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

0.34 MB·English

by  Simone Cifani

ON NUMERICAL METHODS AND ERROR ESTIMATES
FOR DEGENERATE FRACTIONAL CONVECTION-DIFFUSION
EQUATIONS
SIMONECIFANIANDESPENR.JAKOBSEN
2
Abstract. First we introduce and analyze a convergent numerical method
1
foralargeclassofnonlinearnonlocalpossiblydegenerateconvectiondiffusion
0 equations. Secondly we develop a new Kuznetsov type theory and obtain
2 generalandpossiblyoptimalerrorestimatesforournumericalmethods–even
n whentheprincipalderivativeshaveanyfractionalorderbetween1and2! The
a classofequations weconsiderincludes equations withnonlinear andpossibly
J degeneratefractionalorgeneralLevydiffusion. Specialcasesareconservation
laws,fractionalconservationlaws,certainfractionalporousmediumequations,
9
andnewstronglydegenerate equations.
2

]
A
N
1. Introduction
.
h
In this paper we develop a numerical method along with a general Kuznetsov
t
a type theory of error estimates for integro partial differential equations of the form
m
∂ u+divf(u)= µ[A(u)], (x,t) Q ,
[ (1.1) ut(x,0)=u (x), L x R∈d, T
(cid:26) 0 ∈
1
v where QT =Rd (0,T) and the nonlocal diffusion operator µ is defined as
× L
9
7 (1.2) µ[φ](x)= φ(x+z) φ(x) z φ(x)1 (z) dµ(z),
0 L ˆ − − ·∇ |z|<1
|z|>0
6
. for smooth bounded functions φ. Here 1 denotes the indicator function. Through-
1
out the paper the data (f,A,µ,u ) is assumed to satisfy:
0 0
2 (A.1) f =(f ,...,f ) W1,∞(R;Rd) with f(0)=0,
1 d
1 ∈
: (A.2) A W1,∞(R), A non-decreasing with A(0)=0,
v ∈
i (A.3) µ 0 is a Radon measure such that z 2 1 dµ(z)< ,
X ≥ |z|>0| | ∧ ∞
´
r (A.4) u L∞(Rd) L1(Rd) BV(Rd).
a 0
∈ ∩ ∩
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 standardways: (i) f,A can
take any value at u=0 (replace f by f f(0) etc.), and (ii) f,A can be assumed
−
tobelocallyLipschitz. Bythemaximumprincipleand(A.4),solutionsof (1.1)are
bounded, and locally Lipschitz functions are Lipschitz on compact domains.
The measure µ and the operator µ are respectively the L´evy measure and the
L
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
mediumequation, entropysolutions,numericalmethod,convergence rate,errorestimates.
This research was supported by the Research Council of Norway (NFR) through the project
”Integro-PDEs: numericalmethods, analysis,andapplicationstofinance”.
1
2 S.CIFANIANDE.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
(1.3) dµ(z)=c (c >0) and µ ( ∆)λ/2 for λ (0,2).
λ z d+λ λ L ≡− − ∈
| |
Non-symmetric examples are popular in mathematical finance, e.g. the CGMY
model where
Ce−G|z|
dz for z >0,
z 1+λ
dµ(z)= | |
Ce−M|z|dz for z <0,
z 1+λ
| |
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-
≡
tionlaws(A(u)=uandα-stableormoregeneralµ)–seee.g. [6,1]and[7,29,25],
fractional porous medium equations [16] (A= um−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 µ is a fractional derivative of order less than 1, then
L
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
entropysolutiontheoryofscalarconservationlaws[27]wasextendedto coverfrac-
tional conservationlaws in [1], to more generalL´evy conservationlaws 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
(1.4) ∂ u+divf(u)=∆A(u),
t
there is an entropy solution theory due to Carrillo [9].
In recent years, integro partial differential equations like (1.1) have been at the
centerofaveryactivefieldofresearch. Athoroughdescriptionofthemathematical
background for such equations, relevant bibliography, and applications to several
disciplines of interest can be found in [1, 2, 7, 11, 16, 25].
Thefirstcontributionofthis paperis tointroduceanumericalmethodforequa-
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
areallowed,thediscretizationbecomesmorecomplicatedhere. Apartfromitsabil-
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.
Thesecond,andprobablymostimportantcontributionofthepaper,isthedevel-
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-
sionsetting. Aninitialstepinthisanalysiswasperformedin[2],withthederivation
of a so-called Kuznetsov lemma in a relevant form for (1.1). In [2] the lemma is
NUMERICAL METHODS FOR CONVECTION-DIFFUSION EQUATIONS 3
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
dz
(1.5) 0 1 dµ(z) c for c >0 and λ (0,2).
≤ |z|<1 ≤ λ z d+λ λ ∈
| |
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
∆x12 λ (0,1),
∈
ku(·,T)−u∆x(·,T)kL1(Rd) ≤CT  ∆∆xx212−2lλog(∆x) λλ=(11,,2),
∈
where u is the entropysolution of (1.1) andu∆x is the solutionof (3.5). Note that
ourerrorestimatecoversallvaluesλ (0,2),allspacialdimensionsd,andpossibly
∈
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
resultofKuznetsov[28]forconservationlawsfollowsasacorollarybytakingA 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,
∂ u+divf(u)= ∆x( ∆)λ/2u as ∆x 0+,
t
− − →
see e.g. [18, 1], but note that our problem is different and much more difficult.
Thereisavastliteratureonapproximationschemesanderrorestimatesforscalar
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
referencestherein. Inthissettingitisverydifficulttoobtainerrorestimatesfornu-
mericalmethods,andtheonlyresultweareawareofisaveryrecentonebyKarlsen
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. ∂ (A(u)) BV). Whenitcomestononlocalconvection-diffusion
x
∈
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 µ is a nonlocalderivative of order 0. Then fractionalconservationlaws are
L
discretizedin[17,13,12]withfinitedifference,discontinuousGalerkin,andspectral
vanishingviscositymethodsrespectively. In[15,13]Kuznetsovtypeerrorestimates
are given, but only for integrable L´evy measuresor 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.
Linearnon-degenerateversionsof (1.1)frequentlyariseinFinance,andtheprob-
lem of solving these equations numerically has generated a lot of activity over the
lastdecade. Anintroductionandoverviewofthisactivitycanbefoundinthebook
[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
likee.g. the Bellmanequationofoptimalcontroltheory. Suchequationshavebeen
intensively studied overthe lastdecade using viscositysolutionmethods, including
4 S.CIFANIANDE.R.JAKOBSEN
initial results onnumericalmethods anderroranalysis. We refere.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
solutiontheoryofequationslike(1.1),andthe constructionofmonotonenumerical
methods of this paper parallels the one in [8]. However the structure of the two
classesof equations along with their mathematicaland numericalanalysisare very
different.
Thispaperisorganizedasfollows. InSection2werecalltheentropyformulation
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
prioriestimatesforthesolutionsofthenumericalmethodinSection5. Thegeneral
Kuznetsovtype theoryfor derivingerrorestimates is presentedinSection6,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
6≡
give the proof of the main error estimate Theorem 6.1 in Section 8.
2. Preliminaries
Inthissectionwebrieflyrecalltheentropyformulationforequationsoftheform
(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), q (u,k) = η′(u,k)(f (u) f (k)) for
l l l
| − | − −
l=1,...,d, and write the nonlocal operator µ[φ] as
L
µ[φ]+ µ,r[φ]+γµ,r φ,
Lr L ·∇
where
µ[φ](x)= φ(x+z) φ(x) z φ(x)1 dµ(z),
Lr ˆ − − ·∇ |z|≤1
0<|z|≤r
µ,r[φ](x)= φ(x+z) φ(x) dµ(z),
L ˆ −
|z|>r
γµ,r = z 1 dµ(z), l =1,...,d.
l −ˆ l |z|≤1
|z|>r
We also define µ∗ by µ∗(B)=µ( B) for all Borel sets B 0. Let us recall that
− 6∋
ϕ(x) µ[ψ](x) dx= ψ(x) µ∗[ϕ](x) dx
ˆRd L ˆRd L
for all smooth L∞ L1 functions ϕ,ψ, cf. [2, 11].
∩
Definition 2.1. (Entropy solutions) A function u L∞(Q ) C([0,T];L1(Rd))
T
is an entropy solution of (1.1) if, for all k R, r >∈0, and test∩functions 0 ϕ
C∞(Rd [0,T]), ∈ ≤ ∈
c ×
η(u,k)∂ ϕ+ q(u,k)+γµ∗,r ϕ+η(A(u),A(k)) µ∗[ϕ]
ˆ t ·∇ Lr
QT (cid:16) (cid:17)
(2.1) +η′(u,k) µ,r[A(u)]ϕ dxdt
L
η(u(x,T),k)ϕ(x,T) dx+ η(u (x),k)ϕ(x,0) dx 0.
−ˆRd ˆRd 0 ≥
Note that γµ,r 0 when the L´evy measure µ is symmetric, i.e. when µ∗ µ.
l ≡ ≡
From [11] we now have the following well-posedness result.
NUMERICAL METHODS FOR CONVECTION-DIFFUSION EQUATIONS 5
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∞(Q ) C([0,T];L1(Rd)) L∞(0,T;BV(Rd)),
T
∈ ∩ ∩
and the following a priori estimates hold
ku(·,t)kL∞(Rd) ≤ku0kL∞(Rd),
ku(·,t)kL1(Rd)) ≤ku0kL1(Rd),
|u(·,t)|BV(Rd) ≤|u0|BV(Rd),
ku(·,t)−u(·,s)kL1(Rd) ≤σ(|t−s|),
for all t,s [0,T] where
∈
cr if z 1 dµ(z)< ,
σ(r)= |z|>0| |∧ ∞
(cr12 oth´erwise.
Moreover, if also (1.5) holds, then
cr if λ (0,1),
∈
σ(r)= c rlnr if λ=1,

| |
crλ1 if λ (1,2).
∈

Thelastaprioriestimateisslightlymoregeneralthentheonein[11],andfollows
e.g. in the limit fromthe estimates in Lemmas 5.3 and 5.4. We now recallthe new
Kuznetsov type of lemma established in [2]. Let
ω C∞(R), 0 ω 1, ω(τ)=0 for all τ >1, and ω(τ)dτ =1,
∈ c ≤ ≤ | | ˆR
and define ω (τ)= 1ω τ , Ω (x)=ω (x ) ω (x ), and
δ δ δ ǫ ǫ 1 ··· ǫ d
(cid:0) (cid:1)
ϕǫ,δ(x,y,t,s)=Ω (x y)ω (t s)
ǫ δ
− −
for ǫ,δ >0. We also need
(2.2) Eδ(v)= |t−sus|p<δ kv(·,t)−v(·,s)kL1(Rd).
t,s∈[0,T]
In the following we let dw = dxdtdyds and C 0 be a constant depending on
T
≥
time and the initial data u that may change from line to line.
0
Lemma 2.2. (Kuznetsovtype oflemma) Assume (A.1) – (A.4) hold. Let u be the
entropy solution of (1.1) and v be any function in L∞(Q ) C([0,T];L1(Rd))
T
∩ ∩
6 S.CIFANIANDE.R.JAKOBSEN
L∞(0,T;BV(Rd)) with v( ,0)=v (). Then, for any ǫ,r >0 and 0<δ <T,
0
· ·
ku(·,T)−v(·,T)kL1(Rd) ≤ku0−v0kL1(Rd)+C(ǫ+Eδ(u)∨Eδ(v))
η(v(x,t),u(y,s))∂ ϕǫ,δ(x,y,t,s) dw
−¨ ¨ t
QT QT
q(v(x,t),u(y,s)) ϕǫ,δ(x,y,t,s) dw
−¨ ¨ ·∇x
QT QT
+ η(A(v(x,t)),A(u(y,s))) µ∗[ϕǫ,δ(x, ,t,s)](y) dw
¨ ¨ Lr ·
QT QT
η′(v(x,t),u(y,s)) µ,r[A(v(,t))](x)ϕǫ,δ(x,y,t,s) dw
−¨ ¨ L ·
QT QT
η(A(v(x,t)),A(u(y,s)))γµ∗,r ϕǫ,δ(x,y,t,s) dw
−¨ ¨ ·∇x
QT QT
+ η(v(x,T),u(y,s))ϕǫ,δ(x,T,y,s) dxdyds
¨ ˆ
QT Rd
η(v (x),u(y,s))ϕǫ,δ(x,0,y,s) dxdyds
−¨QT ˆRd 0
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 uniform6 space/time grids given by x = α∆x for α Zd and
α
∈
t =n∆tforn=0,...,N = T . Wealsousethefollowingrectangularsubdivisions
n ∆t
of space
R =x +∆x(0,1)d for α Zd.
α α
∈
We start by discretizing the nonlocal operator, replacing the measure µ by the
bounded truncated measure 1 (z)µ and the gradient by a numerical gradient
|z|>∆x
2
(3.1) Dˆ =(Dˆ , ,Dˆ ),
∆x 1 d
···
where Dˆ Dγ are upwind finite difference operators defined by
l ≡ l
D+φ(x):= φ(x+∆xel)−φ(x) for γµ,∆2x >0,
(3.2) Dγφ(x)= l ∆x l
l  φ(x) φ(x ∆xe )
D−φ(x):= − − l otherwise.
l ∆x
Here e1,...,ed is thestandard basis of Rd. This gives an approximate nonlocal
operator
ˆµ[A(φ)](x)
L
(3.3)
=ˆ|z|>∆x A(φ(x+z))−A(φ(x))dµ(z)+γµ,∆2x ·Dˆ∆xA(φ(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
(3.4) ∂ u= ˆµ[A(u)],
t
L
NUMERICAL METHODS FOR CONVECTION-DIFFUSION EQUATIONS 7
by a finite volume method on the spacial subdivision R . I.e. for each t, we
α α
{ }
look for piecewise constant approximate solution
U(x,t)= U (t)1 (x),
β Rβ
βX∈Zd
that satisfy (3.4) in weak form with 1 1 as test functions: For every α Zd,
∆xd Rβ ∈
1 1
∂ Udx= ˆµ[A(U)]dx.
∆xd ˆ t ∆xd ˆ L
Rα Rα
Finally we discretize in time by replacing ∂ by backwardor forwarddifferences
t
D± andU (t)byapiecewiseconstantapproximationUn. Theresultistheimplicit
∆t α α
method
(3.5) Un+1 =Un+∆t ˆµ A(Un+1) ,
α α L h iα
and the explicit method
(3.6) Un+1 =Un+∆t ˆµ A(Un)
α α L h iα
where
1
ˆµ A(Un) = ˆµ[A(U¯n)](x) dx,
L h iα ∆xd ˆ L
Rα
and U¯n(x) = Un1 (x) is a piecewise constant x-interpolation of U. As
β∈Zd β Rβ
initial condition for both methods we take
P
1
U0 = u (x) dx for all α Zd.
α ∆xd ˆ 0 ∈
Rα
Lemma 3.1.
ˆµ A(Un) = GαA(Un)
L h iα β β
β∈Z
X
with Gα =G +Gα,β and
β α,β
1
G = 1 (x+z) 1 (x) dµ(z)dx,
α,β ∆xd ˆRαˆ|z|>∆2x Rβ − Rβ
(3.7)
d
Gα,β = γµ,∆2x 1 Dγ1 (x) dx.
l ∆xd ˆ l Rβ
l=1 Rα
X
Remark 3.2. G is a Toeplitz matrix (cf. Lemma 4.1 (b)) while Gα,β is a tridi-
α,β
agonal matrix. When the measure µ is symmetric, then G is symmetric and
α,β
Gα,β =0.
Proof. Since
A(U¯(x))= A(U )1 (x) and Dˆ U¯(x)= U Dˆ 1 (x),
β Rβ ∆x β ∆x Rβ
βX∈Zd βX∈Zd
8 S.CIFANIANDE.R.JAKOBSEN
we find that
∆xdˆµ A(U) = ˆµ[A(U¯)](x) dx
L h iα ˆ L
Rα
= A(U¯(x+z)) A(U¯(x)) dµ(z)dx
ˆRαˆ|z|>∆2x −
+γµ,∆2x ·ˆ A(Uβ)Dˆ∆x1Rβ(x) dx
RαβX∈Zd
= A(U ) 1 (x+z) 1 (x) dµ(z)dx
βX∈Zd β ˆRαˆ|z|>∆2x Rβ − Rβ !
d
+ A(U ) γµ,∆2x Dγ1 (x) dx .
β l ˆ l Rβ
βX∈Zd Xl=1 Rα !
The proof is complete. (cid:3)
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 defined
β
in (3.7).
Lemma 4.1.
(a) Gα = Gβ =0 for all β Zd.
α∈Zd β α∈Zd α ∈
(b) GPβα =Gαβ++eell fPor all α,β ∈Zd and l =1,...,d.
(c) Gβ 0 and Gα 0 for α=β.
β ≤ β ≥ 6
(d) There is c¯=c¯(d,µ)>0 such that Gβ c¯ and where
β ≥−σˆµ(∆x)
s when z 1dµ(z)< ,
(4.1) σˆ (s)= | |∧ ∞
µ (s2 otherw´ise.
(e) If (1.5) holds, then there is c¯=c¯(d,λ)>0 such that Gβ c¯ for
β ≥−σˆλ(∆x)
sλ for λ>1,
(4.2) σˆ (s)= s for λ=1,
λ |lns|
s for λ<1.
Proof. (a) By the definitions of Gα,β,Gα,β and Fubini’s theorem,
∆xd G = 1 (x+z) dx 1 (x) dx dµ(z)=0,
αX∈Zd α,β ˆ|z|>∆2x (cid:16)ˆRd Rβ −ˆRd Rβ (cid:17)
d µ,∆x
γ 2
∆xd Gα,β = l 1 (x ∆xe )dx 1 (x)dx =0,
αX∈Zd ±Xl=1 ∆x (cid:16)ˆRd Rβ ± l −ˆRd Rβ (cid:17)
and, since 1 (x) 1,
β∈Zd Rβ ≡
∆xd PG = 1 (x+z) 1 (x) dµ(z)dx=0,
βX∈Zd α,β ˆRαˆ|z|>∆2x (cid:16)βX∈Zd Rβ −βX∈Zd Rβ (cid:17)
d µ,∆x
γ 2
∆xd Gα,β = l 1 (x ∆xe ) 1 (x) dx=0.
± ∆x ˆ Rβ ± l − Rβ
βX∈Zd Xl=1 Rα(cid:16)βX∈Zd βX∈Zd (cid:17)
NUMERICAL METHODS FOR CONVECTION-DIFFUSION EQUATIONS 9
Therefore Gα = G +Gα,β =0 and Gα =0.
α∈Zd β α∈Zd α,β β∈Zd β
(b) Let y =Px+e and nPote tha(cid:0)t (cid:1) P
l
∆xd G = 1 (y ∆xe +z) 1 (y ∆xe ) dµ(z)dy
α,β ˆRα+el ˆ|z|>∆2x Rβ − l − Rβ − l
= 1 (y+z) 1 (y) dµ(z)dy
ˆRα+el ˆ|z|>∆2x Rβ+el − Rβ+el
=∆xd G .
β+el,α+el
In a similar fashion we get Gα,β =Gβ+el,α+el.
(c) Note that
∆xd G = 1 (x+z) 1 dµ(z) dx 0.
β,β ˆRβ ˆ|z|>∆2x Rβ − ≤
while by the definition Dγ, see (3.2),
l
∆xd Gβ,β = d γµ,∆2x sgn γµ,∆2x 1Rβ(x) dx 0.
− l l ˆ ∆x ≤
Xl=1 (cid:16) (cid:17) Rβ
For α=β,
6
∆xd G = 1 (x+z) dµ(z)dx 0,
α,β ˆRαˆ|z|>∆2x Rβ ≥
Gα,β =0 for α=β e , and by the definition of Dγ,
6 ± l l
∆xd Gβ±el,β = d γµ,∆2x sgn γµ,∆2x 1Rβ(x±∆xel) dx 0.
l l ˆ ∆x ≥
Xl=1 (cid:16) (cid:17) Rβ±el
Therefore Gβ =G +Gβ,β 0 and Gα =G +Gα,β 0 for α=β.
β β,β ≤ β α,β ≥ 6
(d)TofindthelowerboundonGβ wenotethat 1 (x+z) 1 (x)dx ∆xd,
β Rβ Rβ − Rβ ≥−
and hence ´
z 2
G dµ(z) | | 1 (z)+1 (z) dµ(z).
β,β ≥−ˆ|z|>∆2x ≥−ˆ|z|<1(cid:18)(cid:18)∆2x(cid:19) |z|<1 |z|>1 (cid:19)
The bound then follows since
z 2
∆xGβ,β d z dµ(z) d | | dµ(z).
≥− ˆ∆2x<|z|<1| | ≥− ˆ0<|z|<1 ∆2x
When z 1dµ(z)< , the correspondingboundfollowsby a similarargument.
| |∧ ∞
´
(e) When (1.5) hold we can estimate G in the following way
β,β
z c dz
G | | λ dµ(z)
β,β ≥−ˆ∆2x<|z|<1 ∆2x |z|d+λ −ˆ|z|>1
c 2 σd 1 ∆x 1−λ +C for λ=1,
= λ∆x1−λ − 2 6
− c 2 σ (cid:16)ln∆x(cid:0)+C(cid:1) (cid:17) for λ=1.
− λ∆x d 2

10 S.CIFANIANDE.R.JAKOBSEN
The last equality can be provedusing polar coordinates,and σ is the surface area
d
of the unit sphere in Rd. Similarly we find that
c dz σd 1 ∆x 1−λ for λ=1,
∆xGβ,β d z λ = dc 1−λ − 2 6
≥− ˆ∆2x<|z|<1| | |z|d+λ − λ−σd(cid:16)ln∆2x(cid:0) (cid:1) (cid:17) for λ=1,
andsince 1 (∆x)1−λ islessthan1or(∆x)1−λ whenλ<1orλ>1respectively
− 2 2
(and when(cid:16)∆x<2), th(cid:17)e proof is complete. (cid:3)
From the two facts that Gβ 0 when α =β and sgn(u)A(u) = A(u), we now
α ≥ 6 | |
immediately get a Kato type inequality for the discrete nonlocal operator (3).
Lemma4.2. (DiscreteKatoinequality)If u ,v aretwoboundedsequences,
{ α α}α∈Zd
then
sgn(u v ) Gα(A(u ) A(v )) Gα A(u ) A(v ) .
α− α β β − β ≤ β| β − β |
βX∈Zd βX∈Zd
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
(4.3) c¯L <1 where σˆ is defined in (4.1).
A σˆ (∆x) µ
µ
Herec¯isdefinedinLemma4.1,andL denotestheLipschitzconstantofA. When
A
the L´evy measure µ also satisfies (1.5), we have a weaker CFL condition
∆t
(4.4) c¯L <1 where σˆ is defined in (4.2).
Aσˆ (∆x) λ
λ
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,
Un = U0.
α α
α α
X X
(b) The implicit method is monotone, i.e. if U and V solve (3.5), then
Un Vn Un+1 Vn+1 for n 0.
≤ ⇒ ≤ ≥
(c) If (4.3) (or (4.4) and (1.5)) holds, then the explicit method (3.6) is monotone.
Remark 4.4. The CFL condition (4.3) implies that ∆t C in general (just as
∆x2 ≤
for the heat equation), and ∆t C when z 1dµ(z) < . Condition (4.3) is
∆x ≤ | |∧ ∞
sufficient for all equations considered in th´is paper. In real applications however,
typically (1.5) holds, and the superior CFL condition (4.4) should be used.
Proof. (a) Sum (3.5) or (3.6) over α, change the order of summation, and use
Lemma 4.1 (a):
Un+1 = Un+∆t A(U ) Gα = Un.
α α β β α
αX∈Zd αX∈Zd βX∈Zd (cid:18)αX∈Zd (cid:19) αX∈Zd
(c) Let T [u]=u +∆t GαA(u ), the right hand side of (3.6). By Lemma
α α β∈Zd β β
4.1 (c), Gα 0 for α=β and hence
β ≥ 6 P
∂ T [u] 0 for β =α.
uβ α ≥ 6
Since A non-decreasing and Gα 0, we use the lower bound on Gα in Lemma 4.1
α ≤ α
(c) to find that
∆t
∂ T [u]=1+∆tGαA′(u ) 1 c¯L ,
uα α α α ≥ − Aσˆ (∆x)
µ