On numerical methods and error estimates for degenerate fractional convection-diffusion equations

Abstract

First we introduce and analyze a convergent numerical method for a large class of nonlinear nonlocal possibly degenerate convection diffusion equations. Secondly we develop a new Kuznetsov type theory and obtain general and possibly optimal error estimates for our numerical methods - even when the principal derivatives have any fractional order between 1 and 2! The class of equations we consider includes equations with nonlinear and possibly degenerate fractional or general Levy diffusion. Special cases are conservation laws, fractional conservation laws, certain fractional porous medium equations, and new strongly degenerate equations.

Full text

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arXiv:1201.6079v1 [math.NA] 29 Jan 2012
ON NUMERICAL METHODS AND ERROR ESTIMATES
FOR DEGENERATE FRACTIONAL CONVECTION-DIFFUSION
EQUATIONS
SIMONE CIFANI AND ESPEN R. JAKOBSEN
Abstract. First we introduce and analyze a convergent numerical method
for a large class of nonlinear nonlocal possibly degenerate convection diffusion
equations.Secondly we develop a new Kuznetsov type theory and obtain
general and possibly optimal error estimates for our numerical methods – even
when the principal derivatives have any fractional order between 1 and 2! The
class of equations we consider includes equations with nonlinear and possibly
degenerate fractional or general Levy diffusion. Special cases are conservation
laws, fractional conservation laws, certain fractional porous medium equations,
and new strongly degenerate equations.
1. Introduction
In this paper we develop a numerical method along with a general Kuznetsov
type theory of error estimates for integro partial differential equations of the form
?
where QT= Rd× (0,T) and the nonlocal diffusion operator Lµis defined as
Lµ[φ](x) =
|z|>0
(1.1)
∂tu + divf(u) = Lµ[A(u)],
u(x,0) = u0(x),
(x,t) ∈ QT,
x ∈ Rd,
ˆ
φ(x + z) − φ(x) − z · ∇φ(x)1|z|<1(z) dµ(z),
(1.2)
for smooth bounded functions φ. Here 1 denotes the indicator function. Through-
out the paper the data (f,A,µ,u0) is assumed to satisfy:
(A.1) f = (f1,...,fd) ∈ W1,∞(R;Rd) with f(0) = 0,
(A.2) A ∈ W1,∞(R), A non-decreasing with A(0) = 0,
(A.3) µ ≥ 0 is a Radon measure such that´
(A.4) u0∈ L∞(Rd) ∩ L1(Rd) ∩ BV (Rd).
We use the notation a ∧ b = min(a,b) and a ∨ b = max(a,b).
Remark 1.1. These assumptions can be relaxed in two standard ways: (i) f,A can
take any value at u = 0 (replace f by f − f(0) etc.), and (ii) f,A can be assumed
to be locally Lipschitz. By the maximum principle and (A.4), solutions of (1.1) are
bounded, and locally Lipschitz functions are Lipschitz on compact domains.
|z|>0|z|2∧ 1 dµ(z) < ∞,
The measure µ and the operator Lµare respectively the L´ evy measure and the
generator of a pure jump L´ evy process. Any such process has a L´ evy measure
Key words and phrases. Fractional conservation laws, convection-diffusion equations, porous
medium equation, entropy solutions, numerical method, convergence rate, error estimates.
This research was supported by the Research Council of Norway (NFR) through the project
”Integro-PDEs: numerical methods, analysis, and applications to finance”.
1

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2S. CIFANI AND E. R. JAKOBSEN
and generator satisfying (1.2) and (A.3), see e.g. [4]. Example are the symmetric
α-stable processes with fractional Laplace generators where
dz
|z|d+λ
Non-symmetric examples are popular in mathematical finance, e.g. the CGMY
model where



and where d = 1, λ(= Y ) ∈ (0,2), and C,G,M > 0. We refer the reader to [14] for
more details on this and other nonlocal models in finance. In both examples the
nonlocal operator behaves like a fractional derivative of order between 0 and 2.
Equation (1.1) has a local non-linear convection term (the f-term) and a frac-
tional (or nonlocal) non-linear possibly degenerate diffusion term (the A-term).
Special cases are scalar conservation laws (A ≡ 0), fractional and L´ evy conserva-
tion laws (A(u) = u and α-stable or more general µ) – see e.g. [6, 1] and [7, 29, 25],
fractional porous medium equations [16] (A = |u|m−1u for m ≥ 1 and α-stable µ),
and strongly degenerate equations where A vanishes on a set of positive measure.
If either A is degenerate or Lµis a fractional derivative of order less than 1, then
solutions of (1.1) are not smooth in general and uniqueness fails for weak (distri-
butional) solutions. Uniqueness can be regained by imposing additional entropy
conditions in a similar way to what is done for conservation laws. The Kruzkov
entropy solution theory of scalar conservation laws [27] was extended to cover frac-
tional conservation laws in [1], to more general L´ evy conservation laws in [25], and
then finally to setting of this paper, equations with non-linear fractional diffusion
and general L´ evy measures in [11]. For local 2nd order degenerate convection dif-
fusion equations like
dµ(z) = cλ
(cλ> 0)andLµ≡ −(−∆)λ/2
for λ ∈ (0,2).
(1.3)
dµ(z) =






C e−G|z|
|z|1+λdz
C e−M|z|
|z|1+λdz
for z > 0,
for z < 0,
(1.4)
∂tu + divf(u) = ∆A(u),
there is an entropy solution theory due to Carrillo [9].
In recent years, integro partial differential equations like (1.1) have been at the
center of a very active field of research. A thorough description of the mathematical
background for such equations, relevant bibliography, and applications to several
disciplines of interest can be found in [1, 2, 7, 11, 16, 25].
The first contribution of this paper is to introduce a numerical method for equa-
tion (1.1) and prove that it converges toward the entropy solution of (1.1) under
assumptions (A.1)–(A.4). The numerical method is based upon a monotone fi-
nite volume discretization of an approximate equation with truncated and hence
bounded L´ evy measure. Essentially it is an extension of the method in [11] from
symmetric α-stable to general L´ evy measures, but since non-symmetric measures
are allowed, the discretization becomes more complicated here. Apart from its abil-
ity to capture the correct solution for the whole family of equations of the form
(1.1), the main advantage of our numerical method is that it allows for a complete
error analysis through the new framework for error estimates that we develop in
the second part of the paper.
The second, and probably most important contribution of the paper, is the devel-
opment of a theory capable of producing error estimates for degenerate equations
of order greater than 1. This theory is based on a non-trivial extension of the
Kuznetsov theory for scalar conservation laws [28] to the current fractional diffu-
sion setting. An initial step in this analysis was performed in [2], with the derivation
of a so-called Kuznetsov lemma in a relevant form for (1.1). In [2] the lemma is

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NUMERICAL METHODS FOR CONVECTION-DIFFUSION EQUATIONS3
used in the derivation of continuous dependence estimates and error estimates for
vanishing viscosity type of approximations of (1.1). In the present paper, we show
how it can be used in solving the more difficult problem of finding error estimates
for numerical methods for (1.1).
As a corollary of our Kuznetsov type theory, we obtain explicit λ-dependent
error estimates when µ is a measure satisfying
0 ≤ 1|z|<1dµ(z) ≤ cλ
dz
|z|d+λ
for
cλ> 0 and λ ∈ (0,2).
(1.5)
In this paper we will call such measures fractional measures. For example for the
implicit version of our numerical method (3.5), we prove in Section 6 that
?u(·,T) − u∆x(·,T)?L1(Rd)≤ CT





∆x
∆x
∆x
1
2
λ ∈ (0,1),
λ = 1,
λ ∈ (1,2),
1
2log(∆x)
2−λ
2
where u is the entropy solution of (1.1) and u∆xis the solution of (3.5). Note that
our error estimate covers all values λ ∈ (0,2), all spacial dimensions d, and possibly
strongly degenerate equations! Also note that under our assumptions, the solution
u possibly only have BV regularity in space. Hence the error estimate is robust in
the sense that it holds also for discontinuous solutions, and moreover, the classical
result of Kuznetsov [28] for conservation laws follows as a corollary by taking A ≡ 0
(a valid choice here!) and λ ∈ (0,1). The above estimate is also consistent with
error estimates for the vanishing λ-fractional viscosity method,
∂tu + divf(u) = −∆x(−∆)λ/2u
as∆x → 0+,
see e.g. [18, 1], but note that our problem is different and much more difficult.
There is a vast literature on approximation schemes and error estimates for scalar
conservations laws, we refer e.g. to the books [26, 22] and references therein for
more details. For local degenerate convection-diffusion equations like (1.4), some
approximation methods and error estimates can be found e.g. in [20, 21, 24] and
references therein. In this setting it is very difficult to obtain error estimates for nu-
merical methods, and the only result we are aware of is a very recent one by Karlsen
et al. [24] (but see also [10]). This very nice result applies to rather general equa-
tions of the form (1.4) but in one space dimension and under additional regularity
assumptions (e.g. ∂x(A(u)) ∈ BV ). When it comes to nonlocal convection-diffusion
equations, the literature is very recent and not yet very extensive. The paper [15]
introduce finite volume schemes for radiation hydrodynamics equations, a model
where Lµis a nonlocal derivative of order 0. Then fractional conservation laws are
discretized in [17, 13, 12] with finite difference, discontinuous Galerkin, and spectral
vanishing viscosity methods respectively. In [15, 13] Kuznetsov type error estimates
are given, but only for integrable L´ evy measures or measures like (1.3) with λ < 1.
Both of these results can be obtained through the framework of this paper. In
[12] error estimates are given for all λ but with completely different methods. The
general degenerate non-linear case is discretized in [11] (without error estimates)
for symmetric α-stable L´ evy measures and then in the most general case in the
present paper.
Linear non-degenerate versions of (1.1) frequently arise in Finance, and the prob-
lem of solving these equations numerically has generated a lot of activity over the
last decade. An introduction and overview of this activity can be found in the book
[14], including numerical schemes based on truncation of the L´ evy measure. We
also mention the literature on fractional and nonlocal fully non-linear equations
like e.g. the Bellman equation of optimal control theory. Such equations have been
intensively studied over the last decade using viscosity solution methods, including

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4S. CIFANI AND E. R. JAKOBSEN
initial results on numerical methods and error analysis. We refer e.g. [5, 8, 23] and
references therein for an overview and the most general results in that direction.
In fact, ideas from that field has been essential in the development of the entropy
solution theory of equations like (1.1), and the construction of monotone numerical
methods of this paper parallels the one in [8]. However the structure of the two
classes of equations along with their mathematical and numerical analysis are very
different.
This paper is organized as follows. In Section 2 we recall the entropy formulation
and well-posedness results for (1.1) of [11] and the Kuznetsov type lemma derived
in [2]. We present the numerical method in Section 3. There we focus on the
case of no convection (f ≡ 0) to simplify the exposition and focus on new ideas.
In Section 4 we prove several auxiliary properties of the numerical method which
will be useful in the following sections. We establish existence, uniqueness, and a
priori estimates for the solutions of the numerical method in Section 5. The general
Kuznetsov type theory for deriving error estimates is presented in Section 6, where
it is also used to establish a rate of convergence for equations with fractional L´ evy
measures, i.e. (1.5) holds. In Section 7 we extend all the results considered so far
to general convection-diffusion equations of the form (1.1) with f ?≡ 0. Finally, we
give the proof of the main error estimate Theorem 6.1 in Section 8.
2. Preliminaries
In this section we briefly recall the entropy formulation for equations of the form
(1.1) introduced in [11], and the new Kuznetsov type of lemma established in [2].
Let η(u,k) = |u − k|, η′(u,k) = sgn(u − k), ql(u,k) = η′(u,k)(fl(u) − fl(k)) for
l = 1,...,d, and write the nonlocal operator Lµ[φ] as
Lµ
where
ˆ
0<|z|≤r
ˆ
|z|>r
ˆ
|z|>r
r[φ] + Lµ,r[φ] + γµ,r· ∇φ,
Lµ
r[φ](x) =
φ(x + z) − φ(x) − z · ∇φ(x)1|z|≤1dµ(z),
Lµ,r[φ](x) =
φ(x + z) − φ(x) dµ(z),
γµ,r
l
= −
zl1|z|≤1dµ(z),l = 1,...,d.
We also define µ∗by µ∗(B) = µ(−B) for all Borel sets B ?∋ 0. Let us recall that
ˆ
Rdϕ(x)Lµ[ψ](x) dx =
ˆ
Rdψ(x)Lµ∗[ϕ](x) dx
for all smooth L∞∩ L1functions ϕ,ψ, cf. [2, 11].
Definition 2.1. (Entropy solutions) A function u ∈ L∞(QT) ∩ C([0,T];L1(Rd))
is an entropy solution of (1.1) if, for all k ∈ R, r > 0, and test functions 0 ≤ ϕ ∈
C∞
c(Rd× [0,T]),
ˆ
QT
η(u,k)∂tϕ +
?
q(u,k) + γµ∗,r?
· ∇ϕ + η(A(u),A(k))Lµ∗
+ η′(u,k)Lµ,r[A(u)]ϕ dxdt
ˆ
r[ϕ]
−
ˆ
Rdη(u(x,T),k)ϕ(x,T) dx +
Rdη(u0(x),k)ϕ(x,0) dx ≥ 0.
(2.1)
Note that γµ,r
From [11] we now have the following well-posedness result.
l
≡ 0 when the L´ evy measure µ is symmetric, i.e. when µ∗≡ µ.

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NUMERICAL METHODS FOR CONVECTION-DIFFUSION EQUATIONS5
Theorem 2.1. (Well-posedness) Assume (A.1) – (A.4) hold. Then there exists a
unique entropy solution u of (1.1) such that
u ∈ L∞(QT) ∩ C([0,T];L1(Rd)) ∩ L∞(0,T;BV (Rd)),
and the following a priori estimates hold
?u(·,t)?L∞(Rd)≤ ?u0?L∞(Rd),
?u(·,t)?L1(Rd))≤ ?u0?L1(Rd),
|u(·,t)|BV (Rd)≤ |u0|BV (Rd),
?u(·,t) − u(·,s)?L1(Rd)≤ σ(|t − s|),
for all t,s ∈ [0,T] where
σ(r) =
?
cr
if
´
|z|>0|z| ∧ 1 dµ(z) < ∞,
otherwise.
cr
1
2
Moreover, if also (1.5) holds, then
σ(r) =





cr
c|rlnr|
cr
λ
if λ ∈ (0,1),
if λ = 1,
if λ ∈ (1,2).
1
The last a priori estimate is slightly more general then the one in [11], and follows
e.g. in the limit from the estimates in Lemmas 5.3 and 5.4. We now recall the new
Kuznetsov type of lemma established in [2]. Let
ω ∈ C∞
c(R),
0 ≤ ω ≤ 1,ω(τ) = 0 for all |τ| > 1,and
ˆ
R
ω(τ)dτ = 1,
and define ωδ(τ) =1
δω?τ
δ
?, Ωǫ(x) = ωǫ(x1)···ωǫ(xd), and
ϕǫ,δ(x,y,t,s) = Ωǫ(x − y)ωδ(t − s)
for ǫ,δ > 0. We also need
Eδ(v) =sup
|t−s|<δ
t,s∈[0,T]
?v(·,t) − v(·,s)?L1(Rd).
(2.2)
In the following we let dw = dxdtdy ds and CT ≥ 0 be a constant depending on
time and the initial data u0that may change from line to line.
Lemma 2.2. (Kuznetsov type of lemma) Assume (A.1) – (A.4) hold. Let u be the
entropy solution of (1.1) and v be any function in L∞(QT) ∩ C([0,T];L1(Rd)) ∩

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6S. CIFANI AND E. R. JAKOBSEN
L∞(0,T;BV(Rd)) with v(·,0) = v0(·). Then, for any ǫ,r > 0 and 0 < δ < T,
?u(·,T) − v(·,T)?L1(Rd)≤ ?u0− v0?L1(Rd)+ C (ǫ + Eδ(u) ∨ Eδ(v))
−
QT
QT
¨
QT
QT
¨
QT
QT
¨
QT
QT
¨
QT
QT
¨
QT
¨
QT
¨
¨
η(v(x,t),u(y,s))∂tϕǫ,δ(x,y,t,s) dw
−
¨
q(v(x,t),u(y,s)) · ∇xϕǫ,δ(x,y,t,s) dw
+
¨
η(A(v(x,t)),A(u(y,s)))Lµ∗
r[ϕǫ,δ(x,·,t,s)](y) dw
−
¨
η′(v(x,t),u(y,s))Lµ,r[A(v(·,t))](x)ϕǫ,δ(x,y,t,s) dw
−
¨
η(A(v(x,t)),A(u(y,s)))γµ∗,r· ∇xϕǫ,δ(x,y,t,s) dw
+
ˆ
Rdη(v(x,T),u(y,s))ϕǫ,δ(x,T,y,s) dxdy ds
−
ˆ
Rdη(v0(x),u(y,s))ϕǫ,δ(x,0,y,s) dxdy ds
The proof is given in [2]. The original result of result of Kuznetsov in [28] is a
special case when µ = 0 (or A = 0).
3. The numerical method
In this section we derive our numerical method. Here and in the following sec-
tions we focus on the case f ≡ 0 to simplify the exposition and focus on the new
ideas. The general case f ?= 0 will then be treated at the end, in Section 7.
We will consider uniform space/time grids given by xα= α∆x for α ∈ Zdand
tn= n∆t for n = 0,...,N =
of space
Rα= xα+ ∆x(0,1)d
We start by discretizing the nonlocal operator, replacing the measure µ by the
bounded truncated measure 1|z|>∆x
T
∆t. We also use the following rectangular subdivisions
for
α ∈ Zd.
2(z)µ and the gradient by a numerical gradient
ˆD∆x= (ˆD1,··· ,ˆDd),
lare upwind finite difference operators defined by


Here e1,...,ed is the standard basis of Rd. This gives an approximate nonlocal
operator
ˆLµ[A(φ)](x)
=
|z|>∆x
2
which is monotone by upwinding and non-singular since the truncated measure is
bounded.
A semidiscrete approximation of (1.1) with f ≡ 0 is then obtained by solving
the approximate equation
∂tu =ˆLµ[A(u)],
(3.1)
whereˆDl≡ Dγ
(3.2)
Dγ
lφ(x) =



D+
lφ(x) :=φ(x + ∆x el) − φ(x)
∆x
for γµ,∆x
2
l
> 0,
D−
lφ(x) :=φ(x) − φ(x − ∆x el)
∆x
otherwise.
ˆ
A(φ(x + z)) − A(φ(x))dµ(z) + γµ,∆x
2 ·ˆD∆xA(φ(x)),
(3.3)
(3.4)

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NUMERICAL METHODS FOR CONVECTION-DIFFUSION EQUATIONS7
by a finite volume method on the spacial subdivision {Rα}α. I.e. for each t, we
look for piecewise constant approximate solution
U(x,t) =
?
β∈Zd
Uβ(t)1Rβ(x),
that satisfy (3.4) in weak form with
1
∆xd1Rβas test functions: For every α ∈ Zd,
1
∆xd
ˆ
Rα
∂tU dx =
1
∆xd
ˆ
Rα
ˆLµ[A(U)]dx.
Finally we discretize in time by replacing ∂tby backward or forward differences
D±
method
∆tand Uα(t) by a piecewise constant approximation Un
α. The result is the implicit
Un+1
α
= Un
α+ ∆tˆLµ?A(Un+1)?α,
(3.5)
and the explicit method
Un+1
α
= Un
α+ ∆tˆLµ?A(Un)?α
(3.6)
where
ˆLµ?A(Un)?α=
1
∆xd
ˆ
Rα
ˆLµ[A(¯Un)](x) dx,
and¯Un(x) =?
β∈ZdUn
β1Rβ(x) is a piecewise constant x-interpolation of U. As
initial condition for both methods we take
U0
α=
1
∆xd
ˆ
Rα
u0(x) dx
for all
α ∈ Zd.
Lemma 3.1.
ˆLµ?A(Un)?α=
?
β∈Z
Gα
βA(Un
β)
with Gα
β= Gα,β+ Gα,βand
Gα,β=
1
∆xd
ˆ
Rα
ˆ
|z|>∆x
2
1Rβ(x + z) − 1Rβ(x) dµ(z)dx,
Gα,β=
d
?
l=1
γµ,∆x
l
2
1
∆xd
ˆ
Rα
Dγ
l1Rβ(x) dx.
(3.7)
Remark 3.2. Gα,β is a Toeplitz matrix (cf. Lemma 4.1 (b)) while Gα,βis a tridi-
agonal matrix. When the measure µ is symmetric, then Gα,β is symmetric and
Gα,β= 0.
Proof. Since
A(¯U(x)) =
?
β∈Zd
A(Uβ)1Rβ(x)and
ˆD∆x¯U(x) =
?
β∈Zd
UβˆD∆x1Rβ(x),

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8S. CIFANI AND E. R. JAKOBSEN
we find that
∆xdˆLµ?A(U)?α=
ˆ
Rα
|z|>∆x
ˆ
Rα
ˆLµ[A(¯U)](x) dx
=
ˆ
2
A(¯U(x + z)) − A(¯U(x)) dµ(z)dx
+ γµ,∆x
2 ·
ˆ
Rα
?
β∈Zd
A(Uβ)ˆD∆x1Rβ(x) dx
=
?
β∈Zd
A(Uβ)
?ˆ
Rα
ˆ
|z|>∆x
2
1Rβ(x + z) − 1Rβ(x) dµ(z)dx
?
+
?
β∈Zd
A(Uβ)
?
d
?
l=1
γµ,∆x
l
2
ˆ
Rα
Dγ
l1Rβ(x) dx
?
.
The proof is complete.
?
4. Properties of the numerical method
In this section we show that the numerical methods are conservative, monotone
and consistent in the sense that certain cell entropy inequalities are satisfied. We
start by a technical lemma summarizing the properties of the weights Gα
in (3.7).
βdefined
Lemma 4.1.
(a)?
(b) Gβ
(c) Gβ
(d) There is ¯ c = ¯ c(d,µ) > 0 such that Gβ
α∈ZdGα
α= Gβ+el
β≤ 0 and Gα
β=?
α∈ZdGβ
α= 0 for all β ∈ Zd.
α+elfor all α,β ∈ Zdand l = 1,...,d.
β≥ 0 for α ?= β.
β≥ −
´|z| ∧ 1dµ(z) < ∞,
otherwise.
¯ c
ˆ σµ(∆x)and where
ˆ σµ(s) =
?
s
s2
when
(4.1)
(e) If (1.5) holds, then there is ¯ c = ¯ c(d,λ) > 0 such that Gβ
β≥ −
¯ c
ˆ σλ(∆x)for
ˆ σλ(s) =





sλ
for λ > 1,
for λ = 1,
s
|lns|
s
for λ < 1.
(4.2)
Proof. (a) By the definitions of Gα,β,Gα,βand Fubini’s theorem,
ˆ
|z|>∆x
2
∆xd?
∆xd?
and, since?
∆xd?
∆xd?
α∈Zd
Gα,β=
?ˆ
Rd1Rβ(x + z) dx −
ˆ
Rd1Rβ(x) dx
?
dµ(z) = 0,
α∈Zd
Gα,β= ±
d
?
l=1
γµ,∆x
l
∆x
2
?ˆ
Rd1Rβ(x ± ∆x el)dx −
ˆ
Rd1Rβ(x)dx
?
= 0,
β∈Zd1Rβ(x) ≡ 1,
ˆ
Rα
β∈Zd
Gα,β=
ˆ
|z|>∆x
2
??
ˆ
Rα
β∈Zd
1Rβ(x + z) −
?
β∈Zd
1Rβ(x)
?
dµ(z)dx = 0,
β∈Zd
Gα,β= ±
d
?
l=1
γµ,∆x
l
∆x
2
??
β∈Zd
1Rβ(x ± ∆x el) −
?
β∈Zd
1Rβ(x)
?
dx = 0.

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NUMERICAL METHODS FOR CONVECTION-DIFFUSION EQUATIONS9
Therefore?
(b) Let y = x + eland note that
α∈ZdGα
β=?
α∈Zd
?Gα,β+ Gα,β?= 0 and?
β∈ZdGα
β= 0.
∆xdGα,β=
ˆ
Rα+el
ˆ
Rα+el
= ∆xdGβ+el,α+el.
ˆ
|z|>∆x
ˆ
|z|>∆x
2
1Rβ(y − ∆xel+ z) − 1Rβ(y − ∆xel) dµ(z)dy
=
2
1Rβ+el(y + z) − 1Rβ+el(y) dµ(z)dy
In a similar fashion we get Gα,β= Gβ+el,α+el.
(c) Note that
∆xdGβ,β=
ˆ
Rβ
ˆ
|z|>∆x
2
1Rβ(x + z) − 1 dµ(z) dx ≤ 0.
while by the definition Dγ
l, see (3.2),
∆xdGβ,β= −
d
?
l=1
γµ,∆x
l
2
sgn
?
γµ,∆x
l
2
?ˆ
Rβ
1Rβ(x)
∆x
dx ≤ 0.
For α ?= β,
∆xdGα,β=
ˆ
Rα
ˆ
|z|>∆x
2
1Rβ(x + z) dµ(z)dx ≥ 0,
Gα,β= 0 for α ?= β ± el, and by the definition of Dγ
d
?
Therefore Gβ
l,
∆xdGβ±el,β=
l=1
γµ,∆x
l
2
sgn
?
γµ,∆x
l
2
?ˆ
Rβ±el
1Rβ(x ± ∆x el)
∆x
dx ≥ 0.
β= Gβ,β+ Gβ,β≤ 0 and Gα
(d) To find the lower bound on Gβ
and hence
β= Gα,β+ Gα,β≥ 0 for α ?= β.
βwe note that´
Rβ1Rβ(x+z)−1Rβ(x) dx ≥ −∆xd,
Gβ,β≥ −
ˆ
|z|>∆x
2
dµ(z) ≥ −
ˆ
|z|<1
??|z|
∆x
2
?2
1|z|<1(z) + 1|z|>1(z)
?
dµ(z).
The bound then follows since
∆xGβ,β≥ −d
ˆ
∆x
2<|z|<1
|z| dµ(z) ≥ −d
ˆ
0<|z|<1
|z|2
∆x
2
dµ(z).
When´|z|∧1dµ(z) < ∞, the corresponding bound follows by a similar argument.
(e) When (1.5) hold we can estimate Gβ,βin the following way
Gβ,β≥ −
ˆ


∆x
2<|z|<1
|z|
∆x
2
?
cλdz
|z|d+λ−
1 −?∆x
2+ C
ˆ
|z|>1
dµ(z)
= −

cλ
2
∆x
σd
1−λ2
?1−λ?
+ C
for λ ?= 1,
for λ = 1.
−cλ
2
∆xσdln∆x

Page 10

10S. CIFANI AND E. R. JAKOBSEN
The last equality can be proved using polar coordinates, and σdis the surface area
of the unit sphere in Rd. Similarly we find that
∆xGβ,β≥ −d
ˆ
∆x
2<|z|<1
|z|
cλdz
|z|d+λ= −dcλ



σd
1−λ
?
1 −?∆x
2
2
?1−λ?
for λ ?= 1,
for λ = 1,
−σdln∆x
and since
?
1−(∆x
2)1−λ?
is less than 1 or (∆x
2)1−λwhen λ < 1 or λ > 1 respectively
(and when ∆x < 2), the proof is complete.
?
From the two facts that Gβ
immediately get a Kato type inequality for the discrete nonlocal operator (3).
α≥ 0 when α ?= β and sgn(u)A(u) = |A(u)|, we now
Lemma 4.2. (Discrete Kato inequality) If {uα,vα}α∈Zd are two bounded sequences,
then
?
From Lemma 4.1 it also follows that the explicit method (3.6) and the implicit
method (3.5) are conservative and monotone, at least when the explicit method
satisfies the following CFL condition:
∆t
ˆ σµ(∆x)< 1
Here ¯ c is defined in Lemma 4.1, and LAdenotes the Lipschitz constant of A. When
the L´ evy measure µ also satisfies (1.5), we have a weaker CFL condition
∆t
ˆ σλ(∆x)< 1
Proposition 4.3 (Conservative monotone schemes).
(a) The implicit and explicit methods (3.5) and (3.6) are conservative, i.e. for an
l1-solution U,
?
(b) The implicit method is monotone, i.e. if U and V solve (3.5), then
Un≤ Vn
(c) If (4.3) (or (4.4) and (1.5)) holds, then the explicit method (3.6) is monotone.
sgn(uα− vα)
β∈Zd
Gα
β(A(uβ) − A(vβ)) ≤
?
β∈Zd
Gα
β|A(uβ) − A(vβ)|.
¯ cLA
where ˆ σµis defined in (4.1).
(4.3)
¯ cLA
where ˆ σλis defined in (4.2).
(4.4)
α
Un
α=
?
α
U0
α.
⇒
Un+1≤ Vn+1
for
n ≥ 0.
Remark 4.4. The CFL condition (4.3) implies that
for the heat equation), and
sufficient for all equations considered in this paper. In real applications however,
typically (1.5) holds, and the superior CFL condition (4.4) should be used.
∆t
∆x2 ≤ C in general (just as
∆t
∆x≤ C when´|z| ∧ 1dµ(z) < ∞. Condition (4.3) is
Proof. (a) Sum (3.5) or (3.6) over α, change the order of summation, and use
Lemma 4.1 (a):
?
α∈Zd
Un+1
α
=
?
α∈Zd
Un
α+ ∆t
?
β∈Zd
A(Uβ)
??
α∈Zd
Gα
β
?
=
?
α∈Zd
Un
α.
(c) Let Tα[u] = uα+ ∆t?
β∈ZdGα
βA(uβ), the right hand side of (3.6). By Lemma
β≥ 0 for α ?= β and hence
∂uβTα[u] ≥ 0
Since A non-decreasing and Gα
(c) to find that
4.1 (c), Gα
for
β ?= α.
α≤ 0, we use the lower bound on Gα
αin Lemma 4.1
∂uαTα[u] = 1 + ∆tGα
αA′(uα) ≥ 1 − ¯ cLA
∆t
ˆ σµ(∆x),

Page 11

NUMERICAL METHODS FOR CONVECTION-DIFFUSION EQUATIONS11
which is positive by the CFL condition (4.3).
(b) The proof is similar to and easier than the proof of (c).
?
We then turn to checking the consistency of the method, and to do that we write
Gα,β= Gr
α,β+ Gα,β,rand Gα,β= Gα,β,r+ Gα,β
1
∆xd
Rα
1
∆xd
Rα
r
for r > 0 where
Gr
α,β=
ˆ
ˆ
∆x
2<|z|≤r
1Rβ(x + z) − 1Rβ(x) dµ(z)dx,
Gα,β,r=
ˆ
ˆ
|z|>r
1Rβ(x + z) − 1Rβ(x) dµ(z)dx,
Gα,β,r=
1
∆xd
d
?
l=1
γµ,∆x
l,r
2
ˆ
Rα
Dγr
l1Rβ(x) dx
for
γµ,∆x
l,r
2
= −
ˆ
∆x
2<|z|≤r
zl1|z|≤1dµ(z),
Gα,β
r
=
1
∆xd
d
?
l=1
γµ,r
l
ˆ
Rα
Dγr
l1Rβ(x) dx.
If r <∆x
2, we set Gr
α,β= 0 = Gα,β,r. We also define
Gβ,r
α
= Gr
α,β+ Gα,β,r
and
Gβ
α,r= Gα,β,r+ Gα,β
r
,
and note that Lemmas 4.1 and 4.2 obviously still holds with Gβ,r
Gβ
α.
α
or Gβ
α,rreplacing
Proposition 4.5. (Cell-entropy inequalities)
(a) If U is a solution of the implicit method (3.5), then, for all r > 0 and k ∈ R,
η(Un+1
α
,k) ≤ η(Un
β∈Zd
(4.5)
α,k) + ∆t
?
Gα,r
β
η(A(Un+1
β
),A(k))
+ ∆tη′(Un+1
α
,k)
?
β∈Zd
Gα
β,rA(Un+1
β
).
(b) Assume the CFL condition (4.3) (or (4.4) and (1.5)) holds. If U is a solution
of the explicit method (3.6), then, for all r > 0 and k ∈ R,
η(Un+1
α
,k) ≤ η(Un
β∈Zd
(4.6)
α,k) + ∆t
?
Gα,r
β
η(A(Un
α),A(k))
+ ∆tη′(Un+1
α
,k)
?
β∈Zd
Gα
β,rA(Un
α).
Remark 4.6. In the cell-entropy inequality for the explicit method, the η′-term
appears in the “wrong” time. In Section 6, we will see that this leads to worse error
estimates for the explicit method than for the implicit method.
Remark 4.7 (Convergence to entropy solutions). Proposition 4.5 and a standard
argument show that any C([0,T];L1
solutions ¯ u∆xof (3.5) or (3.6), will converge to an entropy solution of (1.1). We
refer to Theorem 3.9 in [22] and Section 4.2 in [11] for more details. Convergence
to the entropy solution also follows from the error estimates of Section 6.
loc(Rd))-convergent sequence of (interpolated)
Proof. (a) By (3.5) we easily see that for any k ∈ R,
Un+1
α
∨ k ≤ Un
Un+1
α
∧ k ≥ Un
α∨ k + ∆t1(k,+∞)(Un+1
α∧ k + ∆t1(−∞,k)(Un+1
α
)ˆLµ?A(Un+1)?α,
)ˆLµ?A(Un+1)?α.
α

Page 12

12S. CIFANI AND E. R. JAKOBSEN
Subtracting and using η(u,k) = |u − k| and η′(u,k) = sgn(u − k), we find that
η(Un+1
α
,k) ≤ η(Un
For any r > 0, we use Lemmas 4.1 (a) and 4.2 with Gα,r
α,k) + ∆tη′(Un+1
α
,k)ˆLµ?A(Un+1)?α.
replacing Gα
β
βto see that
η′(Un+1
α
,k)
?
β∈Zd
Gα,r
β
A(Un+1
β
)
= η′(Un+1
α
,k)
?
β∈Zd
Gα,r
β
(A(Un+1
β
) − A(k))

since
?
β∈Zd
Gα,r
β
= 0


≤
?
β∈Zd
Gα,r
β
η(A(Un+1
β
),A(k)).
The cell entropy inequality now follows from writing Gα
the above inequalities.
β= Gα
β,r+ Gα,r
β
and using
(b) By (3.6) and monotonicity (Proposition 4.5 (c)) we obtain the following in-
equalities: For all r > 0,
?
+ ∆t1(k,+∞)(Un+1
Un+1
α
∨ k ≤ Un
α∨ k + ∆t
β∈Zd
Gα,r
β
A(Un
β∨ k)
α
)
?
β∈Zd
Gα
β,rA(Un
β),
Un+1
α
∧ k ≥ Un
α∧ k + ∆t
?
β∈Zd
Gα,r
β
A(Un
β∧ k)
+ ∆t1(−∞,k)(Un+1
α
)
?
β∈Zd
Gα
β,rA(Un
β).
Since η(A(U),A(k)) = A(U ∨k)−A(U ∧k), the cell entropy inequality follows from
subtracting the two inequalities.
?
5. A priori estimates, existence, and uniqueness
In this section we state and prove several a priori estimates for the solutions
of the numerical methods (3.5) and (3.6). In what follows, we will use different
interpolants ¯ u of the solutions Un
αof the schemes. For the implicit method (3.5)
we take
¯ u(x,t) = Un+1
α
for all (x,t) ∈ Rα× (tn,tn+1],
(5.1)
while for the explicit method (3.6),
¯ u(x,t) = Un
α
for all (x,t) ∈ Rα× [tn,tn+1).
(5.2)
We now prove the following a priori estimates for ¯ u:
?¯ u(·,t)?L1(Rd)≤ ?u0?L1(Rd),
?¯ u(·,t)?L∞(Rd)≤ ?u0?L∞(Rd),
|¯ u(·,t)|BV (Rd)≤ |u0|BV (Rd).
(5.3)
(5.4)
(5.5)
Lemma 5.1. (A priori estimates)
(a) If U solve (3.5) and ¯ u is defined by (5.1), then the a priori estimates (5.3) –
(5.5) hold for all t > 0.
(b) Assume the CFL condition (4.3) (or (4.4) and (1.5)) holds. If U solve (3.6)
and ¯ u is defined by (5.2), then the a priori estimates (5.3) – (5.5) hold for all t > 0.

Page 13

NUMERICAL METHODS FOR CONVECTION-DIFFUSION EQUATIONS13
Proof. Since the schemes are conservative and monotone, cf. Proposition 4.3, this
is a standard result that essentially follows from the Crandall-Tartar Lemma. For
explicit methods in part (b) we refer to e.g. Theorem 3.6 in [22] for the details.
We did not find a reference for implicit methods, so we give a proof of part (a)
here. See also [17] for the case when A is linear. Let uα= Un+1
write (3.5) as
?
We prove (5.3). Multiply (5.6) by sgn(uα) and use Lemma 4.2 to get
?
which by Fubini’s theorem and the fact that?
?
By the definition of uα,hαand an iteration in n, it follows that
?
By (5.1), ?¯ u(·,t)?L1(Rd)= ∆xd?
To prove (5.5), we subtract two equations (5.6) evaluated at different points,
?
and use the fact that Gα
β+elto see that
?
Then we multiply by sgn(uα−uα−el), use Lemma 4.2, and sum over α, to find that
?
The estimate (5.5) then follows by iteration and the definitions of uα,hα, ¯ u.
It remains to prove (5.4). Note that since?
maximum point: since?
?
Then by the above inequality and (5.6),
?
In a similar way we find that infα∈Zd hα ≤ infα∈Zd uα and (5.4) follow from the
definitions of uα,hα, ¯ u and an iteration in n.
α
, hα= Un
α, and
(5.6)
uα− ∆t
β∈Zd
Gα
βA(uβ) = hα.
|uα| − ∆t
β∈Zd
Gα
β|A(uβ)| ≤ |hα|,
α∈ZdGα
β= 0 implies that
α∈Zd
|uα| ≤
?
α∈Zd
|hα|.
α∈Zd
|Un
α| ≤
?
α∈Zd
|U0
α|.
α∈Zd|Un
α| for t ∈ (tn,tn+1], and (5.3) follows.
uα− uα−el− ∆t
β∈Zd
?
Gα
βA(uβ) − Gα−el
β
A(uβ)
?
= hα− hα−el
β= Gα+el
uα− uα−el− ∆t
β∈Zd
Gα
β
?
A(uβ) − A(uβ−el)
?
= hα− hα−el.
α∈Zd
|uα− uα−el| ≤
?
α∈Zd
|hα− hα−el|.
α|uα| < ∞ by (5.3), there is an
α0 such that supαuα= uα0. Moreover, the parabolic term is nonpositive at the
β∈ZdGα
?
β= 0 and?
β∈Zd|Gα
A(uβ) − A(uα0)
β| < ∞,
β∈Zd
Gα0
βA(uβ) =
β∈Zd
Gα0
β
??
≤ 0.
sup
α∈Zduα= uα0≤ uα0− ∆t
β∈Zd
Gα0
βA(uβ) = hα0≤ sup
α∈Zdhα.
?
Lemma 5.2 (Global existence and uniqueness).
(a) There exists a unique solution Un∈ l1of the implicit scheme (3.5) for all n ≥ 0.
(b) Assume the CFL condition (4.3) (or (4.4) and (1.5)) holds. Then there exists
a unique solution Un∈ l1of the explicit scheme (3.6) for all n ≥ 0.

Page 14

14S. CIFANI AND E. R. JAKOBSEN
Note that Un∈ l1implies that ¯ u(·,t) ∈ L1(Rd).
Proof. (a) Let uα= Un+1
α
and hα= Un
?
α, rewrite (3.5) as (5.6), define
Tα[u] = uα− ǫuα− ∆t
?
β∈Zd
Gα
βA(uβ) − hα
?
,
and let ǫ be such that
ǫ
?
1 + LA¯ c
∆t
ˆ σµ(∆x)
?
< 1.
We first show that Tαis monotone, i.e. u ≤ v implies Tα[u] ≤ Tα[v]. For α ?= β,
Gα
β≥ 0 by Lemma 4.1, and hence since A non-decreasing,
∂uβTα[u] ≥ 0.
Moreover, since A non-decreasing and −
¯ c
ˆ σµ(∆x)≤ Gα
α≤ 0,
?
∂uαTα[u] = 1 − ǫ + ǫ∆tGα
αA′(uα) ≥ 1 − ǫ
1 + LA¯ c
∆t
ˆ σµ(∆x)
?
which is positive by our choice of ǫ.
Since T is monotone and A is nondecreasing,
?
= (1 − ǫ)
α∈Zd
?
α
Tα[u] − Tα[v]
?+
≤
?
α
?
Tα[u ∨ v] − Tα[v]
?
Gα
?
?
(uα∨ vα− vα) + ǫ∆t
?
??
α∈Zd
?
β∈Zd
β
?
A(uβ∨ vβ) − A(vβ)
?
= (1 − ǫ)
α∈Zd
(uα− vα)++ ǫ∆t
?
β∈Zd
α∈Zd
Gα
β
??
A(uβ) − A(vβ)
?+
.
A similar estimate holds for?
α(Tα[u]−Tα[v])−, and since?
|Tα[u] − Tα[v]| ≤ (1 − ǫ)
α∈ZdGα
β= 0, we have
shown that
?
α∈Zd
?
α∈Zd
|uα− vα|.
So Tαis an l1-contraction and Banach’s fixed point theorem then implies that there
exists a unique solution ¯ u ∈ l1of Tα[¯ u] = ¯ uαand hence also of (5.6).
(b) Existence follows by construction and the a priori estimates in Lemma 5.1.
Uniqueness essentially follows by monotonicity and?
and use the Kato inequality (Lemma 4.2) along with?
We have the following regularity estimate in time:
αGα
β= 0: Assume two so-
lutions Unand Vn, subtract the two equations and multiply by sgn(Un− Vn),
?
αGα
β= 0 to show that
α|Un− Vn| ≤?
α|U0− V0|.
?
Lemma 5.3. (Regularity in time)
(a) Assume (A.2) – (A.4) hold, and let U be a solution of the implicit method (3.5)
and ¯ u defined by (5.1). Then
?¯ u(·,s) − ¯ u(·,t)?L1(Rd)≤ σµ(|s − t| + ∆t)
for all s,t > 0, where
σµ(r) =



r
√r
if´
otherwise.
|z|>0|z| ∧ 1 dµ(z) < ∞,

Page 15

NUMERICAL METHODS FOR CONVECTION-DIFFUSION EQUATIONS15
(b) Assume (A.2) – (A.4) and (4.3) (or (4.4) and (1.5)) hold, and let U be a
solution of the explicit method (3.6) and ¯ u defined by (5.2). Then
?¯ u(·,s) − ¯ u(·,t)?L1(Rd)≤ σµ(|s − t| + ∆t)
for all s,t > 0, where σµis defined in (a).
Proof. The two proofs are essentially identical, so we only do the proof for case (a).
1) By (3.5), we find that for any x ∈ Rα,
Un
α− Un−1
α
=
∆t
∆xd
ˆ
Rα
ˆL[A(¯Un)](x)dx.
Take a test function 0 ≤ φ ∈ C∞
?
ˆ
Rd
c
and define φα =
1
∆xd
´
Rαφ(y)dy and¯φ(x) =
αφα1Rα(x). Multiply the equation by ∆xdφαand sum over α to find that
¯φ(x)(¯Un(x) −¯Un−1(x)) dx = ∆t
ˆ
Rd
¯φ(x)ˆL[A(¯Un)](x)dx,
where LetˆL∗be the adjoint ofˆL, then since¯U is constant over Rα,
ˆ
ˆ
Rdφ(x)(¯Un(x) −¯Un−1(x)) dx =
ˆ
Rd
¯φ(x)(¯Un(x) −¯Un−1(x)) dx
ˆ
Rd
= ∆t
Rd(¯φ(x) − φ(x))ˆL[A(¯Un)](x)dx + ∆t
ˆL∗[φ](x)A(¯Un)(x)dx.
2) Let ωε be an approximate unit, i.e. ωε(x) =
´
¯Un
ε
= ∆t(¯ ωε− ωε) ∗ˆL[A(¯Un)] + ∆tˆL∗[ωε] ∗ A(¯Un).
By Fubini we then find that
1
εdω(x
ε) where 0 ≤ ω ∈ C∞
0
and
Rdωdx = 1. Take φ(x) = ωε(y − x) in the equation above and let Un
ε−¯Un−1
ε=¯Un∗ ωε:
1
∆t?¯Un
ε−¯Un−1
ε
?L1 ≤ ?¯ ωε− ωε?L1?ˆL[A(¯Un)]?L1 + ?ˆL∗[ωε] ∗ A(¯Un)?L1 = I1+ I2.
3) To estimate I1, note that by a standard argument
?¯ ωε− ωε?L1 ≤ |ωε|BV∆x =cω
ε∆x,
and then by the definition ofˆL in (3.3), Fubini, the L1∩ BV regularity of Un
(Lemma 5.1), and the regularity of A in (A.2),
?ˆL[A(¯Un)]?L1
=
|z|>∆x
ˆ
|z|>∆x
ˆ
|z|>∆x
ˆ
2
ˆ
RdA(¯Un(x + z)) − A(¯Un(x)) − z ·ˆD∆xA(¯Un(x))1|z|<1dxdµ(z)
≤
2
?
2|A(Un)|BV|z|1|z|<1+ 2?A(Un)?L11|z|>1
?
dµ(z)
≤ C
2
|z| ∧ 1dµ(z) ≤
C
∆x
ˆ
|z|>0
|z|2∧ 1dµ(z).
These estimates along with (A.3) shows that I1≤ Cε−1.