AN INTERFACE TRACKING ALGORITHM FOR THE POROUS ...

TRANSACTIONS of the AMERICAN MATHEMATICAL SOCIETY Volume 284. Number 2. August 1984

AN INTERFACETRACKINGALGORITHM FOR THE POROUS MEDIUM EQUATION1 BY

E. DI BENEDETTOAND DAVID HOFF Abstract. We study the convergence of a finite difference scheme for the Cauchy problem for the porous medium equation u, = («m)AV,m > 1. The scheme exhibits the following two features. The first is that it employs a discretization of the known interface condition for the propagation of the support of the solution. We thus generate approximate interfaces as well as an approximate solution. The second feature is that it contains a vanishing viscosity term. This term permits an estimate of the form ||(""i_1)a.vIIi.r < c/t. We prove that both the approximate solution and the approximate interfaces converge to the correct ones. Finally error bounds for both solution and free boundaries are proved in terms of the mesh parameters.

1. Introduction. In this paper we derive and analyze a finite difference scheme for computing both the solution and the interfaces for the porous medium equation in one space dimension. We demonstrate that the approximate solutions and the approximate interface curves converge to the correct ones, and we obtain L00bounds for the error in terms of the mesh parameter. Consider the laminar flow of a polytropic fluid of density (x, t) -* u(x, t) in a porous medium which is assumed to occupy the whole space, and suppose that at time t = 0 the fluid is contained in the slab £,(0) < x < fr(0). The phenomenon can

be modeled by (1.1)

(1.2)

u,= (um)xx,

(x,t)eST

= Rx(0,T],

ii(-.0)-«o(.)

0 0 if x g (f/(0), fr(0)) and u0(x) = 0 elsewhere. We assume u0 is continuous

inR. Since the problem is degenerate, (1.1)—(1.2)is interpreted in a weak sense and the solution possesses a modest degree of regularity. Precisely (x, t) -* u(x, t) is said to

be a weak solution of (1.1)—(1.2)if u f °. Now given f "+1 > f " and v" forj e Z, we proceed as follows. First define

K(n + 1) = max{fc:x¿+1 < Ç?},

(1.10)

K(n + 1) = min{k:xk_x >tf),

"w+l

'«+it

•

•

•

'„t

•

•

•

• o • Q •

•

* •

—I-HlAT(n+l)

Figure 1 Then for Ä"(« + 1) < k < /f(n + 1) compute v"k+1from the finite difference equation ," + 1 _ i,"

(1.11)

Vk'^~Vk

* A,

At

1 „

\ü* + l _ 2üÄ + Vk-X

* = ffi(t;; + e)-* +

(AxY

+

m

m —1

v'k t+l+ x-v "*-!

,2

2Ax

where e > 0 will be chosen later. Observe that we do not enforce the difference equation across the interface.

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466

E. DI BENEDETTO AND DAVID HOFF

Next let (1-12)

sn+x = s

— xK(n+X),

and observe that

(1.13)

Ax f " +1 and compute f+2 from

(1.15)

f^2.r+i+

m /^^Af.

m - 1

sB+1

We shall prove that p£ > 0 for all « and fcso that by (1.15)T +2 > T+1- Thus the support of the approximate solution increases monotonically in t. In addition, the fact that s„ > Ax insures that numerical instabilities are avoided in the computations (1.14) and (1.15). Introducing the notations

ht

P " 77—2 ;

(Ax)

Avk = uk+ x- 2vk + vk_x,

we can rewrite the difference scheme (1.11) in the form

(1.16)

«r1 = Vk+ mß(vk + e)Av"k+ -¿g^0"*"

¡ ^

)'.

We shall assume throughout that

[Al] [A2]

[A3]

[A4]

0 < v0(x) < M Vx e R; \v0(x)-v0(y)\^y0\x-y\

Vx, >>g R;

e is of the order of Ax and

e>9-rYnAx; m- 1

O,

2mß M + e +-TYnAx m —1

where M and Yoare given positive constants. Since e = 0(Ax), condition [A4] on ß is seen to be a slight strengthening of the usual parabolic stability condition. We let h denote the pair (Ax, Ai), and we construct approximate solutions vh and approximate interface curves f* and £"*by piecewise linear interpolation. Our results may be summarized as follows: [I]

\\vh-v\\œ 0} so that approximate interfaces are not computed. Moreover numerical evidence indicates that the supports of their approximate solutions spread out too rapidly in time. Thus computing the interfaces by "shock capturing" seems to be unsatisfactory. While this paper was in preparation, Mimura and Tomoeda [13] informed us that they have recently derived an interface tracking algorithm for the porous medium equation. Numerical evidence suggests that the approximate interfaces computed by their scheme are accurate, but they are unable to prove this result. In addition, their scheme is somewhat complicated to implement, since it involves solving Riemann problems for the Burgers equation at each mesh point. Both their scheme and ours suffer from the parabolic stability condition Ai = 0[(Ax)2]. The paper is organized as follows. §2 contains the derivation of basic estimates. Specifically we prove the finite difference analog of the following facts, which are known to hold for the exact solution v of (1.6)-(1.7):

(1.17) (1.18) (1.19)

O^v^M, IKIL,sr*(•. Olli** y

(see[5]).

In §3 we demonstrate the convergence of the approximate solutions and interfaces to the correct ones by making use of various compactness arguments. The error estimates are proved in §§4 and 5. Finally in §6 we present and discuss the results of several numerical experiments.

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468

E. DI BENEDETTO AND DAVID HOFF

Throughout the paper we make the convention that C shall denote a generic positive constant depending only on m, M, y0 and some specified time T.

2. Basic estimates. We begin the analysis by establishing maximum principles for v"kand for the discrete space derivative

...» 4-K-x k

Ax

■

We assume throughout that the initial function vQsatisfies assumptions [Al] and [A2], and that the mesh parameters e and ß satisfy [A3] and [A4].

Lemma 2.1. The bounds 0 K(n + 1), these bounds follow from (1.14). We prove (2.2) first for k < K(n + 1). Rewrite (1.16) as follows: (2.4)

v"k+x= v"k+ mß(v"k + e)«+1

- 0. Proof. Denote the variable in question by Zk. That is,

7n _ Avl k

Ax2



>0

by [A3]. Thus Z"k+l is bounded below by the right side of (2.17) with Z"k,Z"k_xand Z"k+X replaced by -K/t„. That is, „„+1

a:

Zr > -7+

m(m + 1)/ä"\\ t„ II Aim --^-1 1

Ä~n - 1

A" i„

m(m + 1) K m- 1 n

A"

K n i„ n + 1

?7I + 1

as required. There are several cases to consider in order to establish the bound (2.15) for Z,71+1 when xk is near f"+1. Now, when Zk +1 > 0, (2.15) is automatically satisfied. We may therefore assume that Zk +1 < 0; that is, that

vît] Jk + X + vl+l 'k-X

2

< V71+

-"*

1

•

But this shows that vk+1 is positive and that v"k+lis not computed from the linear interpolation (1.14). Therefore it must be that k < K(n + 1). Since we already dealt with the case that k ^ K(n + I) - I, v/e may therefore assume that k = K(n + 1). Thus wk+ 1 and wk+¡ satisfy (2.6) and (2.11) respectively. These equations may be rewritten

»k+l = "í+i - c*xZ"k

and ,71 + 1 _

,.,n

+ (a + b)AxZ"k-(a - b)AxZ"k_x, where a, b, and c are as in (2.7), (2.8), and (2.12). Subtracting and dividing by Ax, we thus obtain

(2.18) Zr* = (l-a-bc)Z"k+(a - b)Z"k_x. We checked in the proof of Lemma 2.1 that a - \b\> 0. Using (2.9), (2.10), and (2.13),we have that l-a-b-c>\-

mß(M + e) -

= 1 - 2mß M + £ +

m(m + 1) ßYrjAx -mß(M+< 2 m —1 m+ 3

4(w-

1)

Y0Ax

which is nonnegative by [A4].

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+ £±)

AN INTERFACE TRACKING ALGORITHM

473

Finally, using (2.9), (2.10), (2.13), and (2.14), we have that the sum of the coefficients in (2.18) is l-2b-c^l

+

m(m +H). 1)

mßi YpAx \ —ßy0ax- -y^.-—-J.

m

Using the definition of K, this bound may be rewritten as

1+

/?Y0Ax

K

1+

1 3(w + 1)

mße

Using condition [A3],we then find that 2)8y0Ax

1 - lb - c < 1 -

i

a:

2Yo Ai A" Ax

On the other hand, we have from (2.16) that tn+x > A~Ax/2y0, so that

1 «+ 1^ l/(« + 1) = K n 7-7T+1 > t„ n +

Therefore l-26-c -Cq

Proof. We showed in the proof of Lemma 2.2 that, in all cases, v4i;¡J+1/Ax2is an increasing function of Av"k/Ax2and Av"k±l/Ax2. The bounds (2.19) and (2.21) then

follow easily from (2.17)and (2.18). D Next, we obtain a bound for the discrete time derivative near the interface. Lemma 2.4. There is a constant C such that „71+ 1

- VI

At

0 and k > K(n + 1).

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474

E. DI BENEDETTO AND DAVID HOFF

Proof. We use the symbol 0( ■) to denote dependence on mesh parameters. Let k> K= K(n + 1) and let s'„ = ¿" - xK. Then from (1.14) we have that

Ai

r

Ai = 0(1)

r

^v,+1

■>7I+1

4

„71+ 1

71+1

IK Ai

Ai

- xt

r-**

^71 + 1

JK

^71

The first term on the right can be estimated by using the difference equation, (1.16): „71 + 1

- VI

m(v"K+e)-

At

v K

'

Ax

- + 0(1) = 0(1)

since v\ + e = O(Ax). And the second term on the right is

[r + o(Ai)-xj,;-k V

'

+ o(Ai)](r-x,)

=

s'ns„+At

'

Lemmas 2.2 and 2.4, together with the difference equation (1.11), now imply the following bound for the discrete time derivative. Corollary

2.5. There is a constant C such that „71+1

(2.22)

1+f

Ai

holdsfor all k and all n > 0. In the next lemma we use the one-sided bounds (2.15) and (2.22) to derive L1 estimates for Av\/Ax2 and (vnk+l- v\)/At. Lemma 2.6. (a) For a given T > 0 there is a constant C such that Av"k k

Ax1

„71+ 1

Ax,

£

Ax < C 1 +

Ai

k

holdsfor t„ < T. (b) If the initial function v0 satisfies the hypothesis (2.20), then Av"k k

Ax2

„71+1

Ax, Z

Ax < C

Ai

k

(c) And ifv0 satisfies (2.20) and is concave, then Av"k

„71+1

Ax2 Proof. From Lemma 2.2 we have that Av"k Ax1

Ai Av"k

« C.

2A

< —T + —■

Ax2

t„

We multiply by Ax and sum over k. Since v"kis zero outside an interval of length C(l + i„), we obtain that Av"k Ax t be given. Let Q be the rectangle Q-[xko-p,xko

+ p]x[tHo,tni],

where p is a multiple of Ax to be chosen later. Define the quantities

(2.23)

H=

max \v"ko - oft],

7t0 0 the net {vx(■, i)) ¿sprecompact in Ll[a, b].

Proof. The proof consists of estimating the L1 difference between vx(-,t) and its spatial translates. Given i > 0, choose n so that t„ < t < t„+v Then when h is

sufficiently small, 0 < i - Ai < í„. If we take p = /Ax where / is a positive integer, then it is easy to see that (3.5)

f\vhx(x + p,t) - vhx(x,t)\dx ^

c(EK"+/ - 0. The conclusion of the lemma now follows from [1]. G Lemma 3.1 thus shows that ( vx } has strong L1 limit points. In the next lemma, we prove that these limit points can be identified as the derivatives of limit points of {

— W.)-^)iM,-if^a^}.

m- 1

m

r."

".J'

(™ + l)'/v /'

Now (3.13) follows by letting h -* 0 and using the uniform convergence of vh and f*.

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482

E. DI BENEDETTO AND DAVID HOFF

If dl*(t)/dt > 0, then there is a positive number p such that f*(i + tj) > f*(0 pTjfor small tj. For such tj, then, (3.13) shows that

/

1

*\{Ç(,)-i,s)*>Z

m

-pr¡

Ô2tj

(m + l)i

Dividing by tj and letting tj -» 0, we thus obtain that m

v(tf(t)-8,t)> if 5 < 80 = (m + l)ip/4.

1 ôp m 2

m

(m + l)í

m

la

4

D

Proof of (3.9). Let {f/1} denote any subsequence converging to a curve f* uniformly in [0, T], We shall show that f*(i) = fr(i) for every T. First observe that, since vh(x, t) = 0 for x > ?*(/), e(*> i) must be 0 for x > tf(t); thus fr(i) « fr*(i). Now suppose

that fr < J* on (i, í+tj)

with fr(i) = f*(i).

Then since fr is

increasing, there must be a time i g (í, í +tj) at which dl*/dt exists and is positive. But then (3.14) shows that v(l*(t) - 8, t) is positive for small 8. However, this implies that fr(i) ^ f*(i), which is a contradiction. Therefore there is no maximal time t for which f*(i) = £.(i) for 0 < i < i. Since ff and f* agree at i = 0, they agree for all i. Similar arguments hold for f*(f ). D

4. Error bounds for the approximate solution and interface curves. In this section we prove the following theorem. Theorem 4.1. Fix T > 0. Then there is a constant C such that, for 0 =§ i < T,

(4.1)

H«r*(-,f)-»(■•')

R < Cmin

Ax>gAx|p^i/2

+ ^-

and

(4.2)

\S"(t) - £(0l < CtlAm+1)(Axa\\ogAx\)l/(2(/> + 3))

where f is either f, or fr. //ere

1, l/(w

1).

1 < m < 2, 2 < m,

an
R-

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dxdt.

AN INTERFACE TRACKING ALGORITHM

483

Thus v is a weak solution of (1.6) if and only if J(v, , i,, i2) = 0 for all and all intervals (i,, i2). The weak truncation error associated with the approximate solution vh is then the functional J(vh + e, ■,tx, i2). We have the following estimate for J. Lemma 4.2. Let f satisfy (1.4) and assume that f, andfx are in L°°(R X [0, T]) with

f(x, T) = Oforallx. Thenif. + £)1/(m-1).

In the following lemma we exploit the above estimate for J to obtain a bound for uh - uinLm+1.

Lemma 4.3. There is a constant C such that

(4.5) fTfh\uh-u\m+ldxdt+ f\fT((uh)m-um)xdt]\ dx Jci J„

JaR|/0Jci

c[L(u,uh)

Axa\logAx\+(Ax)mAm~l)\,

where

L(u, u") = \\(uT - OUrx^.t-, + 1 (T((»h)m - W)xd, oo,RX[0, T]

Proof. Let and / be as in Lemma 4.2. Then after integrating by parts in (4.3) and substituting, we obtain

(4.6)

J(vh + e,*,8,T) = (m - 1) f Tf [u^f+ (uh)mxfx] dxdt = (m-l)

-fuh(x,8)f(x,8)dx-

fTf{uhf,-(uh)mxfx)dxdt

We shall replace 8 in these integrals by 0. The resulting error in the second integral on the right will be no more than C|||/|||5. To estimate the change in the first integral, we use the fact that uh is Holder continuous in i, which may be established as follows. When m g (1,2 ] we have from Lemma 2.7 that |ii*(jc, i2) - uh(x, tx)\ < C\vh(x, t2) - v»(x, tx)\ < C\t2 - tx\y'2.

And when m > 2, \uh(x,

t2) - uh(x,

tx)\-1

< \uh(x,

t2)m-1

- uh(x,

tx)m-l\

= \vh(x,t2)-vh(x,tx)\^C\t2-tx\^2.

Thus uh is Holder continuous in i with exponent a/2, and

\f [uh(x,8)f(x,8) \JR

- uh(x,0)f(x,0)] dx

a/2 < cill/HI«

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484

E. DI BENEDETTO AND DAVID HOFF

We thus obtain from (4.6) and Lemma 4.2 that

(uh(x,0)f(x,0)dx+

fTf [uhf ~(uh)xfx] dxdt ^ C\\\f\\\(Axa\log8\ + 8a/2).

JR

J0 JR

Now subtract from this the weak form of the original equation for u, (1.3), and take 5 = Ax2. The result is that

(4.7)

fTf[(uh

J0 JR

- u)f,-{(uh)m

- um)xfx] dxdt < C|||/|||Ax"|logAx|.

We have used here the fact that ||M(.,0)-M"(-,0)||oc,R 2, we have |(d* + e) - o\ = \(uh)m~x

- um-l\ < (m - l)M(r"-2)Am~1)\uh

- u\.

And if 1 < m < 2, then \(vh + e) - v\lAm-l)

< ](vh + e)1Am~1] - vl^m-l)\

Since £ = O(Ax), we have in either case that \vh - v\p ^ C(\uh - u\m + 1 + Ax?).

The conclusion then follows from Lemma 4.3.

D

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= \uh - u\.

AN INTERFACE TRACKING ALGORITHM

485

In order to deduce the L°° bound for v - vh from the above Lp bound, we shall require the following interpolation inequality. Lemma 4.5. Let z be a continuous function on ST = R X [0, T] with support z(-, t) ç [a, b]for t g [0, T]. Assume also that (a) zx g L"°(ST), and (b)z,(-, t) G L\R) with \\z,(-, i)||liR « C0/t. Let p g [ 1, oo) and I > 0 be given. Then there is a constant C independent of I such

that the inequality

II'O.OIU«^^WMW holdsfor 0 < t < T - I. Here l/(/> + 3)

*0O-IWl5fc+1)[ sup \\z(-,s)\\2pp^p^+C0\\zx\\^syp^ .tzíssíT

Proof. For t > 0 we have

(4.8)

\z(x, t)\p+1 ^(p

+ l)f\z(x, JR

t)\p\zx(x, t)\dx

< c\\zx\\„%Qy(ty t wherey(t) = \\z(-, t)\\ R. Differentiating the definition of y(t), we obtain

(4-9)

y(')

p-X

dy_ dt < C\\z(-,OW^WzX-,1%*.

Combining (4.8) and (4.9), we thus obtain

dy

yUY'1f < cXO^'-^-'Vllíf^'^^í-./)!!!„ so that

(4.10)

y(t) \(p-DAp + U

dt Z»A'+V\\z,(-,t)\\1M.

Now let x(i) be the cut-off function

i, x(0

0 1 and use (4.10) to estimate the second term in the above integrand. The result is that y(t)>

,/>(/>+3)/(;>+l)

^1c fTy(tYdt\ sup \y(s) 2p/(p

+ X)

\zxe:si)Ap+1)\\zA-,s)\u,

+

Using hypothesis (b), this may be rewritten as

(4.11) ' y(t)

p(p + 3)Ap+ l)

Çu.up

sup iiz(.,i)n2^+l)

+ Coii^ii^;)/«^1)

Now by (4.8) we have that

(4.12)

r(-»')l oo.R

«

pAp + V c\\zx\\X%+Mt)

The result then follows by substituting (4.11) into (4.12). D We can now prove the first half of Theorem 4.1 by applying the interpolation inequality of Lemma 4.5 to the Lp error bound in Lemma 4.4 as follows. Proof of (4.1). It is clear that vh - v satisfies the first hypothesis of Lemma 4.5. In addition, hypothesis (b) is satisfied because of (3.3) and the results of [5]. (Alternatively, we can observe that the conclusion of Lemma 4.5 remains valid for limits of functions which satisfy hypotheses (a) and (b) uniformly. And v is such a function by the results of §3.) Replacing T - I and T by T and T + 1 respectively, we therefore obtain that \vh(-,t)-v(-,t)\

C

Il h

,!/(/>+ 3) \]V -

oo ,R

||M/> + 3)

V\\p,S'T

,

where S'T = R X [0, T + 1] and p g [ 1, oo). But it we choose p as in the statement of Theorem 4.1, then Lemma 4.4 shows that

vh-v\\p,s,T*iCAx°\logAx\

Therefore \vh(-, t) - v(-, t)\\x„

^ C

/Ax^log_Ax|\1/(í,+3)

On the other hand, the Holder continuity of vh and v in time shows that \\vh(-,t)-v(-,t)\\xM^C(t^2

+ Ax)

for any i. The estimate (4.1) follows from these last two inequalities. D Next, we deduce the bound (4.2) for the error in the approximate interfaces from the above bound (4.1) for the error in vh. Again we drop the subscripts and denote f *

andfrbyfandf. Proof of (4.2). First we refine the result in Lemma 3.4. Divide (3.13) by tj and let tj -» 0. The result is that

(4.13)

v($(t)-y,t)>

m —1

JÍ(')

(m + l)í

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a.e.

AN INTERFACE TRACKING ALGORITHM

487

A similar bound holds for vh: taking q = 1 in (3.16), we have that m —1 VK(n)-j

(/Ax)2

(/Ax)f(iJ

>

(m + l)t„

Since vh is Lipschitz in x and Holder continuous in i, and since fh is piecewise linear, we can conclude that (4.14)

m-

vH{Sh(t)-y,t)>

1

ïh(t)y-

m

(m + l)t

- CAx a.e.

Now let [vx, v2) = {v, vh) and let ^ and f2 be the corresponding interface curves. Assume that, at time i, 0 < f2(i) - f,(i) = y. Then we have, using either

(4.13)or (4.14),that

(4.15) ||t;A(-,i) - v(-, OIL.« > \(v2- Vi)(Si(t), t)\ = v2(

3s

m- 1 m w —1 m

= v2{S2(t)-y,t) yl

fc(O)'(m + l)i (fe fl)(?2 fl)

-CAx

(*2 - fi)'

(m + 1), - CAx.

Now let

*') - [fc(0- fi