 0.50650.1634 0.5118 2.363 0.500

0.1700 0.52200.2300 0.60640.2367 0.6151 2.516 0.600

0.2400 0.61930.3200 0.71380.3242 0.7184 2.625 0.700

0.3300 0.72470.4200 0.81570.4265 0.8218 2.728 0.800

0.4300 0.82510.5400 0.92200.5443 0.9255 2.836 0.900

0.5500 0.93020.6700 1.02330.6783 1.0294 2.942 1.000

0.6800 1.0307(b) At = 0.05

0.0100 0.12400.02561 0.2016 0.790 0.200

0.05816 0.3054 1.816 0.300

0.0600 0.31030.1040 0.4083 2.069 0.400

0.1100 0.42010.1600 0.50650.1634 0.5118 2.360 0.500

0.2100 0.57970.2367 0.6150 2.506 0.600

0.2600 0.64430.3100 0.70270.3242 0.7183 2.619 0.700

0.3600 0.75630.4100 0.80610.4265 0.8218 2.723 0.800

0.4600 0.85270.5100 0.89670.5443 0.9255 2.832 0.900

0.5600 0.93840.6600 1.0159



Table 2. R(t) for numerical example 1 (Continued)

Time, / R(t) Percent error Ref. R(t)1O

(b) tlt = 0.050.6783 1.0294 2.938 1.000

0.7100 1.0523(c) tlt = 0.10

0.0100 0.12400.02561 0.2013 0.661 0.200

0.05816 0.3049 1.641 0.300

0.1040 0.4085 2.114 0.400

0.1100 0.42010.1634 0.5115 2.305 0.500

0.2100 0.57960.2367 0.6150 2.481 0.600

0.3100 0.70270.3242 0.7182 2.606 0.700

0.4100 0.80610.4265 0.8217 2.712 0.800

0.5100 0.89670.5443 0.9254 2.818 0.900

0.6100 0.97800.6783 1.0293 2.929 1.000

0.7100 1.0522(d) tlt = 0.25

0.0100 0.12400.02561 0.2006 0.300 0.200

0.05816 0.3035 1.151 0.300

0.1040 0.4063 1.579 0.400

0.1634 0.5096 1.921 0.500

0.2367 0.6135 2.258 0.600

0.2600 0.64310.3242 0.7169 2.410 0.700

0.4265 0.8209 2.607 0.800

0.5100 0.89680.5443 0.9252 2.801 0.900

0.6783 1.0286 2.857 1.000

0.7600 1.0868(e) tlr = 0.50

0.0100 0.12400.02561 0.1994 0.303 0.200

0.05816 0.3010 0.329 0.300

0.1040 0.4027 0.680 0.400

0.1634 0.5049 0.985 0.500

0.2367 0.6078 1.301 0.600

0.3242 0.7114 1.628 0.700

0.4265 0.8160 1.999 0.800

0.5100 0.89230.5443 0.9209 2.312 0.900

0.6783 1.0248 2.477 1.000

1.0100 1.2452


282 RIZWAN·UDDIN

1.0.------------------::;,.,

0.8 1- exact 1_ .. dt=0.05

-= 0.6it

0.4

0.2

0.2 0.4 0.6

Time

0.8 1.0

Figure I. Exact and approximate positions of themoving boundary versus time for Example 2. HereSte = 1.0, li3 = 0.1, and the time interval is 0.05.

tabulated in Table 3. Here, different values of tin are used for different timeintervals to avoid any reconstruction (interpolation), and the position of the movingboundary is determined at t = 1.0 for all cases. Even for a time interval as large as0.25, the error in the position of the moving boundary for this relatively difficultproblem at t = 1 is less than 2%. In fact, the error in the position of the movingboundary-as in Example I-decreases with larger time intervals. A comparisonbetween the case ilt = 0.5, tin = 0.5 (only one step to reach tfinal = 1) with the caseilt = 0.01, tin = 0.5 (50 steps to reach tfinal = 1) shows that the percent relativeerror in the former is approximately three times less than that in the latter.

In the example above, though the domain-averaged temperature is calculatedat the end of each time interval, we have only compared the position of the. movingboundary with the exact values. While the numerical scheme yields temperatureaveraged over the domain [Eq. (16)], and for the calculation of subsequent stepsneeds only that quantity as the initial condition, the spatial temperature distributionwithin the domain can be easily determined using Eq. (14), which allows for areconstruction of the time-averaged/space-dependent temperature. The tempera-

Table 3. Percent error in position of the moving boundary versus time interval at/ = 1.0 for Example 2 [R~",(I = I) = 1.0]

tJ./ tin 'final No. of steps R{t) Percent error

0.005 0.10 1.00 180 0.978762 2.12380.010 0.10 1.00 90 0.978765 2.12350.020 0.10 1.00 45 0.978781 2.12190.050 0.10 1.00 18 0.978887 2.11130.250 0.25 1.00 3 0.983228 1.67720.500 0.50 1.00 1 1.007000 0.69970.010 0.50 1.00 50 0.979793 2.02070.250 0.50 1.00 2 0.982102 1.7898


NUMERICAL SCHEME FOR STEFAN PROBLEM

2.--------------------,283

o

- Exact...... Reconstructed

Figure 2. Exact solution and the reconstructed domain interior temperature distribution at four different times forExample 2. For t = 1.0, the exact temperature is less than thecalculated temperature for t < 0.34, and becomes greaterthan the calculated temperature for t > 0.34. The other threereconstructed temperatures are indistinguishable from theirexact values.

ture distribution 'ij'(x) for Example 2 is compared in Figure 2 with the exactsolution at t = 0.25, 0.50, 0.75, and 1.0 for the case iiI = 0.05, lin = 0.1. Note thatthe numerically calculated and exact temperature values are averaged over the lasttime interval (0.20 ..,;;; I ..,;;; 0.25,0.45 ..,;;; t ..,;;; 0.50, 0.70 ..,;;; I ..,;;; 0.75, and 0.95 ..,;;; I ..,;;; 1.00,for the four cases). The agreement with the exact solution is so well that for threecases (at I = 0.25, I = 0.50, and I = 0.75) the two curves are indistinguishable. Fortemperature distribution at t = 1.0, a slight difference between the exact andapproximate solution can be seen. For this case the curve for the exact temperature is below the approximate one for x < 0.34, and above that for x > 0.34.

SUMMARY

The semianalytical scheme developed here for the one-dimensional Stefanproblem with time-dependent boundary conditions clearly yields accurate resultsconsidering that, at each time interval, all one is solving is a single transcendentalequation which, in most cases, takes only four to five iterations using the simpleBrent method [15]. Resulting value for the time-interval-averaged derivative of thetemperature at the moving boundary is then used to evaluate directly the positionof the moving-boundary, domain-averaged temperature at the end of the time step,and even the time-interval-averaged temperature distribution within the spatialdomain. The method, as shown by application to reference problems, is veryaccurate even for relatively large time intervals. This approximate, locally analyticsolution will facilitate the analysis of computer-intensive problems that requiresolution for long time durations, such as the Stefan problem with oscillating-temperature boundary condition.


284 R1ZWAN·UDDIN

APPENDIX

The explicit expression for the pseudo source term 5;0 is

5'0 = :b..} <P2

where

_ _( 1 1)<P == -T'(t.} + 1".' 1 + - +--

1 } JJ 82 2a 2C

<P = InO - c1 !:J.t) (_1__ 1) + ~ __1__ .: (1 + ~)2 - C2 C1 !:J.t C2 (aC)2 C 82

81 = 2a 2C

82 = coshrg.) - sinh(81) - 1

C2 = -2C

C = -Ste <Pi

REFERENCES

I. L. S. Yao and J. Prusa, Melting and Freezing, Ado. Heat Transfer, vol. 19, pp. 1-95,1989.

2. J. R. Ockendon and W. R. Hodgkins (eds.), Moving Boundary Problems in Heat Flow andDiffusion, Clarendon Press, Oxford, 1975.

3. A. A. Samarskii, P. N. Vabishchevich, O. P. Iliev, and A. G. Churbanov, NumericalSimulation of Convection/Diffusion Phase Change Problems-A Review, Int. J. HeatMass Transfer, vol. 36, no. 17, pp. 4095-4106, 1993.

4. J. Crank and R. S. Gupta, A Method of Solving Moving Boundary Problems in HeatFlow Using Cubic Splines or Polynomials, J. Int. Math. Appl., vol. 97, pp. 296-304, 1972.

5. R. S. Gupta, Moving Grid Method without Interpolations, Comput. Meth. Appl. Mech.Eng., vol. 4, pp. 143-152, 1974.

6. R. S. Gupta and D. Kumar, A Modified Variable Time Step Method for the OneDimensional Stefan Problem, Comput. Meth. Appl. Mech. Eng., vol. 23, pp. 101-108,1980.

7. R. S. Gupta and D. Kumar, Variable Time Step Methods for One-Dimensional StefanProblem with Mixed Boundary Condition, Int. 1. Heat Mass Transfer, vol. 24, pp.251-259,1981.

8. G. Marshall, A Front Tracking Method for One-Dimensional Moving Boundary Problems, SIAM J. Sci. Stat. Comput., vol. 7, no. 1, pp. 252-263, 1986.

9. R. M. Furzeland, A Comparative Study of Numerical Methods for Moving BoundaryProblems, J. Inst. Math. AppL, vol. 26, pp. 411-429,1980.

10. J. Menning and M. N bZI~lk, Coupled Integral Equation Approach for Solving Meltingor Solidification, Int. J. Heat Mass Transfer, vol. 28, no. 8, pp. 1481-1485, 1985.



11. Y. Y. Azmy and J. J. Doming, A Nodal Integral Approach to the Numerical Solution ofPartial Differential Equations, in Advances in Reactor Computations, vol. II, pp. 893-909,American Nuclear Society, LaGrange Park, IL, 1983.

12. J. P. Hennart, A General Family of Nodal Schemes, SIAM J. Sci. Stat. Comput., vol. 7,no. 1, pp. 264-287, 1986.

13. Rizwan-uddin, An Improved Coarse-Mesh Nodal Integral Method for Partial Differential Equations, Numer. Meth. Partial Differential Equations, vol. 13, pp. 113-145, 1997.

14. Rizwan-uddin, A Nodal Integral Method for Moving Boundary Phase Change Problem,Proc. Int. Symp. on Computational Heat Transfer, Cesme, Turkey, May 1997, in press.

15. W. H. Press, S. A. Teukolsky, W. T. Wetterling, and B. P. Flannery, Numerical Recipes:The Art of Scientific Computing, Cambridge University Press, Cambridge, 1992.


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