We derive the necessary and sufficient condition for the $L^{\infty}-$monotonicity of finite difference $\theta$-scheme for a diffusion equation. We confirm that the discretization ratio $\Delta t = O(\Delta x^2)$ is necessary for the monotonicity except for the implicit scheme. In case of the heat equation, we get an explicit formula, which is weaker than the classical CFL condition.

J. Frederic Bonnans

Xiaolu Tan

English

Research Papers in Economics

apport ￿
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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Monotonicity condition for the -scheme
for diffusion equations
J. Frédéric BONNANS — Xiaolu TAN
N° 7778
Octobre 2011
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1Centre de recherche INRIA Saclay – Île-de-France
Parc Orsay Université
4, rue Jacques Monod, 91893 ORSAY Cedex
Téléphone : +33 1 72 92 59 00
Monotonicity condition for the -scheme
for diusion equations
J. Fr ed eric BONNANSy , Xiaolu TANz
Th eme NUM | Syst emes num eriques
 Equipes-Projets Commands
Rapport de recherche n￿ 7778 | Octobre 2011 | 6 pages
Abstract: We derive the necessary and sucient condition for the L1 monotonicity of nite
dierence -scheme for a diusion equation. We conrm that the discretization ratio t =
O(x2) is necessary for the monotonicity except for the implicit scheme. In case of the heat
equation, we get an explicit formula, which is weaker than the classical CFL condition.
Key-words: Theta-scheme, monotonicity.
 We thank Nizar Touzi (CMAP) for fruitful discussions.
y INRIA-Saclay and CMAP,  Ecole Polytechnique, 91128 Palaiseau, and Laboratoire de Finance des March es
d' Energie, France (Frederic.Bonnans@inria.fr).
z CMAP,  Ecole Polytechnique, 91128 Palaiseau, France (xiaolu.tan@polytechnique.edu), research supported
by the Chair Financial Risks of the Risk Foundation sponsored by Soci et e G en erale, the Chair Derivatives of the
Future sponsored by the F ed eration Bancaire Fran caise, and the Chair Finance and Sustainable Development
sponsored by EDF and CA-CIB.
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1La condition de la monotonie du  sch ema
pour les  equations de diusion
R esum e : Nous nous int eressons  a la condition n ecessaire et sussante de la monotonie du
 sch ema pour l' equation de diusion en dimension un. Notre r esultat conrme que le ratio de
discr etisation t = O(x2) est n ec essaire pour la monotonice sauf le sch ema implicite. Dans
le cas de l' equation de la chaleur, nous obtenons la formule explicite, qui est plus faible que la
condition CFL.
Mots-cl es : Theta-sch ema, monotonie.
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1Monotonicity of theta-scheme 3
1 Introduction
The monotonicity of a numerical scheme is an important issue in numerical analysis. For exam-
ple, in the convergence analysis in Chapiter 2 of Allaire [1], the author uses the L1 monotonicity
to derive the stability of the scheme, which gives a proof of convergence. In the viscosity solution
convergence context of Barles and Souganidis [2], the L1 monotonicity is a key criterion to
guarantee the convergence of the numerical scheme.
We are here interested in the nite dierence  scheme for the diusion equation:
@tv   2(x) D2
xxv = 0; (t;x) 2 R+  R: (1.1)
with initial condition v(0;x) = g(x).
2 The  scheme and CFL condition
Let h = (t;x) 2 (R+)2 be the discretization in time and space, denote tn := nt, xi := ix,
i := (xi) and by un
i the numerical solution of v at point (tn;xi), let N := fxi : i 2 Ng be
a discrete grid on R. The nite dierence  scheme (0    1) for diusion equation (1.1) is
a countable innite dimensional linear system on N:
u
n+1
i   un
i
t
  2
i


u
n+1
i+1   2u
n+1
i + u
n+1
i 1
x2 + (1   )
un
i+1   2un
i + un
i 1
x2

= 0; (2.1)
with initial condition u0
i = g(xi).
Let (un) := (un
i )i2Z be a Z dimentional vector, denote i :=

2
i t
x2 and i := i
1+2i, we dene
Z  Z diensional matrices I, D, T and E as follows: I is the identity matrix, D is a diagonal
matrix with Di;i = i, T is a tridiagonal matrix with Ti;i 1 = Ti;i+1 = i and Ti;i = 0, and
E := [I +2D] 1T which is a tridiagonal matrix with Ei;i 1 = Ei;i+1 = i and Ei;i = 0. Then
the system (2.1) can be written as

I + 2D   T

(un+1) =

I   2(1   )D + (1   )T

(un);
or equivalently

I + 2D
 
I   E

(un+1) =

I   2(1   )D + (1   )T

(un): (2.2)
Proposition 2.1. Suppose that the function g is bounded on N and there is constant   > 0
such that jij    for every i 2 Z, then the Z  Z matrix I   E is invertible and its inversion
B is given by
B := I +
1 X
n=1
En: (2.3)
And therefore, there is a unique solution for system (2.1) (or (2.2)) given by
(un+1) = B

I + 2D
 1 
I   2(1   )D + (1   )T

(un): (2.4)
Proof. First, (i)i2N dened by i =

2
i t
x2 are uniformly bounded by   :=  
2t
x2 since (i)i2Z
are uniformly bounded by  . It follows that i = i
1+2i   :=  
1+2  < 1
2.
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14 J.F. Bonnans and X. Tan
Denote by B(N) the space of all bounded functions dened on N, then E can be viewed as an
operator on B(N) and its L1 norm is dened by
kEk1 := sup
u2B(N);u6=0
jEuj1
juj1
:
Clearly, kEk1  2 < 1, and therefore, B in (2.3) is well dened and one can easily verify that
B is the inverse of

I   E

.
Denition 2.2. A numerical scheme for equation (1.1) given by u
n+1
i = Th[un]i is said to be
L1 monotone if
u
1;n
i  u
2;n
i ; 8i 2 Z ) Th[u1;n]i  Th[u2;n]i; 8i 2 Z:
Remark 2.3. It is well-known that in the case  = 1, system (2.2) is an implicit scheme, and
it is automatically L1 monotone for every discretization (t;x). When  < 1, a sucient
condition to gurantee the L1 monotonicity of system (2.2) is the CFL(Courant-Friedrichs-
Lewy) condition
  :=
 2t
x2 
1
2(1   )
; for   := sup
i2Z
i: (2.5)
The CFL condition is a sucient condition for the monotonicity of  scheme, and it implies
a discretization ratio t = O(x2). We shall conrm that this ratio is necessary to guarantee
the monotonicity in the following.
3 The necessary and sucient condition
Let i :=
(1 )i
1+2i =
(1 )
 i and bi;j be elements of the matrix B, i.e. B = (bi;j)(i;j)2Z2. It is
clear that bi;j  0 for every (i;j) 2 Z2 by the denition of B in (2.3). Therefore, it follows from
(2.4) that the necessary and sucient condition for monotonicity of system (2.1) can be written
as :
bi;j 1j 1 + bi;j
  1
1 + 2j
  2j

+ bi;j+1j+1  0; 8(i;j) 2 Z2: (3.1)
Theorem 3.1. Suppose that jij    < 1 for every i 2 Z, and let  2 (0;1). Then the
necessary and sucient condition of monotonicity for the  scheme in (2.1) is
i =
i
2t
x2 
1
2(1   )
+
bi;i   1
2(1   )
; 8i 2 Z: (3.2)
Proof. First, since B is the inversion of I   E, we have B

I   E

= I, and it follows that
bi;j 1j 1 + bi;j+1j+1 =
(
bij   1; for i = j;
bij; for i 6= j:
Therefore, in case that i 6= j, (3.1) is equivalent to:
bi;j
1   

+
1
1 + 2j
  2j

 0: (3.3)
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1Monotonicity of theta-scheme 5
Since bi;j  0, the inequality (3.3) holds as soon as
(1   )(1 + 2j) +    2(1   )j  0;
which is always true.
In case that i = j, (3.1) is equivalent to:
bi;i
1   

+
1
1 + 2i
  2i

 
1   

 0;
i.e.
i 
1
2(1   )
+
bi;i   1
2(1   )
:
which is the required inequality (3.2).
Remark 3.2. Since bi;i < 1 for every i 2 Z, it follows from Theorem 3.1 that the ratio
t = O(x2) is necessary for the monotonicity of  scheme ( 0 <  < 1) as soon as i 6= 0
for some i 2 Z.
4 The heat equation
In this section, les us suppose that (x)  0 with a positive constant 0, then the diusion
equation turns to be the heat equation:
@tv   2
0 D2
xxv = 0; (t;x) 2 R+  R: (4.1)
In this case, we can compute bi;i and then get an explicit formula for the monotonicity condition
(3.2). Let
A be a Z  Z tridiagonal matrix such that Ai;i 1 = Ai;i+1 = 1 and Ai;i = 0; (4.2)
then clearly, E = A with  = 
1+2 < 1
2,  =

2
0t
x2 and
B =
h
I   A
i 1
:=
1 X
n=0
nAn: (4.3)
Lemma 4.1. Denote by An the n th exponentiation of matrix A in (4.2) for n 2 N, we
rewritten An = (a
(n)
i;j )(i;j)2ZZ. Then,
a
(n)
i;j =
(
C
(n+i j)=2
n ; if
n+i j
2 2 Z \ [0;n];
0; otherwise.
(4.4)
Proof. We proceed by induction. First, it is clearly that (4.4) holds true for n = 1. Suppose that
the (4.4) is true in case that n = m. Since Am+1 = Am A, we then have a
m+1
i;j = am
i;j 1+am
i;j+1.
It follows from Ck
n = Ck
n 1 + C
k 1
n 1 that (4.4) holds still true for the case n = m + 1. We then
conclude the proof.
By Lemma 4.1 and equality (4.3), we get bi;i =
P1
k=0 Ck
2k2k with the convention that C0
0 := 1.
As a result, the monotonicity condition (3.2) of  scheme reduces to
 
1
2(1   )
+
f()
2(1   )
; (4.5)
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16 J.F. Bonnans and X. Tan
where
f(x) :=
1 X
k=1
Ck
2k x2k for  
1
2
< x <
1
2
:
Remark 4.2. We can verify that Ck
2k  1 p
k4k as k ! 1 by Stirling's formula, thus the radius
of convergence of f(x) is 1
2.
Let us now compute the function f(x). Since Ck
2k = 2 2k 1
k C
k 1
2(k 1), it follows that for jxj < 1
2,
f0(x) =
1 X
k=1
2k Ck
2k x2k 1 =
1 X
k=1
4 (2k   1) C
k 1
2(k 1) x2k 1
= 4x +
1 X
k=1
(8k + 4) Ck
2k x2k+1 = 4x + 4x2f0(x) + 4xf(x):
We are then reduced to the ordinary dierential equation
f0(x) =
4x
1   4x2 (f(x) + 1); with f(0) = 0;
whose solution is f(x) = 1 p
1 4x2   1. Inserting this solution into (4.5), and by a direct manip-
ulation, it follows that (4.5) is equivalent to
 
1
2(1   )
+

4(1   )2: (4.6)
We get the following theorem:
Theorem 4.3. The necessary and sucient condition for the L1 monotonicity of  scheme
(0 <  < 1) of the heat equation (4.1) is
 =
2
0t
x2 
1
2(1   )
+

4(1   )2: (4.7)
Remark 4.4. In particular, when  = 1
2, the CFL condition is  =

2
0t
x2  1, and the necessary
and sucient condition of the monotonicity is  =

2
0t
x2  3
2.
References
[1] G. Allaire, Numerical analysis and optimization, Oxford University Press, Oxford, 2007.
An introduction to mathematical modelling and numerical simulation, Translated from the
French by Alan Craig.
[2] G. Barles and P.E. Souganidis, Convergence of approximation schemes for fully nonlinear
second order equations, Asymptotic Anal., 4(3):271-283, 1991.
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1Centre de recherche INRIA Saclay – Île-de-France
Parc Orsay Université - ZAC des Vignes
4, rue Jacques Monod - 91893 Orsay Cedex (France)
Centre de recherche INRIA Bordeaux – Sud Ouest : Domaine Universitaire - 351, cours de la Libération - 33405 Talence Cedex
Centre de recherche INRIA Grenoble – Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier
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615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex
Centre de recherche INRIA Paris – Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex
Centre de recherche INRIA Rennes – Bretagne Atlantique : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex
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Éditeur
INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)
http://www.inria.fr
ISSN 0249-6399
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Convergence of approximation schemes for fully nonlinear second order equations,

Numerical analysis and optimization,

Monotonicity condition for the $\theta$-scheme for diffusion equations

http://hal.inria.fr/docs/00/63/44/17/PDF/RR-7778.pdf

Monotonicity condition for the $\theta$ -scheme for diffusion equations

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Research Papers in Economics

Monotonicity condition for the θ\thetaθ-scheme for diffusion equations

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Research Papers in Economics

Monotonicity condition for the $\theta$ -scheme for diffusion equations