SIAM J. NUMERICAL ANALYSISVol. 16, No. 4, August 1979

1979 Society for Industrial and Applied Mathematics0036-1429/79/1604-0002 $01.00/0

THE ALTERNATING PHASE TRUNCATION METHOD FORNUMERICAL SOLUTION OF A STEFAN PROBLEM*

JOEL C. W. ROGERSf, ALAN E. BERGER AND MELVYN CIMENT

Abstract. A numerical scheme for solving heat conduction problems involving a change of phase ispresented. The numerical solution is obtained for heat flow in a two phase medium by using a method whichtreats each phase alternately. The resulting scheme avoids geometrical front tracking, and requires onlysimple algebraic operations. A maximum principle for the method and ;esults of numerical experiments aregiven. An analytical version of the algorithm for a one dimensional one phase Stefan problem is shown to havean O(At In 1/2 rate of convergence. Numerical experiments indicate that this estimate is sharp.

1. Introduction. Stefan type problems arise in connection with phenomena such ascasting of metals and explosives, crystal growth, fluid flow through porous media,thermal explosions, permafrost behavior, and chemical reactions at an interface 11],[28], [38], [43].

This paper is concerned with the classical Stefan problem modeling heat conduc-tion with a change of phase. The problem is that of finding the temperature u(p, t) (afunction of the position p and time t) and the interface location $(t)between the liquidand solid states of a substance with freezing-melting point z. For convenience we takez 0, and will freely refer to the liquid as water and the solid as ice. Let fl be a boundedconnected open set in R 2 or R with boundary F, and let I(t)={pefl’u(p, t)<0},W(t) {p fl" u(p, t) > 0}, and S(t) {p fl" u(p, t) 0}. The function u is to satisfy

(1.1a) ci(u)ut=7, ki(u)Vu forpI(t),t>O,

(1.1b) cw(u)ut V. kw(u)Tu for p W(t), > O,

where ci(c) is the heat capacity of the ice (water), and ki, kv are the respective thermalconductivities. Appropriate boundary conditions are

(1.2a) u(p, t) g(p, t) for p F, >0,

(1.2b) u(p, O) (p) for p D,.

Proper conditions on the moving interface S(t) are

(1.3a) u(s,t)=O foralls6S(t),t>=O,

and the energy balance equation

(l.3b) -kwu /

+kiu =AV forsS(t),t>0.

Here , is the unit normal to S(t) pointing into W(t), the constant A is the latent heat ofthe change of phase, V is the velocity of the interface in the direction of ,, and for afunction F, F/(s, t) (F-(s, t)) denotes the limit of F as s S(t) is approached from withinW(t) (I(t)).

Received by the editors March 21, 1977, and in revised form September 6, 1978. This research wassupported by the Office of Naval Research under Grants N00014-76-C-0369 and NR334-003, the Naval AirSystems Command under Grant WR01-402-001, and the Naval Surface Weapons Center IndependentResearch Fund.

" Applied Mathematics Branch, Naval Surface Weapons Center, Silver Spring, Maryland 20910.Presently at Applied Physics Laboratory, Johns Hopkins University, Applied Mathematics Research Group,Laurel, Maryland 20810.

t Applied Mathematics Branch, Naval Surface Weapons Center, Silver Spring, Maryland 20910.Applied Mathematics Branch, Naval Surface Weapons Center, Silver Spring, Maryland 20910.

Presently at Mathematical Analysis Division, National Bureau of Standards, Washington, DC 20234.

563

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

564 JOEL C. W. ROGERS, ALAN E. BERGER AND MELVYN CIMENT

Questions concerning existence and uniqueness of the solution of problems of type(1.1)-(1.3) have been .studied extensively (e.g. [8], [9], [10], [12], [25], [26], [27], [28],[32], [39], [45], [46]). For the present, we simply note that in general one is limited toposing existence of weak solutions, and indeed with a heat source term one mayencounter a "slushy" region separating I(t) and W(t), instead of a well defined interfacecurve $(t) [1], [27].

For one space dimension, rather efficient numerical methods are available for suchproblems (e,g. [5], [7], [16], [20], [34], [35]), the approach being based on explicitlytracking the interface location $(t). In two space dimensions, if the physical situationprovides a "nicely behaved" interface, a front tracking approach may still be advan-tageous [30], [33], [37]; however in.general, geometrical complexity can make interfacetracking computationally intractable.

Recently, methods which do not depend on explicit interface tracking have beendeveloped [1], [4], [6], [15], [17], [36], [42], [44], [47]. The underlying idea utilized bymany of these methods [4], [6], [17], [36], [44], [47], has been to smooth out, over asmall temperature range which must be appropriately chosen, the abrupt change inmaterial properties which occurs at the freezing point. This corresponds to replacing(1.1) and (1.3) by one nonlinear parabolic equation on 12, which is then solvednumerically. With other techniques 1], 15 ], [42], the problem is formulated and solvedin terms of temperature, internal heat energy, and/or heat flux. Another approachrelies on posing the Stefan problem as a variational inequality [21], [24]. In such aformulation, however, the independent variable no longer corresponds to temperatureor internal energy.

This paper is devoted to analyzing the alternating phase truncation (APT) solutiontechnique for the Stefan problem which was presented in [3]. In this paper anO(At. In (Zt-1)) 1/2 rate of convergence will be established for a particular case of theAPT method. The APT algorithm allows one to treat the computational problem as if itwere effectively a standard heat equation, and is hence easily applied to problems in two(or three) space dimensions. The APTmethod is in much the same spirit as the so-calledtruncation method that was previously applied to the solution of a model of diffusionand absorption of substrate [2].

2. The APT Method. The APT method is formulated directly in terms of internalenergy (enthalpy) which is defined below, Let di ki/ci, dw kw/cw, and

c(u)=lc(u)’ u<O, /O, u<O, k(u)=/ k(u), u<O,Cw(U), u >=O,

rt(u)=1, u >-_0, kw(u), u >-_O.

Let the enthalpy H(p, t)= H(u(p, t)) be defined by

(2.1) H(u) c(v) dv + Art(u).

Note that limu,0H(u)= 0, and H(0)= . The corresponding equations in terms of Hare

(2.2a) Ht V" diVH in I(t) (i.e. where H < 0), > 0,

(2.2b) Ht V. dwVH in W(t) (i.e. where H > X), > 0,

with boundary conditions

(2.3a) H(p, t) g(p, t) for p 6 F, > 0,

(2.3b) H(p, 0) b(p) for p f,

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

NUMERICAL SOLUTION OF A STEFAN PROBLEM 565

where g and 4 are the enthalpies corresponding to and 4 in (1.2). The. conditions onthe interface become

(2.4a) H-=0, H+=, on S (t), >- 0, and

(2.4b) -dwH+ +diH-=AV forsS(t),t>O.

The APT method for solving (2.2)-(2.4) is analogous to a fractional step method, inthat for each time step it treats the heat flow in the liquid and in the solid in two separatehalf steps, in the first half step, sufficient heat is added at each point of the solid toconvert it to liquid. The heat equation for the resulting all-liquid region is solved, andthen the heat that had been added is subtracted away. In the second half step, sufficientheat is subtracted at each point of the liquid to make it solid, the heat equation for theresulting all-solid region is solved, and then the heat that had been subtracted is addedback in. More formally, given approximate solution values H for (2.2)-(2.4) at a time, one obtains values at the next time level o + At in the following manner.

Choose any convenient numerical method (to be referred to as M) for finding thesolution at of a parabolic problem of the form

(2.5a) v, V. d(v)Vv in D,(d > 0),

(2.5b) v G(p, t) for p 6 F, 6 [t, 1] (G given.Dirichlet data),

(2.5c) v(t) V in f (V given initial data).

Let M be based at mesh points P,..., P, ll (and at Pr+,’’’, Pr on F where theDirichlet data is specified). Let H, Q, Z denote m-vectors of enthalpy values atP1, , P,. The first half step is" use M to solve

(2.6a) v, =V, dwVv in l)

(2.6b) v(p, t) max (h, g(p, t)) for p F, [t, tl],(2.6c) v(pj, o)=max (A,H), j= 1,..., m,

obtaining 0 at time (i.e. enthalpy values 01, , 0,, at the mesh points P1, ’, P,,,,respectively). Then complete the first half step by setting

(2.7) Qj 0. +H max (A, H ), j 1,.. , m.

The second half step is: use M to solve

(2.8a) vt =V. diVv in f,

(2.8b) v(p, t) min (0, g(p, t)) for p 6 F, 6 [t, tl],(2.8c) v(pi, ) min (0, Qi), j 1,..., m,

obtaining values 2 at time . The APT method approximate solution H at is thengiven by

(2.9) H + Qi -min (0, Q), j 1, , m.

Note that the algebraic operations are trivial to implement, while (2.6) and (2.8)are standard heat equations. If the material properties are taken to be constant withineach phase, only one initial matrix decomposition need be done for (2.6) and for (2.8).Furthermore if D, is a rectangle, an alternating direction implicit (ADI) scheme can beused (as was done for Fig. 1 of [31]). In any case, no geometrical front tracking isinvolved.

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

566 JOEL C. W. ROGERS, ALAN E. BERGER AND MELVYN CIMENT

When zero Neumann (heat flux) boundary conditions are imposed (in place of(1.2a), (2.3a), (2.6b), (2.8b)), the APT method tends to be energy conserving (as domost methods for the Stefan problem). This is because the algebraic operations result inno net change in any linear functional defined on m-vectors which is used to approxi-mate the total energy E n H(p, t)dp, so variation in E with time is solely due todissipation (if any) from the numerical method used for solving (2.6) and (2.8).

3. Stability of the APT method. In a case where the material properties areconstant in each phase, problems (2.6) and (2.8) have the form

(3.1a) vt dV2v in fI,

(3.1b) v(p, t) G(p, t) for p s F, [t, tl],(3.1c) v(Pj, o)= V/ for j= 1,..., m.

For any m-vector V let

max V max Vi, min V min V/.

The following standard type discrete maximum principle is easily demonstrated to holdfor the APT method [3].

THEOREM 3.1. Suppose the numerical procedure M used on (2.6) and (2.8) whenapplied to (3.1) with d dw and when applied to (3.1) with d d yields V at o + Atsatisfying the maximum principle

(3.2) min (rain V, inf G) <= min V =< max V max (max V, sup G).

Then the APT method itself satisfies the maximum principle, i.e.;

(3.3) min (H, inf g) -min H -< max H ---max (max H, sup g).

It appears that the algebraic operations used in the APT method disenable theusual 12 type of stability analysis. The requirement that M satisfy the maximumprinciple (3.2) (which in general is a restriction on the size of At) does not seem to be anecessary condition for reasonable numerical behavior of the APT method.

For convenience, we summarize some sufficient conditions for (3.2) to hold whenM is applied to (3.1). Let fI be either an interval in R (with uniform Ax mesh), or arectangle in R 2 (with uniform Ax, Ay mesh). Let M be the standard explicit (0 0),Crank-Nicolson (0 1/2), or implicit (0 1) finite difference method (e.g. [23]). Then Mwill satisfy the maximum principle (3,2) if At satisfies;

(3.4) At -<_ Ax2/(2(1 0) d) (fl a line segment)

(3.5) At=I(1- 0) d(2/Ax2+2/Ay2)]- (fl a rectangle).

If fl is a rectangle (with uniform Ax, Ay mesh) and M is the original ADI scheme ofPeaceman-Rachford [40], then M will satisfy (3.2) if

(3.6) At<- Ax2/d and At_-<Ay2/d.Proofs of the above, and sufficient conditions for (3.2) when M is the finite elementmethod with a general mesh on a general domain in R2 or R can be found in [3].

4. Some numerical results. The APT method was tested by applying it tosolidification problems governed by (1.1)-(1.3) with the initial and boundary conditionschosen so that the analytical solution was known. For one calculation we set I-!

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

NUMERICAL SOLUTION OF A STEFAN PROBLEM 567

(0, 20 cm), ci = 1.90 and Cw 4.19 joules cm-3 C-l, X 306 joules cm-3, ki .023 andkw .0058 joules sec-1 cm-! C-, approximating the material properties of H20 19],[49]. The analytical solution is

s(t) 15.0+vt where v -5 x 10- cm/sec,

H -B +(B +A). exp (a(vt-x + 15))

u =(H-A)/Cw

H -B +B. exp (/3(v t-x + 15))

U =n/ci

x<=s(t),

x<=s(t),

x>s(t),

x > s(t),

where B =-594, ct v/dw, and/ v/d. The APT algorithm consistently follows theinterface position, although with a certain amount of overshoot (cf. Fig. 1). In generalthe error due to time discretization completely dominates the spatial contribution to the

10

0.00 2.50 5.00 7.50 10.0 12.5 15.0 17.5 20.0

DISTANCE, x (cm)FzG. 1. E,xact solution (solid lines) and APT method approximate solution (points) at 0, T/3, 2T/3, T

where T 200,000 sec. The approximate solution values were obtained using the standard implicit finitedifference method with At 38.58 and Ax .5 to solve (2.6) and (2.8).

error even for a very coarse spatial mesh, indicating the desirability of developing avariant of the method having higher order accuracy in time.

For a special case of the APT method which involves no spatial discretization, it isshown in the next section that

(4.1) C(At In (1 + t/At)) 1/2

provides a bound for the error at time (the conatant C depends only on and the dataof the problem). The problem and exact solution given above were used in order to see ifnumerical results (with Ax "small") were consistent with a temporal error behavior ofthe form (4.1). The standard implicit and Crank-Nicolson finite difference schemes withAx .5 and several At values were used to solve (2.6) and (2.8). Mean square (L2(fl))errors were calculated at T/3, 2T/3, T (T 200,000), assuming the APT approxi-mate solution U to be linear within each spatial subinterval [Pj, Pj/I]. The integral

(u(x, t)- U(x, t))2 dx

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

568 JOEL C. W. ROGERS, ALAN E. BERGER AND MELVYN CIMENT

was calculated using 4 point Gaussian quadrature [48] on each subinterval [Pi, Pi/l]. Ifthe exact front location s s(t) was interior to an interval [Pi, Pi/l], it was subdividedinto [P, s] and Is, P/I] and the quadrature was done on each subinterval (so that thefunction being integrated was smooth within each quadrature interval).

For a fixed t, experimental values for the constant C in (4.1) were obtained bysetting

C L2-error/(At In (1 + flat)) 1/2.

The results obtained (Table 1) are markedly consistent with an error behavior of theform (4.1). For this particular problem it happened that using Crank-Nicolson resultedin decidedly smaller errors than occurred with the implicit method (cf. Table 1, Fig. 1,and Fig. 1 of [3]); however, the opposite behavior was. observed in.a different testproblem.

TABLEL2-error and C calculated at T/3, 2T/3, T (T 200,000) for several values of At.

Crank-Nicolson Implicit

At L2-error C L2-error C

T/3 617.28 4.6143 .0857 5.9758 .1110T/3 308.64 3.4632 .0849 4.5887 .1126T/3 154.32 2.7071 .0884 3.4655 .1132T/3 77.16 1.9911 .0871 2.7046 .1183T/3 38.58 1.3279 .0782 1.9895 .1173

2T/3 617.28 2.9601 .0513 4.2531 .07382T/3 308.64 2.4459 .0565 2.9487 .06812T/3 i54.32 1.7584 .0544 2.4368 .07542T/3 77.16 1.4258 .0594 1.7587 .07332T/3 38.58 ,9455 .0533 1.4253 .0803

T 617.28 2.5859 .0432 4.5081 .0754T 308.64 1.9596 .0438 2.6002 .0581T 154.32 1.3374 .0402 1.9626 .0590T 77.16 .9127 .0370 1.3397 .0543T 38.58 .7433 .0409 .9127 .0502

5. An error estimate. In this section a rate of convergence for an "analytical"version of the APT method when applied to the following one dimensional one phaseStefan problem is obtained:

(5.1a) ut Uxx, 0 < x < s(t), (0, T],

(5.1b) u(x, O)= Uo(X), O<-x <=s(O),

(5.1c) ux(0, t) 0, (0, T],

(5.1d) u(s(t), t) L, [0, T],

(5 le) Lds-= -ux(s(t), t), (0, T],

(5.1f) s(0) So.

Here u denotes enthalpy, L is the latent heat of the change of phase, So is a givenpositive number, the initial data Uo(X) C 110, So], u0 NL on [0, So], and Uo(So) L. From

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

NUMERICAL SOLUTION OF A STEFAN PROBLEM 569

[25], there exists a solution of (5.1) in the. sense that ut and u,x are continuous for0 < x < s(t), 0 < <- T; u is continuous on 0 -<_ x _-< s(t), 0 -<_ -< T; u, is continuous whenO<-_x<-_s(t), O<t<-_T, and when 0<x_<-s0 for t=0; s(t)Cl[O,T], and (5.1) issatisfied. This solution is unique. Also, s (t) is monotone nondecreasing, To embed the,

problem on a fixed interval define, for each -> 0, u(x, t) 0 for x > s(t).We now describe the analytical APT method (referred to as A1) which will be

studied. Pick a time step zt T/N (N a positive integer), and let f denote (0, eo). LetW(x) Uo(X) for x - [0, oo). Given W"(x) approximating u(x, nat) on ,w"/X(x) is ,obtained as f,ollows. Let ff,,/l be the exact solution at At of the heatequation

c,=Cxx, x6f, 6 (0, At],

c(x, 0) max (W"(x),L), x f,

cx (0, t) O, (0, At].

Then

W"+(x) =- ff’"+a(x) + W"(x)-max (Wn(x),L),

The approximate front location Sn+l is defined by

Sn+I=SUp{X W+I(x)>-_L}.x>_O

Then the following error estimate holds.THEOREM 5.1. There are constants c and K depending on Uo, T, and L but

independent of At such that when At <--c, then for n 1,..., N, and for O<-x <-<_

min (s,, s(t"));

(5.2) Is.-s(t")l/lw"(x)-u(x, t")I<-_K(+/-t In (1 +T/+/-t)) 1/.

Furthermore, for n 1, 2, , N; W >= 0 on , W"(x) >- L on [0, s, ], s,_ <= s,, W" ismonotone nonincreasing on [So, o), W is C except at s,-1, and the integral over f of(W"-uo) equals zero (i.e. the method is "energy conserving"). One may define the"temperature" corresponding to enthalpy value u by

O, u <-_ Ltemp (u)=

u-L, u>L"

Then for x ef and for n 1,.. ,N;

(5.3) Itemp (W"(x))- temp (u(x, t"))[<=K(At In (1 + T/At)) 1/2.

Before proceeding with the proof, a brief outline of the strategy and techniquesused may be helpful. First, a front tracking method (called A3) is defined and shown toyield a front location s,(t) (for any time step - > 0). As z goes to 0, the approximate fronts,(t) converges with rate O("/2) to a function r(t) which indeed is s(t). An algorithmA2, which is a convenient variant of A1, is defined (and is eventually demonstrated tobe identical to A1). A family A4 of front tracking methods (which includes A3 as aspecial case) is then defined. By bounding the front locations obtained by A2 and A3using front locations obtained by special choices of members of A4, the error estimate isobtained. The principal analytical tools used throughout the proof are the energypreserving character of all the algorithms, and basic properties of the simple heatequation. In particular, extensive use is made of certain Green’s functions. Basic results

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

570 JOEL C. W. ROGERS, ALAN E. BERGER AND MELVYN CIMENT

on the heat equation, whose proofs are independent of the exposition in this section,will be stated (in the order they are used) in 6.

We now present the sequence of lemmas used to obtain Theorem 1 along withindications of their significance. Except when brief, complete proofs are deferred to theend of the section.

Algorithm A3. We begin with the definition and examination of scheme A3. Let Fvdenote the class of functions f defined on f such that: there is a real number z = z(f) > 0(the front location associated with f) satisfying f >___ L on [0, z] and f(x) 0 when x > z;for x and x2 in [0, z], [f(xt)-f(x2)[ Vlx-x2[; and f(z)=L. If fFv, u(x, t,f) willdenote the solution of

(5.4a) c, c**, 0 < x < z z (f), > O,

(5.4b) c(x, 0)=/(x), 0=x -z,

(5.4c) c,, (0, t) 0, t > 0,

(5.4d) c(z,t)=L, t>-O.

For a given time step - > 0, define

(5.5) s(z, f)= z(f)-L-1 ux(z, t, f) dt.

By the maximum principle, u(x, t,f)>.L, so ux(z, t, f) must be nonpositive for t>0.Also from Remark 6.1, V u, so

(5.6) 0 s(’, f)- z <= Vr/L.The operator T, "Fv Fv is defined by

T,f u (x, z, f) for0-x=<z,

(5.7) Tf L for z <_- x <_- s(r, f),

Tf O for x > s(’, f).

From Remark.6.1 it follows that Tf is indeed in Fv, and clearly z(Tf) s(z, f). Finally,given positive numbers ’1," , z,,, s(zn, ., ’1, f) is defined to be s(-,, T._ TIf).Given n >0, the "front tracking" algorithm A3 approximates u(x, t) and s(t) (thesolution of (5.1)) by (T)"Uo (where r t/n) and s(z,. ., -, Uo) (with - understood to berepeated n times), respectively. Henceforth, for f Fv, m a positive integer, and - > 0;s(m ’, f) will denote s(z, -, f) (" repeated rn times).

Properties of A3 which will be used in Theorems 5.2 and 5.3 to estimate thedifference between s(m-) and s(n * ’, Uo) will now .be developed. Given a function

f Fv, the energy functional E(f) is defined by

(5.8) E(f) [. f(x) dx.

Let f e Fv, and let - > 0. Then we haveLEMMA 5.1. E(Tf) E(f).Proof. Integrate ut Uxx (u u(x, r, f)) over 0_-<x -<_.z and 8 <-t<=r and let 50+

(i.e. let 8 approach 0 from above). The result follows by recalling (5.5) and (5.7).In a similar fashion, the following useful expression for s(z, f) can be obtained;

(5.9) s(’, f) z +L- (f T,f) dx.

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

NUMERICAL SOLUTION OF A STEFAN PROBLEM 571

The sequence of results given below on monotonicity properties of A3 will be used toshow the convergence of s(n (t/n), f) as n --> oo. The proofs use standard properties ofthe heat equation (cf. 6).

LEMMA 5.2. lff Fv and ’1 and ’2 are positive, then

(5.10a)

where

0 S(T1 + 7"2, f)

(5.10b) D(x2, x1)’Xl\"t--+ 2 ]’or x, x2 >= O.

LEMMA 5.3. Iff and f2 are in Fv, fx >= rE, and the time steps z,..., z, satisfy

(5.11) O<2V(’/r)a/E<-L fori l, n,

then

T, T,lfl >= T. T,lf2 and s(’rn, ’1, fi) >= s(’rn, zl, f2).LEMMA 5.4. Let feFv and assume (5.11) is valid, Then or n >=3, and using the

definition (5.10b))

(5.12) [s(,,..., z,f)-s(r,,’.., r3, zx + 2,/’)l_-<O(z2, ).

We are now ready to proveTHEOREM 5.2. Let f Fv. For each > 0 there exists the limit

(5.13) g(t)m tim S(. ,).Setting t/n, i 2 V(/)/NL, then we obtain

(5.14) s(n r,[)-(t) N2V. t. (r/)//L.It will later be demonstrated (in Theorem 5.3) that when uo (in (5. lb)) then (t)

is indeed s(t) (the front in the Stefan problem (5.1)).Pro@ It will first be demonstrated that the sequence in (5.13) is Cauchy. Let > 0,

m > O, n > O, r t/m, r t/n, t/(mn). Repeated use of Lemma 5.4 shows that

(5.15a) Is(n ,z,,f)-s((nm),zi, f)[nmD(z,rl),

(5.15b) Is(m ,z,f)-s((mn),z,f)lmnD(z, z),

which shows that g(t) in (5.13) exists. Letting m approach in (5,15a) gives (5.14). Wenote that an O(&t1/) rate of convergence has been obtained for a similar front trackingalgorithm (Huber’s method) in [22].

We now develop sufficient facts to prove that g(t) s(t). Let f Fv, and let a(t) bea monotone nondecreasing Lipschitz continuous function with a(O)= z(f). Let T > 0,and consider the following problem; find c(x, t) satisfying

(5.16a) c,=cxx, O<x<a(t), O<t<-_T,

(5.16b) cx(0, t) 0, O<t<-_T,

(5.16c) c(a(t), t) L, 0 <- <= T,

(5.16d) c(x, 0) =/(x), O<=x_z(f)=a(O).

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

572 JOEL C. W. ROGERS, ALAN E. BERGER AND MELVYN CIMENT

The solution u(x, t) of (5.1) satisfies (5.16) (with a s, f Uo). In addition;

f0(

fo(5.17) c(x,t)dx= f(x)dx for0t-< T

is satisfied when a s, c u, [ u0. It will be shown that o-(t) is nondecreasing andLipschitz continuous and satisfies (5.17). Then using arguments similar to those in [32]it will be proven that given fFv, there is at most one monotone nondecreasingLipschitz continuous function a(t) with a(0)= z(f) such that if c solves (5.16) then(5.17) holds. Thus vhen [ Uo, one ill have o-(t) s(t). One thus has;

LEMMA 5.5. Letf Fvand let tr(t) be given by (5.13) (and define tr(O) z(f)). Thenfor >- 0 and 8 > 0;

(5.18) 0 <_- tr(t + 6)-o’(t) <- V6/L.

THEOREM 5.3. Letf Uo and let or(t) be given by (5.13). Then or(t) is the Stefan freeboundary s(t) in (5.1).

We complete the study of A3 by giving an estimate of (T)"Uo(X) u(x, mr). Letf= Uo; then fFv with V =max {lug(x)[ :0=<x <--So}. The construction in the proof ofThm. 5.3 of {un} converging to u shows that u (., t) Fv for each 0. Let T > 0, n > 0,and set z Tin. Using Lemma 6.3, and Corollary 6.1 with {un, an} and u, s, along with(5.6), (5.18), (5.14), and the maximum principle, one can obtain;

COROLLARY 5.1. Let 2V(’/r)I/2<-_L. Then for m= 1,... ,n,and for O<=x <-

min (s(mz), s(m * ’, Uo));

(5.19) I(T,)"Uo(X) u(x, m’)l <=E3 V(2 V2mT"(’r/Tr)l/2L-2 + 3 V’rL-1).

Furthermore, for 0 <-_ x <

(5.20) Itemp ((T,)’Uo(X))-temp (u(x, m’))l <-E3.

Algorithm A2. An equivalent formulation of the analytical APT method will nowbe studied. Suppose f Fv and z(f) So; let T > 0, let N be a positive integer, and setthe time step At TIN. Let f continue to denote (0, oo), and set u(x)=f. Given u"(x)approximating the solution at time -= n At of the one phase Stefan problem with initialdata f, u"+l(x) is obtained by A2 as follows. Define Sn =sup {X:Un(X)>--L}, and

x0

(5.21) An(x)=l un(x)’L,

Let U (x, t) be the solution of

(5.22a) c, cxx, x

(5.22b) c(x, O)= An(x),

(5.22c) cx(O, t) O,

0 X Sn

X >Sn.

O<t<-At,

O<t<=At,

set B" (x)= U (x, At) for x f, and define

(5.23a)Bn(x)n+lu (x)B"(x)+u"(x)-L,

0 <- X Sn,

X >Sn,

(5.23b) Sn+l sup {X" un+I(x) >- L}.

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

NUMERICAL SOLUTION OF A STEFAN PROBLEM 573

It will be demonstrated below that for each n, un(x)>=L when 0_-<x <=s,,, whichshows that A2 is equivalent to A1. We now begin developing some of the properties ofA2 to be used in proving Theorem 5.1. When convenient, Zkt will be denoted by -.

LEMMA 5.6. For n 1, 2,. , N;i(n)ii(n) u n is monotone nonincreasing for x >=sn-1,iii(n B >= L on (, and u >= L for 0 <= xiv(n) Am(x) max (un(x), L) >-L for xv(n) u <- B n-1 onvi(n) A Bn-1 on [0,vii(n) A <_ Bn-1 on ,viii(n) A (x) is monotone nonincreasingix(n) u is C except at sn-1, u" restricted to [0, s,-1] can be extended to be C

at s,-x, and similarly with u restricted to (sn-1, c),x(n) u >=0 on f, and un(x)O as x c.

Proof. Except perhaps for (ii), the proof follows easily by induction using thedefinition of A2. For (ii) at n 1, use Remark 6.1 and explicitly express (U-L)x bydifferentiating G with respect to x in (6.1). Since (A-L)>=O on [0, So], and (A-L)vanishes for y > So, one finds that B (x) < 0 for x > So. One obtains (ii) at n + 1 similarly.

Remark 5.1. It can quickly be seen that if m is the first integer for which s, > Sothen u" continuous at s,, and u is continuous on 1) for n > m.

LZMMa 5.7. For n O, 1,. , N, extendA n, u", and U (., t) to be defined on R byreflection (i.e. An(-x) An(x) etc.). Then

i(n) an(so-6)>=a"(so+6) for6>=O,ii(n) U (So 6, t) >= U (So + 6, t) when 6 >->_ 0 and >= O,iii(n) An(x) is monotone nonincreasing for x >=So,iv(n) U (., t) is monotone nonincreasing for x >= So, >= O,v(n) un+(x) is monotone nonincreasing for x >-So.LEMMA 5.8. For n 1, 2,. N; a u a f.Proof. The result quickly follows by replacing (5.22b) with c(x, O)=An(x)-L

(whose integral over fl is finite) and considering the corresponding modification of(5.23a).

We have now proven the qualitative assertions in Theorem 5.1. The error estimaterequires further quantitative estimates on {u (x), sn} which are developed below.

LEMMA 5.9. Suppose f Fv and z(f) So. Set e 271/2 and V’rl/2/3. Then forn =0, 1,... ,N;

(5.24)

(5.25)

(5.26)

U (Sn, t) <= 37’1/2 V +L for 0 <= <= ’,

An(x)<-L+c+ V(s,+e-x) )CorO<--x<<-s,,

t)<=L+c, x >-sn +e, O_t<=z,U"(x,

L+c+ V(s,+e-x), O<=x<-sn+e,Ot<=".

LZMMa 5.10. Given 1 Fv and z (]) So. Then

(5.27) sn so <-_ 3nrV/L.

In the next lemma, a bound for the total enthalpy which has passed beyond thefront sn at each step of the algorithm A2 is established.

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

574 JOEL C. W. ROGERS, ALAN E. BERGER AND MELVYN CIMENT

LEMMA 5.11. Given fFv, z(f)= So, and 3 V’rl/2<-L. Then for n =0, 1, 2,..

(5.28) u"(x) dx <-_ 3L’/2(ln (1 + n))/2.

Algorithm A4. At this point, we introduce a tamily ot energy preserving algorithmscalled A4. Two particular members of this family will be used to link together the frontlocation obtained by the APT method with that obtained by A3 (for which the errorestimates (5.14) has been established).

Suppose fFv and/ is a nonnegative constant. We say that fFv() if f(x)= Lfor z (f) -/3 =< x

_z (f). Henceforth, it will be assumed that a constant time step denoted

by z or by zt has been chosen. When f Fv(B), v(x, t, , ) will denote the solution of

(5.29a) v, v,,,, 0 < x < z (f)-/, > 0,

(5.29b) v(x, O) [(x), 0 < x < z([)-,(5.29c) vx(0, t) 0, > 0,

(5.29d) v(z(f)- 3, t)-- L, O.

Analogously to (5.5), (5.7) define

(5.30) r(r, f, B)= z(f)-L- v,(z(f)-B, t, f, B) dt,

T(r,)f(x)=v(x, r,f,B) for0-<-x=<z(f)-B,

(5.31) T(r, )f L for z (f) -/ =< x =< or(r, f, fl),

T(’, fl)f= 0 for x > or(r, f, B).

Given /o_->0 and given feFv(o), define tro=z(f), o’=o’(r,f,/o), andT(r,/o)f. As in (5.6) and Lemma 5.1, one has fie Fv(Bo), 0 <= cr-cro<= Vz/L, and

Let fo Fv and let a sequence of nonnegative numbers/0,/31, m be given.Suppose fFv(flo), and suppose for i=l,...,m that feFv(i) whereT(’,Bi_x)f-1 and trmo’(’,/-, fli..1); then the sequence /o,""" ,/3 is said to beadmissible for fo. With appropriate choices of the/, {tri, fi} may be used to approximatethe solution of the Stetan problem (5.1) having initial data fo (indeed, note that A3 is A4with/i 0). We will use two special choices of {/} to estimate the difference between s,(from A1) and s(n * -, Uo) (from A3), and thus obtain Theorem 5.1.

The next lemma provides a comparison between A3 and A4 for a special class of{/,}. For fo fv, po will denote z(f) and tg s(n * 7", fo).

LEMMA 5.12. Assume fo Fv, Bo,""’,/ is admissible for f, o <-_ <=,, and2 V(z/r)/2 <__ L. Then for m 1, 2, ., n + 1,

(5.32) or,, -/L,- =< p,, < or,,,.

The first specific choice for {/i} will now be considered. Suppose f0 e Fv andz(f) So. Recalling the notation (5,23), we define

(5.33) /30 0, fli tri si for 1, 2,., .These fl will be shown to be admissible for fo, and hence {fli} estimates the differencebetween the front locations obtained by the APT method and by a special case of A4.

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

NUMERICAL SOLUTION OF A STEFAN PROBLEM 575

LEMMA 5.13. Assume 3 Vx/-zL. Then (5.33) defines an admissible sequenceand for n O, 1,

(5.34) n<-_3Vzl/EL-l(2(nz/cr)l/E+3nzVL-l)+3zX/E(ln (l +n))a/z.Henceforth denote the {fli, cri, fi} obtained in Lemma 5.13 by {/*, or/*, f}. When

fo(x) Uo(X); Theorem 5.2 bounds s(n z, Uo)-S(t), Lemma 5.13 estimates s,,-cr,*,while Lemma 5.12 measures or,, s(n * r, Uo) for nondecreasing {/}. The missing link isa specific choice of nondecreasing "small’ {/} for which the resulting c is ’near"for n 1, 2,. .. The following lemma provides such a choice when the total elapsedtime, nr, is sufficiently small.

LEMMA 5.14. Let z > O, fo Fv, z(f) Zo, and let {/i} be an admissible sequenceforfo (the functions andfront locations obtainedfrom fo, {/8i} using A4 are denoted, as usual,by fi and cri). Assume that for n 1, 2,

(5.35) O’n-l [n-l O’n [n.

Define(5.36) o=0, and , m’VL- +o for n l,,2,

and let M be the largest integer for which

(5.37) 2 VL-a(Mr/r)/+ VL-I"rlM_I 1/2.

Define a new sequence o, ", M as follows (with corresponding functions and frontsdenoted by and );

(5.38) /o =/30, and/,, min (2. max/, ,, tn-1%" n-1) for n 1,..., M.Oin

Note that o Zo, , o Zo, , and fl Then for n 1,... M"

i(n) g, ft, (Zo- flo) ,,ii(n) ,_iii(n) Fv() and_l---,iv(n) ,v(n) , ,,vi(n) ,-, g,-,,vii(n) "(x)f"(x)forOx,-,viii(n) f(x)-"(x) V,forOx,-,,ix(n) , 2. max {g/Oi n}.

We now have sufficient results in hand to establish convergence of the APT methodwhen it is applied to (5.1). Pick a positive integer N and set the time step z T/N.Define f= Uo; then fFv where V =max {[u’(x)[’Ox So. It will be convenient tohave the notation (ef. (5.34));

B(z, T)3 vl/ZL-X(2(T/)1/2+ 3TVL-)+ 3zx/Z(ln (1 + Tz-a)) /z.

The estimate of Is,- s(nz)[ claimed in Theorem 5.1 is contained inTnOM 5.4. Assume z is small enough to satisfy

(s.39) 4 v. (z, T) .Then for n 1,. , N;

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

576 JOEL C. W. ROGERS, ALAN E. BERGER AND MELVYN CIMENT

where s, is the front location obtained by the APTmethod (i.e. A2), and s(t) is the frontlocation in (5.1).

The estimate of un(x) u(x, nT.) in (5.2), and (5.3) follow from a line of reasoningsimilar to that used to obtain Corollary 5.1 (note Lemma 5.9 and (iv) and (v) of Lemma5.7 compensate for the fact that u" is not necessarily in Fv).

From examining the extent to which the lemmas of this section can be improved,we feel that the rate of convergence in At given in Theorem 5.1 for the analytical APTmethod is sharp. The numerical experiments presented in Table 1 indicate that the samerate of convergence is a sharp estimate for the temporal error in the numerical APTmethod. The requirement that Uo e Fv for some V can no doubt be relaxed, since afteran arbitrarily small time the solution will be "smooth."

We believe similar convergence behavior obtains in the case of a multifrontone-dimensional one phase problem as long as the various fronts remain separatedfrom each other. The analysis would probably involve treating solutions locally neareach front. The two phase problem should also be amenable to a similar type ofconvergence proof. In higher dimensional problems similar types of convergenceresults may be obtainable if the free boundaries are known to have a priori a certainamount of regularity. When, for example, fronts run together the error behavior islikely to be altered.

Proof ofLemma 5.2. Setting Sl s(7.1, f) and using (5.9), one obtains

(5.41) S(7.1 "[- 7"2, f)- S(7.2, 7.1, f) L-1 (u(x, 7.2, T.rlf)- u(x, 7.1 --b 7.2, f)) dx

(u(x, ,+L-1 T.f) L) dx.

Since for f Fv, the solution of (5.4) is no smaller than L, it follows that the last term in(5.41) is nonnegative. Let

J(x, t) =- u(x, t, T,f)- u(x, 7.1 + t, f).

Then J satisfies the heat equation in 0 < x < z, 0 < _<- 7.2, with J(x, 0) 0 for 0 <_- x <_- z,with J, 0 at x 0, and with J(z, t) >- 0 for 0 =< _-< r2.

Hence J >= 0 for 0 _-< x _-< z, 0 -< -< 7.2. Thus the left inequality in (5.10a) is true.Upper bounds for the two integrands in (5.41) will now be given, commencing with thesecond. Using Remark 6.1, we see that

(5.42) u(x,t, Tlf)-L < V(sl-x) forO<=x<-sl, t>-O,

and so (recalling (5.6)), the last term in (5.41) can be bounded by-L-l[V(sl-x)2/2]2 <-L-1V( VT.I/L)2/2. Again using (5.42) and (5.6) we find

O<-J(z,t)=u(z,t, T,f)-L <- V(sl-z) <- V27.1/L for0_--<t<_--7.2.

By application of Lemma 6.1 to J in (5.41), the proof of (5.10) is concluded.ProofofLemma 5.3. It clearly suffices to prove the result for n 1. Let 7. 7.1. From

the definition of Fv; z=-z(fl)>=z2=-z(f2). For z2<-s<-z define f(x,s) in Fv byf(x,s)=f2(x) for O<-_x<-_z2, f(x,s)=L for z2<-x<-_s, and f(x,s)=O for x>s.

It will be shown (first) that r.rfx r.rf(’, z1), and also that when (5.11) is valid,T,f(., z)>-T,f2. By the maximum principle for the heat equation, for 0-_<x _-< z l,

0 _-< -<_ 7., one has

U (X, t fl) ’ U (X, t f(’, Z 1)) L.

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

577

O<-x S(7"2, 7"1,(5.46) U(x) =--min (T,T,f(x), T,+,f(x))=

0, x > s(z2, zx, f).

Applying (5.44) with fi T,/, f and f U yields

(5.47) E(f)-E(f2)>=L(-s(r,, z3, U))>-O.

Using (5.46), Lemma 5.1, and (5.10), one has

(5.48) E(ft)- E(f2) L(s(’ + 7"2, f)- S(T2, 7"!, f)) LD (’r2, 7"1)

which with (5.47) gives

(5.49) s(z, z3, U) <- fl <-_ s(z,, , z3, U) + D(z2, Zx).

(5.45)

Define

NUMERICAL SOLUTION OF A STEFAN PROBLEM

Since both functions equal L when x Zl,

-Ux(Zl, t, fl)>-_-Ux(Zl, t,f(.,Zl)) for 0<t<_-z.

Thus from (5.5) and (5.7) it follows that T,fl >-_ T,f(., z 1). The proof will be completedby showing that when t _-> 0 and z2 --< s <- s + 8 _-< z 1, then

(5.43) T,f(. s) <_ T,f(. s + 6).

By the maximum principle, for 0 -< x -< s, 0 <= -< r, one has

c(x, t) =- u(x, t, f(. s + 6))- u(x, t, f(. s)) >-_ O.

The result (5.43) then follows if one can show that a s(z, f(., s + 8)) is at least as big asB =- s(z, f(., s)). Using Remark 6.1,

u(x,t,f(.,s+6))<=L+ V6 whens<-x<=s+6,0<=t<=z.

Applying Lemma 6.1 to c(x, t) shows that

ioA =_ c(x, z) dx

Now recalling (5.7), (5.8), and Lemma 5.1, we find

LS E(T,f(. s + 8))-E(T,f(. s)) <- As + (L + VS)8 + L(a -(s + 8))-L(13 s)

and so

O < 2(’r/ "rr2 vt + Vt2 Lt +L a

which with (5.11) gives a-B >- Vt2/L. The result easily follows.ProofofLemma 5.4. Denote s(zn," Zl,f) by a and s(z,,, za, 71 + z2, f) by/3.

Let fl and f2 be in Fv and let fl --> f2 (specific choices for fl and f2 will be made below).Let sl=s(z,,"’, z3,fl) and s2 =s(z.,..., z3, f2), and let T denote the operatorT, T,3. Then by Lemma 5.1 and Lemma 5.3,

(5.44) E(fl)-E(f2)=E(Tfl)-E(Tf2) >-_ J Tfl>-L(Sl-S2)>-O.

By (5.10), s(’2, Zl, f)<--s(zl + z2, f), and so by using the definition (5.7) of T and themaximum principle,

T,2T,lf(x) >= T,+,zf(x) for0_<-x_-<s(z2, zl, f).

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

578 JOEL C. W. ROGERS, ALAN E. BERGER AND MELVYN CIMENT

Now let fl T,2 T,lf and fz U. Then (5.48) is again valid, and (5.44) yields

E([1)-E(h)>-L(a-s(r,, ’3, U))>--O.

From (5.50) and (5.48) one sees that (5.49) is true when/ is replaced by a, and thus(5.12) is valid.

Proo ofLemma 5.5. When 0 the result is an immediate consequence of (5.6).Now let > 0. Let n be a positive integer, and set r t/n. There is a unique integer m >-- nsuch that mr <- + 6 < (m + 1)r. Define y by m(r + y) + 6; then 0 y < r/m. By (5.6),

0 <= s(m r, )- s(n * r, t) <-_ V6/L.

The proof is completed by demonstrating that as no; s(m *r,f)r(t+6) ands(n r,])tr(t). Theorem 5.2 asserts the latter; and that s(m (’+y),[)tr(t+6).But by repeated use of Lemma 5.4,

[s(m * (r + y), f)- s(my, m r, f)[--- mD(my, r) + mD(r + y, my),

and from (5.6), ]s(my, m. r,f)-s(m * r,f)]_-< Vmy/L. As n (and thus m), theresult (5.18) is obtained.

Proof of Theorem 5.3. We first show that tr(t) satisfies (5.17). Let fFv and let Tbe a fixed positive number. For n a positive integer, set r T/n. Consider the piecewiselinear function an(t) defined such that the polygonal curve (an(t), t) passes through thepoints (z, 0); (z, r); (s(rn r, f), mr + r), m 1,. ., n 1. The functions u(x, t, f) (see(5.7)) used in obtaining s(rn r, f) together yield a function un(x, t) satisfying (5.16a),(5.16b), (5.16d) (with a an). By using (5.14), (5.18), the maximum principle, andLemmas 6.3 and 6.2, one finds that {un(x, t)} converges (in 0<-x < tr(t), 0<=t<= T) to afunction w(x, t) which solves (5.16) (with a =tr). Because of the construction (5.7)which gives Lemma 5.1, it follows that the pair w, o- satisfies (5.17).

We now use the general line of reasoning in [32] to show uniqueness for (5.16),(5.17), under the condition that a is monotone nondecreasing and Lipschitz continu-ous. Suppose a, ca and fl, c are two such solutions. Let/.(t)--- min (a(t), fl(t)) and letc(x, t)c(x, t)-ce(x, t) for O<=x<-_(t), O<-_t<-- T. Define e(t)=la(t)-B(t)], E(t)=max {e(6) :0 6 -< t}, and let V1 be a bound for the Lipschitz constants of c and ft. ByCorollary 6.1

]c(x, t)l <- V. E(t) for 0_-< x -< (t), 0 <= =< T.

Let tl max {t [0, T]/E(t)= 0}. If tl T then a =/3, so let us suppose t < T. Lettz (tx, T]. Now there is a (t, tz] such that e(t) E(tz) E(t) > 0. Without loss ofgenerality suppose a(t)</3(t). Now by (5.17)

0 c(x, t) dx + c(x, t) dx- c(x, t) dx,aO a,(tD as(t)

which by use of Lemma 6.1 implies

(5.51) O<=2((t-t)/rr)l/2v E(t)+ V(t-t)V. E(t)-e(t)L.

Dividing (5.51) by E(t)= e(t) yields a contradiction for t2 sufficiently close to t.Proof ofLemrna 5.7. The proof is by induction, beginning with n 0. Now A>-L

and A(so+ 8) L when 6 >0, so i(0) is true. For (ii), use (6.1) to express Un(so+ 8, t),and then perform the change of variable z y So. Similarly treat U (s0- 8, t), and soobtain

Un(so-6, t)-Un(so+6, t)= fo (An(s-z)-An(s+z))(G(z-6’ t)-G(z +8, t)).

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

NUMERICAL SOLUTION OF A STEFAN PROBLEM 579

Since the second factor in the integrand is nonnegative, ii(n) follows from i(n). At n 0;(iii) is obvious, while for (iv) and (v) see the proof of (ii) of Lemma 5.6. Assume (i)-(v)are true at 0, 1,. , n. Now A"+l(x) u"+l(x) U" (x, At) for s, <= x <= sn, so by ii(n),A"+l(so-3)>=A"/(So+3) when So<=So+3 <=s,. In the case s, <s0+3 =<s,/x, one has

A"+ (So + 3) U" (So + 3, At) + u" (So + 3) L =< U" (So + 3, At) U" (So 3, At).

If-s, <=So-3 then the latter term above equals A"+a(So-3). Otherwise one must haves, <3-So <=So+3 <-s,+, and so

A"+(So-6)=A"+a(6-So)= U"(6-So, At)+ u(3-So)-L

>= U"(so + 6, At) + u"(so + 6)-L A"+(So + 6).

A"+a(So+6)=L<A"(so 3) completing the proof of i(n + 1). AsWhen So + 3 > s,+,above, ii(n + 1) follows from i(n + 1). By v(n), u "+1 is nonincreasing for x >=So, whichimplies iii(n + 1). For iv(n + 1), use (6.1) to write

U+ (x, t)= I0 1/2zt-XG(z’ t)(a"+l(x +z)-a"+(x-z)) dz.

By i(n + 1) and iii(n + 1), the term in parentheses is nonpositive when x => So, so iv(n + 1)is true. Finally, v(n), iv(n + 1), and (5.23a) yield v(n + 1).

Proof of Lemma 5.9. Note that (5.24) follows from (5.26). The proof of (5.25),(5.26) is by induction. At n 0, since f Fv, (5.25) is obvious. We show that (5.25)implies (5.26). Use Remark 6.1 to express

U"(x, t)=L+ Io (G(x-y, t)+G(x + y, t))(An(y)-L) dy.

Then (5.25) gives

U" (x, t) <= L + G(y, t)(a + V(y x + sn + e)) dy + G(y, t)Srt

(a + V(x + s. + e y)) dy.

When x

_s, + e this shows that

U" (x, t) =< L + / G(y, t)(a + Vy) dy

(5.52)L + a J0 G(z + e, t) dz + V(t/Tr) /2 exp (-e2/(4t)).

Using the fact that exp (-zE/(4t)) 1, we observe that (5.52) yields

:2/U"(x,t)<-L+(e-+ V)(t/r)a/2 exp(-e (4t)) forx>s,+e,t>O.

With e and a as given, the first part of (5.26) follows. The function c(x, t) =-L + a + V(s, + e x) for 0 <_- x <= s, + e, 0 <_- <_- z, is a solution of the heat equation whoseinitial and boundary values majorize those for U" (x, t), and so (5.26) is a consequenceof (5.25). The proof of Lemma 5.9 is completed by assuming (5.25), (5.26) (at n) andshowing that (5.25) is true at (n + 1). By vii(n + 1) of Lemma 5.6, it suffices to show that

Un(x, 7-)<=L+a+V(Sn+l+e-x) for O-<x -<S.+l.

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

580 JOEL C. W. ROGERS, ALAN E. BERGER AND MELVYN CIMENT

But by i(n + 1) of Lemma 5.6; sn <--Sn+l and so

(5.53) L+o+V(sn+l+e-x)>-max(L+o+V(sn+e-x),L+o) for0<--x<=Sn+l.

The result follows from (5.53) and (5.26) at n.

Proof o[Lemma 5.10. It will be established by induction that

(5.54) J un(x) dx <-3m’V-L(sn-so),Sn

which implies (5.27) since u is nonnegative. Now (5.54) is clearly true when n 0;suppose it is true at n. By (5.25), An(x)<-A(x) where

A(x)-=L+3V’I/2+ V(sn-x) for

A(x) L for sn < x.

Recalling the definition B (x)- U (x, ’), we see

(5.55) (Bn(x)-An(x)) dx U (sn, t) dt.Sn

Let U(x, t) be defined by Ut Uxx for x eR 1, t>0, and U(x, 0)= A(x). Take An(x) tobe extended onto (-oo, 0) by reflection (i.e. A (-x)= A (x)). Then using (6.1), andnoting that A (x) A" (x) is nonnegative for oe __< x _-< sn, and is zero when x > sn, we get

(5.56) u’ (s,, t) dt <- U(s,, t) dr.

By direct calculation using (6.1), the quantity on the right side of (5.56) is equal to

(- 3 Vrl/(4rt)-1/- V/2) dt 3 V’r

From the definitions (5.23a) and (5.21), and the above,

(5.57) | (un+(x) un(x)) dx | (Bn(x)-An(x)) dx <- aVe-.

Using (5.57), (5.54), and Lemma 5.6, we obtain

n+l n+lu N3Vz+ u u N3Vz+3Vzn-L(s,-so)-L(S,+l-S,),Sn Sn

giving (5.54) at n + 1, and thus (by induction) Lemma 5.10 has been proven.Proof of Lemma 5.11. Set r 2zx/Z(ln (1 + n)) 1/2 and p s,+a + r. We will first

bound

Using the maximum principle, Remark 6.2 (5.24), and the change ol variablex/(r s)-/ in (6.2), one finds that for x s,

(5.59) B(x)-L U(x, r)-LNJ(x) 3r/V(r)-/ exp (- /(4r)) d.

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

NUMERICAL SOLUTION OF A STEFAN PROBLEM 581

From (5.59), and setting rt sn+l- sn + r, we obtain

(S"(x)-L) dx <- (x) dx

67"1/2 VG(, r) dx dj <- 67"/2 VG(, r)(j- r) dlj.aSn+l+r

Using the change of variable y r(lj-r)/(2r) in the latter term, and noting thatexp (- ryZr-z) _-< 1, one obtains

1/2 2 -1 -2(5.60) (B -L) dx <-_K(r, r, V)=- 3r V 4r (Trr) /2r exp (-r2/(4r)).

From (5.23a), Lemma 5.6, (5.58), and (5.60) it follows that (for n _>- 1);

i+1u (x) dx <- (u (x)- u (x)) dx <-_ ng(r, r, V).n+r i=0 i+l+r

Recalling the definition of r, for n -> 1 we obtain

u"(x) <-Lr + nK(z, r, V) Lr + + 1). In (1 + n)).dx 3n Vrzr-1/2/((n

If 3 Vr/2 <=L, this implies (5.28).ProofofLemma 5.12. From/3 => 0 and f Fv define $(/3)f by $()f(x) f(x) when

O<-x <-_z(f)-, and S(B)f(x)=O when x > z(f)-3. Observe that (5.31)is equivalentto

(5.61)T(r, )f(x)= TS()f(x) for O<=x <--zl-- z(TS()f),

T(r,B)f=L forzl<=x<-zl+, T(z,)f=O forx>zl+/3.

Define

(5.62) fl T,S(/3o)f, and fi+l T,S(i-i-1)fi for i= 1,..., n.

By a straightforward induction argument, one can show that for 1,. , n + 1;

(5.63) z(f/) o- -/3_a,

(5.64a) [i(x)=fi(x) forO<_x<=z(fi),

(5.64b) fi Fv( -l),

where/3,,+1 ---/3,. From (5.62) and Lemma 5.3 it follows (by induction) that

(5.65) f_-<(T,)gf for i= 1,..., n + 1,

and the left inequality in (5.32) is implied by (5.65) and (5.63). The right inequality in(5.32) will be demonstrated using the following form of (5.44): let f and g be in Fv andsuppose f_-> g, then

(5.66) s(r, f)<=s(% g)+L-l(E(f)-E(g)).Letting f f and g S(/3o)f in (5.66) gives px z (]c1) "+" flO O’1. Letting f (T,)if andg fi gives

(5.67a) Pi+l S(7" fi)q-L-X(E(f)-E(fi))

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

582 JOEL C. W. ROGERS, ALAN E. BERGER AND MELVYN CIMENT

while letting f= fi and g S(i)fi= S(i- i-1)fi gives

(5.67b) s(r, fi) <= zi+l) + L-l(Ei)-(E(f)- CiL)),

and (5.67) and (5.63) complete the proof of (5.32).Proof ofLemma 5.13. Using conservation of energy, it will first be demonstrated

(by induction) that (5.33) defines an admissible sequence and that

(5.68) ui(x)>fi(x) forO<--x<--si, i=0,1,....

This is true by definition at 0. Assume/30, ,/3,-1 is admissible, and (5.68) is truefori=0,1,...,n-l. Then

n-1(5.69) E(f) E(f") (or. s.-1)L + f’,

and so

(x) dx (s, Sn-1).(5.70) 3, "-:---Crn--Sn L-l’(f)-L-1 fn

Noting that a u" E(u =fo), (5.70) yields

I IoS - u_(5.71) , =L- u"+L- (u"-L)+L- f"Sn

Using the definition (5.29)-(5.31) (with 8 8_, f -), the definition (5.21(5.68) at i-n- 1, and the maximum principle, one finds that U-(x, t) (x, t) for01, 0, d so ()() for 0,-1. Tho (rooll o0, ]),sin othrwis () would fail to b true. Sinc

_ ,thn (5.6) is tru at , and (). It rmains to obtain (5.3) (for 0) withth aid of (5.71). Th first trm in (5.71) is bounded using Lmma 5.11, whil () ofLmmu 5.6 und (5.26) imply

(5.72) L-1 (un-L) N(3Wra/2)L-(sn-sn_l).

To estimate the third integral in (5.71), we prove (by induction) that for 0, 1,.

(5.73) u (x) ff (x) 3 Vra/2 for 0 x si.

The case 0 is trivial. Assume (5.73) at n. Then using (5.29)-(5.31), (5.21)-(5.23),(5.24), and the maximum principle,

u"+(x)-f"+(x)3Vr/2 for 0x s.Now s, s,+a ,+, ["+= L on Is,, ,+], and so by (5.24), v(n) of Lemma 5.6, andiv(n) of Lemma 5.7, (5.73) at n + 1 is valid. Also note that {U"(x, t)}, n 0, 1,...(from (5.21)-(5.23)) "piece together" to form a solution c(x, t) of

(5.74a) c, c, (x, t) 6 (0, s,) x (nr, nr +n=0

(5.74b) c(x, O) (x), 0 x So,

(5.74c) c(0, t) 0, > 0,

such that c(x, nr) u"(x) on [0, s,], n 1, 2,.... Similarly the functions v in (5.29)used to obtain the fi together yield cz(x, t) satisfying (5.74) and cz(x, nr) =f"(x) on

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

NUMERICAL SOLUTION OF A STEFAN PROBLEM 583

[0, s,], n 1, 2,.... Setting c(x, t)=Cl(X, t)-c2(x, t), the arguments yielding (5.73)show that c(x,t)<-3.V-1/2. We apply Lemma 6.1 to (x,t) restricted to (x,t)[0, So] [0, nz], and find that

2(nr/Tr) Vr(5.75) (u"-f" < /2(3 /z).

Using (5.75) and (5.73) one obtains

/2L- 1/2 1/2L-1 So)(5.76) L- (u -f") < 3 Vr 2(nr/) + 3V (s,_-

The result (5.34) follows from (5.71), (5.72), (5.76), (5.27), and (5.28).Proo of Lemma 5.14. From the two sentences below (5.31) one has ,-z0N

nrVL- from which i(n) follows, and ix(n) is part of (5.38). The proof of the remainingstatements is by induction. The case n 1 can be quickly verified. Now assume (ii)-(viii)are true at 1,..., n(NM-1). Then (ii) and (iii) at n + 1 are again straightforward.From the definition of A4, viii(n), vii(n), vi(n), and using the maximum principle oneascertains that 0N/n+l-ff+lN Vn on [0, 5,-], and thus also ff+l fn+l on[0, g,+x]. Since En+l) E()= E(fn+X), it must be true that +1N+I. One hasf,+l Fv, so f,+1_+1N V, on [0, g+l]. Now vi(n), (5.35), and iv(n + 1) yield theassertion that

from which it follows that ,+,+-, +,, and this with (5.38) implies v(n + 1).Observe that the first inequality in vi(n + 1) is equivalent to

(5.77) n+l-- n+l n+l-- n+l.Now (iii) and (viii) at I,. , n and an argument similar to that which produced (5.75)shows that

[z-O (fn+l n+)=2((n + 1),/)l/2vfin.(5.78a) <aO

One has from i(n) that

(5.78b) (["+a N V(-fl,-(Zo-flo))N V,,,O--O

and conservation of energy, (5.78), and (5.37) give

"L- (,+ "+)< ,VL-(2((n + 1)/)/2 + ,)(5.79) 6,+-,+

If ,+ 2. max {" 0i n + 1}, then (5.79) and (5.38) yield vi (n + 1). Otherwise(5.38), vi(n), and (5.5) give vi(n + 1), and then vii(n + 1) and viii(n + 1) are easilyverified.

Proo[ o[ Theorem 5.4. Set =G Uo, and let {, ,[} denote the resultsobtained using (5.33) and algorithm A4. By utilizing Lemma 5.14 it will be shown belowthat

(5.80) ]s(n , o)-( -)[2(72V=TL-2+2)B(, T) for n 1,... ,N.

From (5.33) one has , , s,, and so (5.80) and Theorems 5.2 and 5.3 yield (5.40).

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

584 JOEL C. W. ROGERS, ALAN E. BERGER AND MELVYN CIMENT

We now obtain an M for which (5.37) is valid in order to be able to apply Lemma 5.14.For x R let g(x) be the greatest integer which does not exceed x, and defineM=g(L2/(36V2r)).

From (5.39) it follows that

2)1/26 V’F 1/2 N 4 V" 3r /Z(ln -<_ L

which implies that M=>I. Since g(x)>-x/2 for x->l, it is also true that Mr>-_

L2/(72 V2). For each use of Lemma 5.14, the/30 in (5.36) will be equal to/3’ for some/ {0,..., N}. Lemma 5.13 shows that

f<--BO’,T) for/=0,-..,N;

and hence (5.37) will be valid for the value of M defined above.We now obtain (5.80) by repeated application of Lemma 5.14. Set too=0,

m min (M, n), and define Bi =/3" for 0, m Apply Lemma 5 14 to thesemo+i

and with fo Uo, obtaining {ii, , ]ri}, 0, , m. The results of Lemma 5.12 will nowbe used on {ii, ,/zi}. With the definition (5.62), and (5.63)-(5.65) one can show that

S(m)fTM S(m --[m-X)Tm (T,)"f,and vii(m) of Lemma 5.14 yields S([,)/ <-f. Conservation of energy and iii(m) ofLemma 5.14 give

E(fTM E(S(d,.)[") + d,.L E(f) E((T,)f) E(f’).

Apply the result indicated in (5.44) to (T)f>=S(1,,) and get

(5.82) O<=L(s((n -m) * z, () )-s((n -m) z, S(,,)’))<-,,L,and apply it to f’ => S(/m)/r’ and obtain

(5.83) O<=L(s((n-m),z,f’)-s((n-m),z,S(,,)’))<=,L,where s(0 z, f) denotes z(f) for f Fv. Multiplying (5.83) by -1 and then adding it to(5.82), and recalling (5.81) and ix (m) of Lemma 5.14 yields

(5.84a) [s(n z,f)-s((n-m), z,f’)[<=[,,, <=2B(r, r).

If m n, this yields the result (5.80). Otherwise the procedure is repeated: the kth stepis implemented with too-(k-1)M, m-rain (M, n-(k-1)M), fo=f,-)u, and fig=fl* for =0,... m, and producesmo+i

m#k-1)M)(5.84b) Is((n-(k 1)M m) r, (r,) ,,-s((n-(k- 1)M- m) z, f,k-)u+")l <= /3 (_1)M+,, --<2/30; T).

The final repetition occurs at the Kth step where (K + 1)M + min (M, n (K 1)M)n, and the inequalities in (5.84) combine to yield

[s(n * r, Uo)-r. [_-<2K. S(r, T).

Recalling that Mr ->_ L2/(72 V2) and noting that K <_- 1 + Nr/(Mr) concludes the proofof (5.80).

6. Appendix of basic results on the heat equation. In this section some standardproperties of the heat equation are stated in forms which are convenient for use withinthe convergence proof of 5.

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

NUMERICAL SOLUTION OF A STEFAN PROBLEM 585

Remark 6.1. The Green’s function G(x, y) for the initial value problem ct Cxx,

c (x, O) f(x) for x e D R is

(6.1) G(x, t) (4zrt)-1/2 exp (-x2/(4t)).If f satisfies a Lipschitz condition (with Lipschitz constant V) then for each >-_ 0, so doesc(x, t). The solution of ct =Cxx, c(x, 0)=f(x) on D with D a semi-infinite or finiteinterval, and with c 0 or cx 0 specified on the endpoint(s) of D, can be obtained using(6.1) by appropriately extending " to R 1. If " satisfies a Lipschitz condition (withLipschitz constant V), then for each so does c(x, t), assuming f-0 at any endpointwhere the condition c =0 is imposed. See, e.g. [13], [14], [18], [29], [41], [50].

Remark 6.2. Given a piecewise continuous function f(t), the solution of ct Cxfor x > 0, > 0; c(x, 0) 0 for x > 0, and c(0, t) f(t) for > 0, can be obtained in theform (e.g. 13])

(6.2) c(x, t) H(x, t- s)[(s) ds with H(x, t) xt-’ exp (-x/(4t)).

LEMMA 6.1. Consider the problem c cor 0 < x < b, > 0 with zero initial data,and with c(O, t) 0 and c(b, t) co or > O. Here b and Co are given positive numbers.Then or O,

/(6.3) c(x, t) dx 2(t/ Co.

Pro@ Let u be the solution of c c on {x >-b, > 0} with 0 initial data andc(-b, t) co for > 0; and let ua be the solution of c c on {x < b, > 0} with 0 initialdata and c(b, t) co for > 0. Then u(x, t) + u(x, t) c(x, t) when 0 < x < b, and so

bo c(x, t) dt I-b Ul(X, t).

The latter term may be calculated directly using (6.2) and yields (6.3).LFMMA 6.2. Let {cn (x, t)} be a sequence offunctions defined on an open setD in R

which are solutions of the heat equation in D. Assume for each compact set K in D, {cn}converges uniformly on K to a function c(x, t). Then (e.g. [32] with [41]) ct Cxx in D.

LEMMA 6.3. Let feFv, and let a(t) be a monotone nondecreasing Lipschitzcontinuous function with a(0)= z(f). Let T>0, and suppose ca(x, t) satisfies (5.16).Then (e.g. [10])

(6.4) L<-c(x,t)<-V(a(t)-x)+L forO<-x<-a(t),O<=t<-T.

COROLLARY 6.1. Let f eFv, and let a(t) and 3(t) be monotone nondecreasingLipschitz continuous ]’unctions with a(0)=/3(0)= z(f). Let T>0, suppose ca(x, t)satisfies (5.16), and suppose co(x, t) solves (5.16) when a is replaced by . Let E(t)=-max {[a (6) -/3 (6)1:0 <- 6 <- t}. Then

(6.5) Ic(x,t)-co(x,t)l<-V.E(t) forO<-_x<-min(a(t),(t)),O<-t<-T.

Proof. Apply Lemma 6.3 and the maximum principle.

REFERENCES

[1] D. R. ATTHEY, A finite difference scheme for melting problems, J. Inst. Math. Appl., 13 (1974), pp.353-366.

[2-] m. E. BERGER, M. CIMENT AND J. C. W. ROGERS, Numerical solution of a diffusion consumptionproblem with a free boundary, this Journal, 12 (1975), pp. 646-672.

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

586 JOEL C. W. ROGERS, ALAN E. BERGER AND MELVYN CIMENT

[3] Numerical solution ofa Stefan problem by a technique ofalternating phase truncation, SeminairesIRIA, Analyse et Contr61e des Syst/mes, 1976, pp. 21-34.

[4] C. BONACINA, G. COMINI, A. FASANO AND M. PRIMICERIO, Numerical solution of phase-changeproblems, Internat. J. Heat Mass Transfer, 16 (1973), pp. 1825-1832.

[5] R. BONNERO’r AND P. JAME’r, A second order finite element method for the one-dimensional Stefanproblem, Internat. J. Numer. Meth. Engrg., 8 (1974), pp. 811-820.

[6] B.M. BUDAK, E. N. SOBOL’EVAAND A. B. USPENSKII, A difference method with coefficient smoothingfor the solution of Stefan problems, U.S.S.R. Computational Math. and Math. Phys., 5 (1965), pp.59-76.

[7] B. M. Bt;DAI< AND A. B. USPENSIII, A difference method with front straightening for solvingStefan-type problems, Ibid., 9 (1969), pp. 83-103.

[8] J. R. CANNON, J. DOtGLAS, JR. AND C. D. HILL, A multi-boundary Stefan problem and thedisappearance ofphases, J. Math. Mech., 17 (1967), pp. 21-34.

[9] J. R. CANNON, D. B. HENRY AND D. B. Ko’rLOW, Continuous differentiability of the free boundary forweak solutions of the Stefan problem, Bull. Amer. Math. Soc., 80 (1974), pp. 45-48.

10] J. R. CANNON AND C. D. HILL, Existence, uniqueness, stability, and monotone dependence in a Stefanproblem for the heat equation, J. Math. Mech., 17 (1967), pp. 1-19.

[11] --------., On the movement of a chemical reaction interface, Ibid., 20 (1970), pp. 429-454.12] J. R..CANNON AND M. PRIMICERIO, A two phase Stefan problem, regularity of the free boundary, Ann.

Mat. PurR Appl., 88 (1971), pp. 217-228.[13] H. S. CARLSLAW AND J. C. JAEGER, Conduction ofHeat in Solids, 2nd ed., Clarendon Press, Oxford,

1959.[14] R. V. CHURCHILL, Fourier series and boundary value problems, McGraw-Hill, New York, 1941.[15] J. F. CIAVALDINI, Analyse numerique d’un problme de Stefan d deux phases par une mthode

d’glments finis, this Journal, 12 (1975), pp. 464-487.16] M. CIMENT AND R. B. GUENTHER, Numerical solution of a free boundary value problem for parabolic

equations, J. Appl. Anal., 4 (1974), pp. 39-62.[17] G. COMINI, S. DEE GUIDICE, R. W. LEWIS AND O. C. ZIENKIEWICZ, Finite element solution of

nonlinear heat conduction problems with special reference to phase change, Internat. J. Numer. Meth.Engrg., 8 (1974), pp. 613-624.

[18] J. CRANK, The Mathematics of Diffusion, Clarendon Press, Oxford, 1956.[19] J. A. DEAN, ED., Lange’s Handbook of Chemistry, 11th ed., McGraw-Hill, New York, 1973.[20] J. DOUGLAS, JR. AND T. M. GALLIE, JR, On the numerical integration of a parabolic differential

equation subject to a moving boundary condition, Duke Math. J., 22 (1955), pp. 557-571.[21] G. DUVAUT, Resolution d’un problme de Stefan (fusion d’un bloc de g!ace gt zero degrd), C. R. Acad.

Sci. Paris S6r. A, 276 (1973), pp. 1461-1463.[22] A. FASANO AND M. PRIMICERIO, Convergence ofHuber’s method for heat conduction problems with

change ofphase, Z. Angew. Math. Mech., 53 (1973), pp. 341-348.[23] G. E. FORSYTHE AND W. R. WASOW, Finite-difference methods for partial differential equations, John

Wiley, New York, 1960.[24] M. FREMOND, Variational formulation of the Stefan problem-coupled Stefan problem-frost propagation

in porous media, Computational Methods in Nonlinear Mechanics, J. T. Oden et al., eds., The TexasInstitute for Computational Mechanics, 1974, pp. 341-350.

[25] A. FRIEDMAN, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ,1964.

[26], One dimensional Stefan problems with nonmonotone free boundary, Trans. Amer. Math. Soc.,133 (1968), pp. 89-114.

[27], The Stefan problem in several space variables, Ibid., 133 (1968), pp. 51-87, Correction, 134(1969), p. 557.

[28] W. B. FULKS AND R. I. GUENTIqER, A free boundary problem and an extension of Muskat’s model,Acta Math., 122 (1969), pp. 273-300.

[29] G. HELLWIG, Partial Differential Equations an Introduction, E. Gerlach, Transl., Blaisdell, New York,1964.

[30] P. JAMET AND R. BONNEROT, Numerical computation of the free boundary for the two-dimensionalStefan problem by space-time finite elements, J. Computational Phys., 25 (1977), pp. 163-181.

[31] G. B. KOLATA, Moving boundary problems: advances on several fronts, Science, 191 (1976), p. 841 ft.[32] W. T. KYNER, An existence and uniqueness theorem for a nonlinear Stefan problem, J. Math. Mech., 8

(1959), pp. 483-498.[33] A. LAZARIDIS, A numerical solution of the multidimensional solidification (or melting) problem,

Internat. J. Heat Mass Transfer, 13 (1970), pp. 1459-1477.

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

NUMERICAL SOLUTION OF A STEFAN PROBLEM 587

[34] M. LOTKIN, The calculation of heat flow in melting solids, Quart. Appl. Math., 18 (1960-1961), pp.79-85.

[35] G. H. MEYER, A numerical methodfor two-phase Stefan problems, this Journal, 8 (1971), pp. 555-568.[36] Multidimensional Stefan problems, this Journal, 10 (1973), pp. 522-538.[37] An alternating direction method for multi-dimensional parabolic free surface problems, Internat.

J. Numer. Meth. Engrg., 11 (1977), pp. 741-752.[38] J. OCKENDON AND W. HODGKINS, EDS., Moving Boundary Problems in Heat Flow and Diffusion,

Clarendon Press, Oxford, 1975.[39] O. A. OLEINIK, A method of solution of the general Stefan problem, Soviet Math. Dokl., (1961), pp.

1350-1354.[40] D. W. PEACEMAN AND H. H. RACHFORD, JR., The numerical solution of parabolic and elliptic

differential equations, J. Soc. Indust. Appl. Math., 3 (1955), pp. 28-41.[41] I. G. PETROVSKII, Partial Differential Equations, W. B. Saunders, Philadelphia, 1967.[42] M. ROSE, A methodfor calculating solutions ofparabolic equations with a free boundary, Math. Comput.,

14 (1960), pp. 249-256.[43] L. I. RUBINSTEIN, The Stefan Problem, American Mathematical Society Translations, vol. 27, Provi-

dence, RI, 1971.[44] A. A. SAMARSKII AND B. D. MOISEYENKO, An economic continuous calculation scheme for the Stefan

multidimensional problem, U.S.S.R. Computational Math. and Math. Phys., 5 (1965), pp. 43-58.[45] A. SCHATZ, Free boundary problems ofStefan type with prescribed flux, J. Math. Anal. Appl., 28 (1969),

pp. 569,580.

[46] B. SHERMAN, General one-phase Stefan problems and free boundary problems for the heat equation withCauchy data prescribed on the free boundary, SIAM J. Appl. Math., 20 (1971), pp. 555-570.

[47] A. SOLOMON, Some remarks on the Stefan problem, Math. Comp., 20 (1966), pp. 347-360.[48] A. H. STROUD AND O. SECREST, Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs,,

NJ, 1966.[49] R, C. WEAST, ED., Handbook of Chemistry and Physics, 53rd ed., The Chemical Rubber Co.,

Cleveland, 1972.[50] D. V. WIDDER, The Heat Equation, Academic Press, New York, 1975.

Dow

nloa

ded

08/2

8/12

to 1

62.1

29.2

48.3

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php