A Convergent Finite Difference Scheme for the Variational Heat Equation

1.2 MB·

by  G. M. Coclite

A CONVERGENT FINITE DIFFERENCE SCHEME FOR
THE VARIATIONAL HEAT EQUATION
G. M. COCLITE, J. RIDDER, AND N. H. RISEBRO
7
1
0
2
Abstract. Thevariationalheatequationisanonlinear,parabolicequa-
n
tionnotindivergenceformthatarisesasamodelforthedynamicsofthe
a
J director field in a nematic liquid crystal. We present a finite difference
5 scheme for a transformed, possibly degenerate version of this equation
and prove that a subsequence of the numerical solutions converges to a

] weak solution. This result is supplemented by numerical examples that
A
show that weak solutions are not unique and give some intuition about
N how to obtain the physically relevant solution.
.
h
1. Introduction
t
a
m In this paper we investigate the Cauchy problem
[ (cid:40)
(1) ut = c(u)(c(u)ux)x, x ∈ Ω, t > 0
1
u(x,0) = u (x), x ∈ Ω,
v 0
5 where Ω = R or Ω = [0,1] with periodic boundary conditions. We assume
6
that
2
1 (H.1) c ∈ C2(R), c ≥ 0, |{ξ|c(ξ) = 0}| < ∞, and, w.l.o.g., c ≤ 1,
0
(H.2) u ∈ W1,1(Ω)∩W1,∞(Ω), u ∈ BV(Ω).
. 0 0,x
1
We call (1) the “variational heat equation”, because it can be derived from
0
7 a variational principle, similar to the variational wave equation [13, 21, 11,
1 4, 5], see (3) below.
:
v The variational heat equation arises in the context of the continuum the-
i ory for nematic liquid crystals as a model for the dynamics of the director
X
field. Liquid crystals are materials in a state of matter between the solid
r
a and the liquid state. In the case of uniaxial nematic liquid crystals, this
means that the elongated molecules can move freely like in a fluid, but tend
to align along the same direction like in a crystal. On a macroscopic scale
such a state can be described by two vector fields, the velocity field and
the so-called director field, which are governed by the Ericksen-Leslie equa-
tions [22, 24, 8, 16, 17, 18, 9]. The director field is a unit vector field that
gives the average direction of the molecules at each point.
Date: January 6, 2017.
G. M. Coclite is member of the Gruppo Nazionale per l’Analisi Matematica, la Prob-
abilita` e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica
(INdAM).
1
2 G.M.COCLITE,J.RIDDER,ANDN.H.RISEBRO
To arrive at equation (1), we assume the simplified setting of a uniaxial
nematic with no flow and a director field n that lies in the x − y plane
and varies only in x-direction. Then the director can be described by an
angle u as n = (cos(u),sin(u),0). The Oseen-Frank energy, which models
the tendency of the director to align along the same direction everywhere,
reduces to ˆ
E = (c(u))2(u )2dx,
x
where
(cid:112)
(2) c(u) = k cos(u)2+k sin(u)2,
1 2
and k and k are the Oseen-Frank elastic constants corresponding to bend
1 2
andsplaydeformations[22,8,24,19,10]. Inaddition,thedirectorissubject
to the dissipation ˆ
D = κ (u )2dx,
x
where κ is the rotational viscosity coefficient. Together, a variational prin-
ciple applied to the energy law
d
E = D,
dt
and scaling κ = 1 gives (1), see [2, 1].
A similar model is the variational wave equation [13, 21],
(3) u = c(u)(c(u)u ) ,
tt x x
which is derived in the same way from the Oseen-Frank energy, but ne-
glecting dissipation and instead including inertia in the form of the kinetic
energy ˆ
σ(u )2dx,
x
where σ is the rotational inertia of the director, scaled to 1 in (3). Typical
valuesfortheelasticconstantsk andk in(2)areoforder10−11–10−12, the
1 2
dissipation κ is of order 10−1–10−3, and the rotational inertia σ is of order
10−13, [22, 25]. On small length scales, the term from the elastic energy
and the dissipation can be of the same order. The inertia term however is
usually dominated by the dissipation [2], therefore (1) is a more suitable
model than (3) in most physical settings.
From a mathematical point of view, if k and k are strictly positive, i.e.,
1 2
c > 0, equation (1) is a nonlinear, uniformly parabolic equation. While (3),
and also the combination of (1) and (3) where both u and u are included,
t tt
does not possess a unique classical solution [11, 4, 5], standard theory of
nonlinear parabolic equations guarantees well-posedness of (1), see [14].
Wearethereforeinterestedinthedegeneratecaseof (1)wherecisallowed
to vanish at some points, i.e., if c is given by (2), in the case that k = 0
1
or k = 0. Solutions of degenerate parabolic equations are not necessarily
2
smooth or unique, therefore new concepts of solutions, e.g., weak solutions,
A CONVERGENT SCHEME FOR THE VARIATIONAL HEAT EQUATION 3
entropy solutions, or viscosity solutions are required. In the case of (1), a
formal calculation shows that there is no maximum principle for u , but
x
for c(u)u (see Section 3). At points where c(u) vanishes, this allows for
x
gradient blow-up.
The goal of this paper is to design a convergent numerical scheme for (1).
The form of the right-hand side and the resulting lack of a gradient bound
suggests that one should transform (1) first.
One possibility to do this is to define
ˆ
u 1
(4) v = k (u) = dξ.
v
c(ξ)
Then (1) becomes
(5) v = (c2(k¯ (v))v ) ,
t v x x
where k¯ is the inverse of k (u). For this equation it is straightforward to
v v
obtain an L2 bound and one can also show uniqueness of weak solutions. If
we assume c > 0, a simple finite difference scheme based on central differ-
ences and averages in space can be shown to converge to a weak solution
using Aubin-Lions lemma, see also [15, 20] for examples in a similar setting.
If however c = 0 for some u, then (4) is not necessarily finite and a bound
on v does not follow directly from the L2 bound.
x
An alternative transformation of (1) is
ˆ
u
(6) w = k (u) = c(ξ)dξ,
w
so w satisfies
(7) w = c2(k¯ (w))w .
t w xx
The transformation k and its inverse k¯ are well-defined for any c ≥ 0 if c
w w
vanishesonlyonsinglepoints. Itisalsopossibletoshowa priori boundsfor
both w and w in L∞ and BV (functions of bounded total variation), see
x
Section 3. However, (7) does not guarantee uniqueness of solutions. Indeed,
Ughi et al. [23, 7, 3] showed that for the special case where c2(k¯ (w)) = w,
w
weak solutions of (7) (defined in a standard way, see Section 2) are not
unique. To choose the physically relevant solution, they define “viscosity
solutions” which are obtained by taking the limit of classical solutions of
the equation with c > 0 or suitable initial data. In the setting of (2),
these viscosity solutions correspond to sending k or k to 0 or choosing
1 2
the solution that corresponds to a solution of (5). Ughi et al.’s concept of
viscosity solutions is not generally the same as Lions’ theory of viscosity
solutions for degenerate parabolic equations [6, 3]. The uniqueness theory
ofthelatterisnotapplicablehere, becausetheright-handsideof (7), or(1),
is not proper.
The scheme that we will present in this paper discretizes (7). Based on
discrete versions of the L∞ and BV bounds on w and w , we use Kol-
x
mogorov’s compactness theorem to show that the numerical approximations
4 G.M.COCLITE,J.RIDDER,ANDN.H.RISEBRO
for both w and w converge strongly in L1(Ω). The strong convergence of
x
the derivative is important, because the weak formulation of (7) includes
nonlinear terms in w . Passing to the limit in the definition of the scheme,
x
we prove that a subsequence of the numerical solutions converges to a weak
solution as ∆x,∆t → 0.
Our numerical experiments confirm the nonunqiueness properties dis-
cussed above. If k = 0 in (2) and the grid is chosen such that c(u (x))
1 0
is positive at every grid point, then the numerical solutions converge to
Ughi et al.’s viscosity solution. This solution is the same as the one ob-
tained by a method based on (5) and as the limit k → 0 of solutions of the
1
w-based scheme for any set of grid points. If however one of the grid points
coincides with a zero of c(u (x)), we get another solution which corresponds
0
to a classical solution of (7), “glued together” at the zeros of c(u (x)) with
0
Dirichlet boundary conditions. Interpreted as solutions of (1), this type of
solutions shows clearly that the gradient is unbounded.
The rest of this paper is structured as follows: In Section 2 we will de-
fine the scheme for (7), introduce the notion of weak solutions, and state
our convergence result. Section 3 contains discrete a priori bounds, which
are based on Harten’s lemma and motivated by formal calculations in the
continuous case. Time continuity is shown in Section 4 and the convergence
proof is carried out in Section 5. In Section 6 we present a series of nu-
merical experiments that confirm the convergence result and highlight the
nonuniqueness properties of (7).
2. A numerical scheme for w and the main result
To be precise, let us restate (7) in the form that will be the basis of our
scheme. Assume that
(H.3) B ∈ C2(R) and 0 ≤ B ≤ 1,
(H.4) w ∈ W1,1(Ω)∩W1,∞(Ω), w ∈ BV(Ω).
0 0,x
Then we want to solve
(cid:40)
w = B(w)w , t > 0,x ∈ Ω,
t xx
(8)
w(x,0) = w (x), x ∈ Ω,
0
on Ω = R or [0,1] with periodic boundary conditions.
Equation (1) can be transformed to (8) by defining w as in (6). If u sat-
0
isfies (H.2), then w will satisfy (H.4), but not vice versa. Similarly, (H.3)
0
follows from (H.1). As an example, if we choose c according to (2) with
k = 0 and k = 1, then k (u) = |sin(u)| and B(w) = 1 − w2, see also
1 2 w
Section 6.
To define the scheme, let Ω be discretized by the equidistant grid points
x = j∆x, j = 0,...,N, and let tn = n∆t denote the time steps. If
j
Ω = [0,1], we set periodic boundary conditions. We will implicitly assume
that all functions are periodically extended outside of the domain, so that
no boundary terms occur.
A CONVERGENT SCHEME FOR THE VARIATIONAL HEAT EQUATION 5
A straightforward discretization of (8) is
(9) D+wn = B(wn+θ)D2wn+θ,
t j j j
where we used the difference quotients
1 1
D a = (a −a ), D a = (a −a ),
+ j j+1 j − j j j−1
∆x ∆x
1
D+an = (an+1−an), D2a = D D a ,
t ∆t j + − j
and the convex combination
wn+θ = θwn+1+(1−θ)wn, where θ ∈ [0,1].
j j j
For θ = 0, the scheme is explicit, for θ = 1, it is fully implicit, and for θ = 1
2
we have the Crank-Nicholson time discretization. In the fully implicit case
of θ = 1, the scheme is unconditionally stable. Otherwise, we require that
the time step ∆t and grid size ∆x satisfy the CFL condition
∆t 1
(10) λ = < .
(∆x)2 2(1−θ)
For the discrete derivatives zn = D wn and yn = D zn, the scheme defined
j + j j − j
by (9) becomes
(11) D+zn = D (B(wn+θ)D zn+θ),
t j + j − j
(12) D+yn = D2(B(wn+θ)yn+θ).
t j j j
We will use these forms below to get a priori bounds on wn.
j
For given initial data w ∈ W2,1∩W1,∞, define the discrete initial data
0 ˆ
(13) w0 = 1 xj+12 w (x)dx.
j ∆x 0
xj−12
To get from the discrete approximations wn back to continuous functions,
j
we use the piecewise linear and piecewise constant interpolations
x −x x−x
(14) w∆t(x,t) = j+1 wn+ jwn ,
∆x j ∆x j+1
for x ∈ [x ,x ), t ∈ [tn,tn+1),
j j+1
(15) w∆t(x,t) = wn,
j
for x ∈ [x ,x ), t ∈ [tn,tn+1),
j−1 j+1
2 2
(16) z∆t(x,t) = w∆t(x,t) = D wn = zn,
x + j j
for x ∈ [x ,x ), t ∈ [tn,tn+1).
j j+1
Our main result is the convergence of the numerical scheme. Since B(w)
is allowed to vanish, equation (8) is a degenerate parabolic equation and
solutions are not necessarily smooth. In particular, the derivative of w may
not be defined at every point. We will therefore prove convergence to weak
solutions of (8).
6 G.M.COCLITE,J.RIDDER,ANDN.H.RISEBRO
Definition2.1(Weaksolutionsof (8)). Afunctionw ∈ L∞(0,∞;H1(Ω))×
L∞(Ω×(0,∞)) is a weak solution of (8) if it satisfies
ˆ ˆ ˆ
∞
(17) wφ −B(w)w φ −B(cid:48)(w)(w )2φdxdt+ w (x)φ(x,0)dx = 0,
t x x x 0
0 Ω Ω
for all φ ∈ C∞(Ω×[0,∞)).
c
The convergence result, which we will prove in Section 5, reads as follows.
Theorem 2.1. A subsequence of the interpolations w of the solutions of
∆t
the scheme defined by (9) converges in C([0,∞),W1,1(Ω)) to a weak solution
of (8) as defined in Definition 2.1.
Note that only a subsequence of w converges, because weak solutions
∆t
of (8) are not unique. We will comment more on this in Section 6.
For the a priori bounds in the next section, we will use the discrete norms
(cid:88) (cid:88)
(cid:107)an(cid:107) = sup|an|, (cid:107)an(cid:107) = ∆x |an|, |an| = |an−an |,
∞ j 1 j BV j j−1
j
j j
3. A priori bounds
In the following, we will show discrete maximum principles and BV
bounds for wn and zn = D wn. Here, note that the original equation (1)
j j + j
only possesses a maximum principle for u, but not for u , since in
x
1
u = (c(u))2u +4c(u)c(cid:48)(u)u u + (c2(u))(cid:48)(cid:48)(u )3,
tx xxx x xx x
2
the third term can lead to growth of local maxima in u . Our numerical
x
examples in Section 6 confirm this. One advantage of the transformation to
w is that for equation (8) both w and z = w are bounded in L∞.
x
The BV bound for z will be important in the convergence proof, because
strong convergence for both w and its first derivative is needed to pass to
the limit in the third term of the weak formulation (17). Before turning to
the discrete setting, let us show formally how L1 bounds for z and y = z
x
(i.e., BV bounds for w and z) can be obtained in the continuous case.
For z, multiply
z = (Bz )
t x x
by η(cid:48)(z), where η is some convex smooth function, and integrate in space to
get
ˆ ˆ
d
η(z)dx = − B(w)(z )2η(cid:48)(cid:48)(z)dx ≤ 0.
x
dt
Ω Ω
Letting η → |·|, we get an L1 bound for z.
For y, the formal continuous equivalent of equation (12) is
(18) y = (B(w)y) .
t xx
A CONVERGENT SCHEME FOR THE VARIATIONAL HEAT EQUATION 7
Again, let η ∈ C2(R) be convex and multiply (18) by η(cid:48)(y). Then,
η(y) = (By +2B y +B y)η(cid:48)(y)
t xx x x xx
≤ (y )2Bη(cid:48)(cid:48)(y)+By η(cid:48)(y)+2B η(y) +B yη(cid:48)(y)
x xx x x xx
= Bη(y) +2B η(y) +B yη(cid:48)(y)
xx x x xx
= (Bη(y) ) +B η(y) +B yη(cid:48)(y)
x x x x xx
= (Bη(y) ) +(B η(y)) −B η(y)+B yη(cid:48)(y)
x x x x xx xx
= (Bη(y)) +B (η(cid:48)(y)y−η).
xx xx
Integrating over Ω and taking η(y) = |y| such that it converges to |y| as
(cid:15)
(cid:15) → 0, we get
ˆ
d
|y| ≤ 0.
dt
Ω
In the discrete case, we will base our proofs on an extended version of
Harten’s Lemma [12, p. 118].
Lemma 3.1. Let v be given by
j
(19) v = u −A ∆ u +B ∆ u −C ∆ v +D ∆ v ,
j j j−1/2 − j j+1/2 + j j−1/2 − j j+1/2 + j
where ∆ u = ±(u −u ).
± j j±1 j
(i) If A , B , C , and D are nonnegative for all j, and
j+1/2 j+1/2 j+1/2 j+1/2
A +B ≤ 1 for all j, then
j+1/2 j+1/2
|v| ≤ |u| .
BV BV
(ii) If A , B , C , and D are nonnegative for all j, and
j+1/2 j+1/2 j+1/2 j+1/2
A +B ≤ 1 for all j, then
j−1/2 j+1/2
minu ≤ v ≤ maxu
i j i
i i
Proof. From (19), we get
(1+C +D )∆ v = (1−A −B )∆ u
j+1/2 j+1/2 + j j+1/2 j+1/2 + j
+A ∆ u +B ∆ u
j−1/2 − j j+3/2 + j+1
+C ∆ v +D ∆ v .
j−1/2 − j j+3/2 + j+1
Hence, under the assumptions of (i),
(cid:88) (cid:88)
(1+C +D )|∆ v | ≤ (1−A −B )|∆ u |
j+1/2 j+1/2 + j j+1/2 j+1/2 + j
j j
(cid:88)
+ A |∆ u |+B |∆ u |
j−1/2 − j j+3/2 + j+1
j
(cid:88)
+ C |∆ v |+D |∆ v |
j−1/2 − j j+3/2 + j+1
j
(cid:88)
= |∆ u |+(C +D )|∆ v |,
+ j j+1/2 j+1/2 + j
j
8 G.M.COCLITE,J.RIDDER,ANDN.H.RISEBRO
from which the BV bound follows.
For the maximum principle, we can write (19) as
(1+C +D )v = (1−A −B )u +A u +B u
j−1/2 j+1/2 j j−1/2 j+1/2 j j−1/2 j−1 j+1/2 j+1
+C v +D v .
j−1/2 j−1 j+1/2 j+1
Thus, if the assumptions of (ii) hold, v = max v satisfies
j(cid:48) i i
(1+C +D )v ≤ (1−A −B )maxu
j(cid:48)−1/2 j(cid:48)+1/2 j(cid:48) j(cid:48)−1/2 j(cid:48)+1/2 i
i
+A maxu +B maxu
j(cid:48)−1/2 i j(cid:48)+1/2 i
i i
+C v +D v ,
j(cid:48)−1/2 j(cid:48) j(cid:48)+1/2 j(cid:48)
and hence, max v = v ≤ max u . Similarly, min v ≥ min u , which
i i j(cid:48) i i i i i i
concludes the proof. (cid:3)
The L∞ and BV bound for wn and zn follow directly from the above
j j
lemma.
Lemma 3.2. Let wn be the solution of (9) and zn = D wn. Then
j j + j
minw0 ≤ wn ≤ maxw0, |wn| ≤ |w0| ,
i j i BV BV
i i
minz0 ≤ zn ≤ maxz0, |zn| ≤ |z0| .
i j i BV BV
i i
Proof. Rewriting (9), we get
wn+1 = wn+(1−θ)∆tB(wn+θ)D2wn+θ∆tB(wn+θ)D2wn+1.
j j j j j j
To apply Harten’s lemma, set v = wn+1, u = wn, and
j j j j
A = (1−θ)λB(wn+θ), C = θλB(wn+θ),
j−1/2 j j−1/2 j
B = (1−θ)λB(wn+θ), D = θλB(wn+θ),
j+1/2 j j+1/2 j
where λ = ∆t/(∆x)2. Because λ satisfies the CFL condition (10) and θ and
B(w) take values in [0,1], the assumptions of Harten’s lemma hold and we
get the maximum and BV bound for wn.
j
For z, write (11) as
zn+1 = zn+(1−θ)∆tD (B(wn+θ)D zn)+θ∆tD (B(wn+θ)D zn+1).
j j + j − j + j − j
Set v = zn+1, u = zn, and
j j j j
A = (1−θ)λB(wn+θ), C = θλB(wn+θ),
j−1/2 j j−1/2 j
B = (1−θ)λB(wn+θ), D = θλB(wn+θ),
j+1/2 j+1 j+1/2 j+1
in Harten’s lemma. Again, due to the CFL condition and the bounds on B,
the conditions are satisfied and the claim follows. (cid:3)
A CONVERGENT SCHEME FOR THE VARIATIONAL HEAT EQUATION 9
4. Continuity in time
In order to show compactness, we will need continuity in time of both
w∆t and z∆t. For w∆t this follows directly from the definition of the scheme
and the BV bound for z above.
Lemma 4.1. Let w∆t be the interpolation (14) of the solutions wn of (9).
j
Then, for any t,t+τ ≥ 0,
ˆ
|w∆t(x,t+τ)−w∆t(x,t)|dx ≤ (τ +O(∆t))|z0| +O(∆x)|w0| .
BV BV
Ω
Proof. Using the piecewise constant interpolation w∆t, we get
ˆ ˆ
|w∆t(x,t+τ)−w∆t(x,t)|dx ≤ |w∆t(x,t+τ)−w∆t(x,t+τ)|
Ω Ω
(20) +|w∆t(x,t)−w∆t(x,t)|
+|w∆t(x,t+τ)−w∆t(x,t)|dx.
Regarding the first two terms on the right-hand side, note that for t ∈
[tn,tn+1),
ˆ ˆ
(cid:88) xj
|w∆t(x,t)−w∆t(x,t)|dx = |(x −x)D wn|dx
j − j
Ω j xj−ˆ12
+ xj+12|(x−x )D wn|dx
j + j
(21) xj
(∆x)2 (cid:88)
= |D wn|
4 + j
j
∆x ∆x
= |wn| ≤ |w0| ,
4 BV 4 BV
where the last inequality is due to Lemma 3.2. For the last term in (20),
let m, n be such that t+τ ∈ [tn,tn+1) and t ∈ [tm,tm+1). Using the BV
bound on z from Lemma 3.2, we get
ˆ ˆ
|w∆t(x,t+τ)−w∆t(x,t)|dx = (cid:88) xj+21|wn−wm|
j j
Ω j xj−12
n−1
(cid:88) (cid:88)
= ∆x ∆t|D+wk|
t j
j k=m
n−1
(cid:88) (cid:88)
= ∆x∆t |B(wk+θ)D zk+θ|
j − j
j k=m
≤ ∆t(n−m)|z0| = (τ +O(∆t))|z0| ,
BV BV
and the claim follows. (cid:3)
10 G.M.COCLITE,J.RIDDER,ANDN.H.RISEBRO
Forz∆t,wewilluseaversionofKruˇzkov’sinterpolationlemma[12,p.208,
Lemma 4.11], which gives continuity in time if for all t ,t ≥ 0 and φ ∈
1 2
C∞(B ), where B = [−r,r]∩Ω,
0 r r
(22ˆ)
(cid:12) (cid:12)
(cid:12) ((z∆t(x,t )−z∆t(x,t ))φ(x)dx(cid:12) ≤ C (cid:107)φ(cid:48)(cid:107) (|t −t |+O(∆t)),
(cid:12) 2 1 (cid:12) r L∞(Br) 2 1
Br
in addition to the L∞ and BV bound from Lemma 3.2.
Lemma 4.2. Let z∆t be the piecewise constant interpolation of zn = D wn,
j + j
where wn is the solution of (9). Then z∆t satisfies for any t,t+τ ≥ 0, r > 0,
j
ˆ
(cid:112) ∆t
|z∆t(x,t+τ)−z∆t(x,t)| ≤ C max(|z0| ,1)( |τ|+ ),
r BV (cid:112)
|τ|
Br
where B = [−r,r]∩Ω.
r
Proof. ToapplyKruˇzkov’sinterpolationlemma,weneedtoshow(22). First,
note that for any time step n,
ˆ ˆ
(cid:12) (cid:12) (cid:12)(cid:88) xj+1 (cid:12)
(cid:12) (z∆t(x,tn+1)−z∆t(x,tn))φdx(cid:12) = (cid:12) (zn+1−zn) φdx(cid:12)
(cid:12) (cid:12) (cid:12) j j (cid:12)
Ω j xj ˆ
(cid:12)(cid:88) xj+1 (cid:12)
= (cid:12) ∆tD (B(wn+θ)D zn+θ) φdx(cid:12)
(cid:12) + j − j (cid:12)
j ˆ xj
(cid:12)(cid:88) 1 xj+1 (cid:12)
= (cid:12) ∆tB(wn+θ)D zn+θ φ(x)−φ(x−∆x)dx(cid:12)
(cid:12) j − j ∆x (cid:12)
j xj
(cid:88)
≤ ∆t∆x|D zn+θ|(cid:107)φ(cid:48)(cid:107)
− j L∞(Ω)
j
≤ ∆t(cid:107)φ(cid:48)(cid:107) |z0| .
L∞(Ω) BV
For given t ,t > 0, let n,m be such that t ∈ [tn,tn+1) and t ∈ [tm,tm+1).
1 2 1 2
The above estimate yields
ˆ
(cid:12) (cid:12)
(cid:12) (z∆t(x,t )−z∆t(x,t ))φdx(cid:12) ≤ (cid:107)φ(cid:48)(cid:107) |z0| (tm−tn)
(cid:12) 2 1 (cid:12) L∞(Ω) BV
Ω
≤ (cid:107)φ(cid:48)(cid:107) |z0| (t −t +2∆t).
L∞(Ω) BV 2 1
Kruˇzkov’s interpolation lemma [12, p. 208, Lemma 4.11] then implies
ˆ
|τ|+2∆t
|z∆t(x,t+τ)−z∆tx,t|dx ≤ C ((cid:15)+(cid:15)|z0| +|z0| ),
r BV BV (cid:15)
Br
for any (cid:15) > 0. Choosing (cid:15) = (cid:112)|τ|, we arrive at the claim. (cid:3)
5. Convergence
Finally, we are able to prove the convergence of the scheme, Theorem 2.1.