In M. Fey and R. Jeltsch, editors, "Hyperbolic problems: theory, numerics, applications", Int. Series of Numerical Mathematics, Vol. 129 (1999), 285-294, Birkhäuser Verlag
Degenerate Convection-Diusion Equations
and Implicit Monotone Dierence Schemes
Steinar Evje, Kenneth Hvistendahl Karlsen
Abstract. We analyse implicit monotone nite dierence schemes for nonlinear, possibly strongly degenerate, convection-diusion equations in one spatial
dimension. Since we allow strong degeneracy, solutions can be discontinuous
and are in general not uniquely determined by their data. We thus choose to
work with weak solutions that belong to the
class and, in addition, satisfy
an entropy condition. The dierence schemes are shown to converge to the
unique
entropy weak solution of the problem. This paper complements
our previous work 8] on explicit monotone schemes.
BV
BV
1. Degenerate Convection-Diusion Equations
We are interested in nite dierence schemes for nonlinear, possibly strongly degenerate, convection-diusion problems of the form
@t u + @x f(u) = @x(k(u)@x u) u(x 0) = u0 (x)
(1)
where (x t) 2 QT = R (0 T ) and u0 f k are given, suciently smooth functions.
For later use, we need a conservative-form version of (1),
Z u
2
@t u + @x f(u) = @x K(u) K(u) = k() d:
(2)
0
By the term `strongly degenerate' we mean that there are two numbers and such that k(u) = 0 for all u 2 ]. Hence, the class of equations under
consideration is very large and contains, to mention a few, the heat equation,
the porous medium equation, the two-phase ow equation and conservation laws.
Strongly degenerate equations will in general possess discontinuous { shock wave {
solutions. Furthermore, discontinuous weak solutions are not uniquely determined
by their data. In fact, an entropy condition is needed to single out the physically
relevant weak solution of the problem. We call a bounded measurable function
u(x t) an entropy weak solution if
@tju ; cj + @x sgn(u ; c)(f(u) ; f(c)) + @x2 jK(u) ; K(c)j 0 (weakly):
It is not dicult to construct an entropy weak solution of (1), even in several
space dimensions, see 12]. To the authors knowledge, the main open question
seems to be the uniqueness of such solutions, even in one space dimension. This
2
S. Evje, K. H. Karlsen
has motivated us to seek solutions in the (signicantly) smaller class containing the
BV entropy weak solutions. Before introducing this notion of a solution, we recall
that uniqueness of weak solutions for the purely parabolic case (no convection
term) in the class of bounded integrable functions has been proved by Brezis and
Crandall 1], and that uniqueness of entropy weak solutions for hyperbolic problems
(no diusion term) is a classical result due to Kruzkov 9].
Denition 1.1. A bounded measurable function u(x t) is said to be a BV entropy
weak solution of the initial value problem (1) if
1
(a) u(x t) 2 BV (QT ) and K(u) 2 C 1 2 (QT ).
(b) For all non-negative 2 C01(QT ) and any c 2 R,
ZZ QT
ju ; cj@t + sgn(u ; k)(f(u) ; f(c) ; @x K(u))@x dt dx
Z
+ ju0 ; cj(x 0) dx 0:
R
(3)
What makes this denition interesting is that uniqueness of BV entropy
solutions follows from the work of Wu and Yin 13]. We mention here that the
jump conditions proposed by Volpert and Hudjaev 12] are in general not correct,
and thus the uniqueness proof presented there is incomplete, see 13] for more
details. One should note that it is rather restrictive to require BV (in space and
time) regularity of solutions to parabolic equations. In particular, for @t u to be a
(locally) nite measure on QT , @x u and @x2 K(u) need to be (locally) nite measures
on QT . This fact immediately implies that the diusion term K(u( t)) needs to
possess a certain amount of smoothness, which in turn indicates that it should be
harder (than for conservation laws) to establish the analog of the Crandall and
Majda theory 5] for strongly degenerate parabolic equations. The convergence of
a scheme to the desired BV solution (in the sense of Denition 1.1) is not an
immediate consequence of a BV estimate (in space), as is the case with hyperbolic
conservation laws.
It is possible to use the theory developed in 13] to treat strongly degenerate
boundary value problems as well, see Burger and Wendland 2] (and the references
therein). In 2], the authors analyse their recently proposed model for the settling
and consolidation of a occulated suspension under the inuence of gravity. We
refer to Concha and Burger 4] for an overview of the activity centring around this
and related sedimentation models. Cockburn and Gripenberg 3] have recently
shown that solutions of degenerate equations also depend continuously on the
nonlinear uxes of the problem, see 3, 8] for more details.
It is important to realize that solutions of strongly degenerate parabolic equations (1) in general have a more complex structure than solutions of conservation
laws. The following example demonstrates this. Let f(u) = u2 (referred to as the
Burgers ux), and let k(u) = 0 for u 2 0 0:5], 2:5u ; 1:25 for u 2 (0:5 0:6) and
Degenerate Convection-Diusion Equations
0.8
0.8
0.6
0.6
u−axis
1
u−axis
1
3
0.4
0.4
0.2
0.2
0
−1
0
−0.8
−0.6
−0.4
−0.2
0
x−axis
0.2
0.4
0.6
0.8
1
−1
−0.8
−0.6
−0.4
−0.2
0
x−axis
0.2
0.4
0.6
0.8
1
Figure 1. Left: The solution (solid) of the inviscid Burgers equation.
Right: The solution (solid) of Burgers' equation with a strongly degenerate diusion term and the corresponding diusion function K (u( t))
(dashed). The initial function is shown as dotted.
0:25 for u 2 0:6 1:0]. Note that k(u) is continuous and degenerates on the interval 0 0:5]. In Fig. 1 we have plotted the initial function, the solution of the
corresponding conservation law, i.e., k 0 in (1), and the solution of (1) at time
T = 0:15. An interesting observation is that the solution of (1) has a `new' increasing jump, despite of the fact that f is convex. Thus the solution is not bounded
in the so-called Lip+ norm, as opposed to the solution of the conservation law.
Moreover, while the speed of a jump in the conservation law solution is determined
solely by f, the speed of a jump in the solution of (1) is in general determined
by the jumps in both f(u) and @x K(u), see 13, 8] for precise statements of these
jump conditions. Here it suces to that say the speed s of a jump is
f(ur ) ; f(ul ) ; @x K(u)r ; @x K(u)l
(4)
s=
u r ; ul
where ul and ur denote the usual left and right limits of u( t) respectively. Furthermore, the entropy condition requires that the following inequalities hold for
all c 2 int(ul ur ):
f(ur ) ; f(c) ; @x K(u)r s f(ul ) ; f(c) ; @x K(u)l :
(5)
ur ; c
ul ; c
See Fig. 2 for an illustration of (5). Finally, we mention here that the techniques
developed by Kruzkov (stability) and Kuznetsov (error estimates) for rst order
equations are not straight on adaptable to second order problems such as (1).
In this paper we are interested in implicit monotone dierence schemes for
(1). A convergence analysis of explicit monotone schemes was given recently in
8]. In view of the classical monotone theory for conservation laws 5], the main
diculty in obtaining a convergence theory for (1) is to show that the (strongly
degenerate) discrete diusion term is Holder continuous (Denition 1.1 (a)), see
Lemmas 2.3 and 2.4 in this paper. To the authors knowledge, there exists no
general nite dierence theory for strongly degenerate parabolic equations, except
4
S. Evje, K. H. Karlsen
f (c)
f (c)
l
K(u)x
f(u r )
f(u l )
K(u) rx
f(u l )
f(u r )
ul
ur
ur
c
ul
c
Figure 2. Geometric interpretation of the entropy condition (5) for
the solution shown in Fig. 1 (right). Left (the left jump): Note that
K (u) = 0 K (u) > 0, see Fig. 1 (right). Condition (5) requires that
the graph of f restricted to the interval u u ] lies above or equals the
straight line between (u f (u )) and (u f (u ) ; K (u) ). Right (the
right jump): Note that K (u) < 0 K (u) = 0. Condition (5) requires
that the graph of f restricted to u u ] lies below or equals the straight
line between (u f (u )) and (u f (u ) ; K (u) ).
l
x
r
x
l
l
l
l
x
r
r
r
r
r
l
l
r
r
x
l
r
x
l
x
for 8]. The main purpose of this paper is show that the theory developed in 8]
can be easily extended, using the theory of Crandall and Liggett 6], to implicit
schemes as well. An accurate and ecient operator splitting scheme for (1) has
been proposed and analysed in 7]. However, for this approximation it is in general
impossible to prove that the discrete diusion term is Holder continuous, see 7]
for details. Finally, we are currently looking into the issue of devising higher order
dierence schemes for strongly degenerate parabolic equations. In particular, we
are investigating to what extent the `higher order' theory/schemes developed for
hyperbolic conservation laws can be taken over to strongly degenerate equations.
2. Implicit Monotone Dierence Schemes
We will follow the work 8] closely and refer the reader to it for details not found
here. Introduce the dierence operators D; Uj = Uj ;Uxj;1 and D+ = Uj+1;x Uj . We
then consider the three-point monotone dierence schemes of the form
;
Ujn+1 ; Ujn
n+1 U n+1) ; D+ K(U n+1 ) = 0
+
D
F
(U
(6)
;
j
j +1
j
t
where F (u u) = f(u) and fUj0g is some discretization of u0. In what follows, we
assume, without loss of generality, that u0 has compact support and f(0) = 0. The
Degenerate Convection-Diusion Equations
0.8
0.8
0.8
0.6
0.6
0.6
u−axis
1
u−axis
1
u−axis
1
0.4
0.4
0.4
0.2
0.2
0.2
0
−1
0
−0.8
−0.6
−0.4
−0.2
0
x−axis
0.2
0.4
0.6
0.8
1
5
−1
0
−0.8
−0.6
−0.4
−0.2
0
x−axis
0.2
0.4
0.6
0.8
1
−1
−0.8
−0.6
−0.4
−0.2
0
x−axis
0.2
0.4
0.6
0.8
1
Figure 3. The solutions produced by the conservative scheme (6)
(solid) and the non-conservative scheme (8) (dashed) plotted at the
times T = 0:0625, T = 0:25 and T = 1:0. The initial function is shown
as dotted see the text for a further description of the problem.
assumption of monotonicity in the case of implicit schemes reads
1 k(r ) ; F (r r ) 0 8(r r r ): (7)
1 k(r ) 0
Fu(r1 r2) + x
3
v 1 2
1 2 3
x 3
Note that a sucient condition for (7) to hold is that Fu 0 and Fv 0. Consequently, any monotone numerical ux F for conservation laws will produce a
monotone scheme for (1). To keep the notation simple and making the arguments
more transparent, we consider only three-point schemes in this paper.
The monotone schemes (6) are based on dierencing the conservative-form
equation (2), and not the equation in its original form. One can also devise schemes
based on dierencing (1) directly, yielding, for example, schemes of the form
Ujn+1 ; Ujn
;
n+1 U n+1) ; k(U n+1 )D+ U n+1 = 0
+
D
F
(U
(8)
;
j
j +1
j
j +1=2
t ;
+1
1 U n+1 + U n+1 . Although it is possible to prove that (8) conwhere Ujn+1
=
j +1
=2 2 j
verges to a limit, we have not been able to show that this limit satises an entropy
condition. In fact, we do not believe that (8) will converge to the physically correct solution in the case of strong degeneracy. We now present a simple numerical
~ = 41 u2
example intended to support this view. For this purpose we use uxes f(u)
and k~(u) = 4k(u), where k is the one used above. In Fig. 3 we have plotted the solution produced (using small grid parameters) by (6) and (8) at three dierent times.
+1
The convective numerical ux was the upwind ux F (Ujn+1 Ujn+1
) = f(Ujn+1 ) in
these calculations. Clearly, the non-conservative scheme (8) produces an incorrect
solution. We are currently investigating this phenomenon and will come to back
to it in a separate report.
As an aid in the following analysis we shall view the equation (6) in terms
of an m-accretive operator and an associated contraction solution operator, i.e.,
we shall use the Crandall and Liggett theory 6]. A similar treatment of implicit
dierence schemes for conservation laws has been given earlier by Lucier 11]. If X
6
S. Evje, K. H. Karlsen
is a Banach space, a duality mapping J : X ! X has the properties that for all
x 2 X, kJ(x)kX = kxkX and J(x)(x) = kxk2X . A possibly multi-valued operator
A, dened on some subset D(A) of X, is said to be accretive if for every pair of
elements (x A(x)) and (y A(y)) in the graph of A, and for every duality mapping
J on X, J(x ; y)(A(x) ; A(y)) 0. If, in addition, for all positive , I + A is a
surjection, then A is m-accrective. For a xed n, let us now rewrite the dierence
equation (6) as (suppressing the x dependence)
Ujn+1 + tA(U n+1! j) = Ujn
(9)
;
+1
where A(U! j) = D; F(Ujn+1 Ujn+1
);D+ K(Uj ) . Let (" d) be a measure space.
1
Recall that, since the dual
of L (") is L1 ("), any duality mapping J in L1 (") is
R
^
of the form J(u)(v) = J(u)(x)v(x)
d, where
8
>
<
1
if u(x) > 0
^
(10)
J(u)(x) = kukL1 () >;1 if u(x) < 0
:
(x) if u(x) = 0
where (x) is any measurable function with j(x)j 1 for almost every x 2 ".
We shall rely heavily on the following well-known results (see e.g. 6, 11]):
Theorem 2.1. Let (" d) be a measure space. Suppose that the nonlinear and possibly multi-valued operator A : L1 (") ! L1 (") is m-accretive. Then for any
> 0 and any u 2 L1(") the
R equation T (u) + A(T (u)) = u has a unique
solution T (u). If A satises A(u) d = 0 and commutes with translations,
then the solution
operatorR T : L1 (") ! L1 (") possesses the following propR
erties: (a) T (u) d = u d, (b) kT (u) ; T (v)kL1 () ku ; vkL1 (), (c)
jT (u)jBV () jujBV (), (d) u v ) T (u) T (v), (e) kT (u)kL1() kukL1() .
The following lemma deals with the question of existence, uniqueness and
properties of the solution of the (nonlinear) system (6).
satisfying
Lemma 2.2. If (7) is satised,
then
for
any
U
there
exists
a
unique
U
;
the equation Uj;tUj + D; F(Uj Uj+1) ; D+ K(Uj ) = 0 8j 2 Z. Furthermore,
j Vj 8j 2 Z, (b) U L1 ( ) we
have
the
properties:
(a)
U
V
8
j
2
Z
)
U
j
j
U U ; V U U ; V ,
(c)
,
(d)
1
1
1
L ( )
L( )
L( )
BV ( ) U BV ( ).
Z
Z
Z
Z
Z
Z
Proof. We will rst show that the operator A is accretive. As a rst step to achieve
this goal, we observe that
X
;
sgn(Uj ; Vj ) A(U! j) ; A(V ! j)
j2
Z
;
X
cWj ; A(U! j) ; A(V ! j) + c
Z
j2
;
(11)
X
Z
j2
Wj Degenerate Convection-Diusion Equations
7
where Wj denotes Uj ; Vj and c = c(x) > 0 is a number chosen so that c 2
1
x (Fu (r1 r2) ; Fv (r3 r1)) + x2 k(r4) 8(r1 r2 r3 r4). Next, we write
1 ;(F ( U )W + F (V )W )
A(U! j) ; A(V ! j) = x
u j j +1 j
v j j +1 j +1
; (Fu(j ;1 Uj )Wj ;1 + Fv (Vj ;1 j )Wj )
; x1 2 ;k(j ;1 )Wj ;1 ; 2k(j )Wj + k(j +1 )Wj +1 for some numbers j j between Uj and Vj . Inserting this into (11) yields
X
;
sgn(Uj ; Vj ) A(U! j) ; A(V ! j)
Z
j2
1 F ( U ) + 1 k( )iW u j ;1 j
j ;1
j ;1
2
x
x
j2
j2
Xh
1 (F ( U ) ; F (V )) ; 2 k( )iW ; c ; x
u j j +1
v j ;1 j
j
x2 j
j2
Xh
1 F (V )iW 0
; x1 2 k(j +1 ) ; x
(12)
v j j +1
j +1
j2
which shows that A is accretive. Observe that A is Lipschitz continuous, which
implies that A is not only accretive but also m-accretive. We can now invoke
Theorem 2.1 to conclude the existence of a unique solution operator
S of (6), i.e.,
P
Uj = S (U! j), which proves the rst part of the lemma. Since j 2 A(U! j) = 0
and A commutes with translations, the second part of the lemma follows.
The next lemma plays a key role in our analysis and has no counterpart in
the theory of monotone dierence approximations for conservation laws.
Lemma 2.3. If (7) is satised, we have
F (U 0 U 0 ) ; D K(U 0 )
F (U n+1 U n+1) ; D+ K(U n+1)
+
1
j
j
+1
j
j +1
j
j
L ( )
L1 ( )
(13)
n
+1
n
+1
n
+1
0
0
0
F (U
j Uj +1 ) ; D+ K(Uj )BV ( ) F (Uj Uj +1 ) ; D+ K(Uj )BV ( ):
c
X
Wj ;
Z
Xh
Z
Z
Z
Z
Z
Z
Z
Z
n
n;1 P
Proof. From the equation (6), it follows that Vjn = x ji=;1 Ui ;Uti
satises
;
+1
Vjn+1 = ; F(Ujn+1 Ujn+1
) ; D+ K(Ujn ) :
(14)
Next, we derive an equation for the quantity fVjng. For this purpose consider the
dierence equation (6) evaluated at ix and subtract the corresponding equation
at time t = nt. Multiplying the resulting equation by x and then summing
over i = ;1 : : : j, yields the following equation
;
; n+1
;
+1
Vj ; Vjn + F (Ujn+1 Ujn+1
) ; F (Ujn Ujn+1) ; D+ K(Ujn+1 ) ; K(Ujn) = 0:
8
S. Evje, K. H. Karlsen
After observing that Uj ;t Uj = D; Vjn+1 , we can write
+1
F (Ujn+1 Ujn+1
) ; F (Ujn Ujn+1 ) = tanuj D; Vjn+1 + tavj D; Vjn+1+1 (15)
+1
where anuj = Fu(nj Ujn+1
) avj = Fv (Ujn ~ nj+1) and nj ~ nj are some numbers
between Ujn and Ujn+1 . Similarly, we can write
K(Ujn+1 ) ; K(Ujn ) = tbj D; Vjn+1
where bnj = k(jn ) and j is a number between Ujn and Ujn. Summing up, the
sequence fVjn g satises the following linear system of equations
Vjn+1 ; Vjn ; n
n+1 + an D; V n+1 = D+ ;bn D; V n+1 :
+
a
D
V
(16)
;
uj
vj
j
j
j +1
j
t
Observe that this system can be written as
n n+1 + C nV n+1 = V n Anj Vjn;+1
(17)
j j +1
j
1 + Bj Vj
h
i
h
;
;
i
where Anj = ; xt anuj + xt2 bnj Bjn = 1 + xt anuj ; anvj + xt2 bnj + bnj+1
h
i
t n
t
n
n
and Cj = ; x2 bj +1 ; x avj . Because of (7), the linear system (17) is strictly
diagonal
there exists a unique solution V n+1 of (17) satisfying
dominant.
Hence,
n
V n+1
1 V L1( ). An argument similar to the one in 8] will also reveal
L ( )
that V n+1 BV ( ) V n BV ( ). The lemma now follows by induction.
A direct consequence of (13) is that the approximations are L1 Lipschitz
continuous in the time variable, and thus in BV in space and time.
Lemma 2.4. If (7) is satised, we have
t
F(U 0 U 0 ) ; D+ K(U 0 )
U m ; U n 1
j
j
+1
j
BV ( ) x jm ; nj:
L( )
Proof. The result follows directly from (6) and (13), see also 8].
n+1
Z
Z
n
Z
Z
Z
Z
Lemma 2.5. If (7) is satised, we have
p
;
K(U m ) ; K(U n ) = O(1) j(i ; j)xj + j(m ; n)tj :
i
j
K(U m ) ; K(U n ) Q + Q , where Q = K(U m ) ; K(U m )
Proof. First,
1
2
1
i
j
i
j
and Q2 = K(Ujm ) ; K(Ujn)j. In view of (13), D+ K(U m )L1 ( ) = O(1) and
thus Q1 = O(1)j(i ; j)xj. Kruzkov 10] has developed a technique for deriving
a modulus of continuity in time from a known modulus of continuity in space of
certain parabolic equations. To estimate Q2 we apply a discrete version of this
technique to the parabolic dierence equation (16). To this end, let (x) be a test
function, put j = (jx) and let m < n. Using the dierence equation (16) and
summation by parts (on the right-hand side of (16)), we easily nd that
X ;
;
j Vjm ; Vjn = O(1) kkL1( ) + k0kL1 ( ) t(m ; n)
x
Z
j2
Z
R
R
Degenerate Convection-Diusion Equations
9
Z
since, for all l, aluj , alvj and V l+1 BV ( ) are uniformly bounded quantities. From
this weak estimate and the BV regularity of V n , it now follows that
X
p
x Vjm ; Vjn = O(1) jm ; njt
Z
j2
see 8] for details. On the other hand, from (14) and Lemma 2.4, we also have
X
X
x Vjm ; Vjn = O(1)(m ; n)t + x D+ K(Ujm ) ; D+ K(Ujn ) :
Z
j2
P
Z
Z
j2
p
We thus conclude that x j 2 D+ K(Ujm ) ; D+ K(Ujn ) = O(1) (m ; n)t.
From this the desired Holder estimate in time follows, since
X
p
Q2 = K(Ujm ) ; K(Ujn ) x D+ K(Uim ) ; D+ K(Uin ) = O(1) (m ; n)t:
Z
i2
This concludes the proof of the lemma.
Lemma 2.6. If (7) is satised, then the following cell entropy inequality holds
U n+1 ; c ; U n ; c
;
j
j
n+1 _ c U n+1 _ c) ; F (U n+1 ^ c U n+1 ^ c)
+
D
F(U
;
j
j +1
j
j +1
t
n
+1
; D+ K(Uj ) ; K(c) 0:
Proof. The proof is similar to the one presented in 8], see also 5, 11].
Let u (where = (x t)) be the interpolate of degree one associated
with the discrete data points fUjng, see 8]. Note that u is continuous everywhere
and dierentiable almost everywhere. In view of Lemmas 2.2 - 2.4, we conclude
that there is a constant C = C(T) > 0 such that
ku kL1 (QT ) + ju jBV (QT ) C
p
p ;
jK(u (y )) ; K(u (x t))j C jy ; xj + j ; tj + x + t :
for all (x t) (y ) 2 R 0 T]. Consequently, since BV is compactly imbedded
into L1 on compacta, there is a subsequence of discretization parameters and a
function u 2 L1 (QT ) \ BV (QT ) such that uj ! u a.e. in QT . Furthermore, via
the Ascoli-Arszela theorem, K(uj ) ! K(u) uniformly on compact sets K QT ,
and K(u) 2 C 1 12 (QT ). Repeating the proof of the Lax-Wendro theorem (with
Lemma 2.6 in mind), it follows that u satises the entropy condition (3).
Summing up, we have proven the following main theorem:
Theorem 2.7. The sequence fug, which is built from the implicit monotone difference schemes (6), converges a.e. to the BV entropy weak solution of (1).
Acknowledgement
We thank Raimund Burger, Bernardo Cockburn, Magne S. Espedal and Wolfgang
L. Wendland for interesting discussions. This work has been supported by VISTA,
a research cooperation between the Norwegian Academy of Science and Letters
and Den norske stats oljeselskap a.s. (Statoil).
10
S. Evje, K. H. Karlsen
References
1] H. Brezis and M. G. Crandall, Uniqueness of solutions of the initial value problem
for u ; '(u) = 0, J. Math. Pures et Appl., 58 (1979), 153{163.
2] R. Burger and W. L. Wendland, Existence, uniqueness and stability of generalized
solutions of an initial-boundary value problem for a degenerating parabolic equation,
J. Math. Anal. Appl., 218 (1998), 207{239.
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Department of Mathematics, University of Bergen,
Johs. Bruns. gt. 12, N-5008, Bergen, Norway
E-mail address : [email protected], [email protected]