Journal of Inequalities in Pure and
Applied Mathematics
NORMIC INEQUALITY OF TWO-DIMENSIONAL
VISCOSITY OPERATOR
volume 6, issue 4, article 121,
2005.
YUEHUI CHEN
Department of Mathematics
Zhangzhou Teachers College
Zhangzhou 363000, P.R. China.
Received 22 February, 2005;
accepted 01 September, 2005.
Communicated by: S.S. Dragomir
EMail: yuehuich@21cn.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
A relation between the coefficients of Legendre expansions of two-dimensional
function and those for the derived function is given. With this relation the normic
inequality of two-dimensional viscosity operator is obtained.
2000 Mathematics Subject Classification: 65N35.
Key words: Legendre expansion, Two-dimension, Viscosity operator.
Normic Inequality of
Two-Dimensional Viscosity
Operator
The author wants to express his deep gratitude to Prof. Benyu Guo for research inspiration and is greatly indebted to the referee for improving the presentation. The work
is supported by the Science Foundation of Zhangzhou Teachers College (JB04302).
Yuehui Chen
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Contents
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Notations and Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
4
6
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1.
Introduction
Spectral methods employ various orthogonal systems of infinitely differentiable
functions to represent an approximate projection of the exact solution sought.
The resulting high accuracy of spectral algorithms was a major motivation behind their rapid development in the past three decades, e.g., see Gottlieb and
Orszag [1] and Guo [2] .
For nonperiodic problems, it is natural to use Legendre spectral methods or
Legendre pseudospectral methods. More attention has been paid to these two
methods recently due to the appearance of the Fast Legendre Transformation. In
studying the spectral methods for nonlinear conservation laws, we have to face
those equations whose solutions may develop spontaneous jump discontinuities,
i.e., shock waves. To overcome these difficulties, the spectral viscosity (SV)
method was introduced by Tadmor [3] . Maday, Ould Kaber and Tadmor [4]
firstly considered the nonperiodic Legendre pseudospectral viscosity method
for an initial-boundary value problem, and Ma [5] , Guo, Ma and Tadmor [6]
recently developed the nonperiodic Chebyshev-Legendre approximation. So
far, however, few works have been done in multiple dimensions. This paper will
study the relation between the coefficients of Legendre expansions of the twodimensional function and those for the derived function, and gives the normic
inequality of the two-dimensional SV operator, which plays important roles in
the SV method.
Normic Inequality of
Two-Dimensional Viscosity
Operator
Yuehui Chen
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2.
Notations and Lemmas
Let x = (x1 , x2 ), Λ = (−1, 1)2 . We define the space Lp (Λ) and its norm k · kLp
in the usual way. If p = 2, we denote the norm of space L2 (Λ) by k · k, that is
Z
kvk =
2
21
|v(x)| dx
.
Λ
Let N be the set of all non-negative integers and PN be the set of all algebraic
polynomials of degree at most N in all variables. Let l = (l1 , l2 ) ∈ N 2 , |l| =
max{l1 , l2 }, the Legendre polynomial of degree l is Ll (x) = Ll1 (x1 )Ll2 (x2 ).
The Legndre transformation of a function v ∈ L2 (Λ) is
Sv(x) =
∞
X
v̂l Ll (x),
Yuehui Chen
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|l|=0
with the Legendre coefficients
Z
1
1
v̂l = l1 +
l2 +
v(x)Ll (x)dx,
2
2
Λ
Normic Inequality of
Two-Dimensional Viscosity
Operator
Contents
|l| = 0, 1, . . . .
Lemma 2.1 ([2]). For any φ ∈ PN , 2 ≤ p ≤ ∞ and |k| = m, we have
k∂xk φkLp ≤ cN 2m kφkLp .
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Here, c is a generic positive constant independent of any function and N .
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A viscosity operator Q is defined by
(2.1)
Qv(x) :=
N
X
q̂l v̂l Ll (x),
|l|=0
v=
∞
X
v̂l Ll (x).
|l|=0
Here, q̂l are the so-called viscosity coefficients,

for |l| ≤ m
 q̂l = 0
 q̂l ≥ 1 −
m2
for m < |l| ≤ N.
l12 +l22
Observe that the Q operator is activated by only the high mode numbers, ≥ m.
In particular, if we let m −→ ∞, the Q operator is spectrally small (in the sense
that kQvkH −s ≤ cm−s kvk).
Let R denote the corresponding low modes filter
(2.2)
Rv(x) :=
N
X
r̂l v̂l Ll (x),
here r̂l = 1 − q̂l .
|l|=0
Normic Inequality of
Two-Dimensional Viscosity
Operator
Yuehui Chen
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Clearly,

 r̂l = 1
 r̂l ≤
m2
l12 +l22
for
|l| ≤ m
for
m < |l| ≤ N.
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3.
Main Results
Firstly , we consider the relation between the coefficients of Legendre expansions of v(x) and those for ∂x v(x).
(1)
Theorem 3.1. For any v(x) ∈ H 1 (Λ), let v̂k , v̂k be the coefficients of Legendre
expansions of v(x), ∂x v(x), |k| = 0, 1, 2, . . . ,
(1)
v̂k
=
(1)
v̂(k1 ,k2 )
= (2k1 + 1)
∞
X
v̂(p,k2 ) + (2k2 + 1)
p=k1 +1
k1 +p odd
∞
X
v̂(k1 ,q) .
q=k2 +1
k2 +q odd
Proof. By the property of the one-dimensional Legendre polynomial:
(2k + 1)Lk (x) = L0k+1 (x) − L0k−1 (x),
we have
k ≥ 1,
Normic Inequality of
Two-Dimensional Viscosity
Operator
Yuehui Chen
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L0k (x) =
k−1
X
(2l + 1)Ll (x),
k = 0, 1, 2, . . . .
l=0
k+l odd
Then, for x = (x1 , x2 ),
=
l1 =0
k1 +l1 odd
(2l1 + 1)Ll1 (x1 )Lk2 (x2 ) +
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∂x Lk (x) = L0k1 (x1 )Lk2 (x2 ) + Lk1 (x1 )L0k2 (x2 )
kX
1 −1
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kX
2 −1
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(2l2 + 1)Ll2 (x2 )Lk1 (x1 );
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l2 =0
k2 +l2 odd
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∂x v =
∞
X
v̂k ∂x Lk (x)
|k|=0
=
∞
X
∞
X

k1 −1
 X
v̂(k1 ,k2 ) 
(2l1 + 1)Ll1 (x1 )Lk2 (x2 )
l1 =0
k1 +l1 odd
k1 =0 k2 =0
kX
2 −1
+


(2l2 + 1)Ll2 (x2 )Lk1 (x1 ) .
Normic Inequality of
Two-Dimensional Viscosity
Operator
l2 =0
k2 +l2 odd
On the other hand,
∂x v =
∞
X
Yuehui Chen
(1)
v̂k Lk (x) =
|k|=0
∞ X
∞
X
(1)
v̂(k1 ,k2 ) Lk1 (x1 )Lk2 (x2 ).
k1 =0 k2 =0
Contents
We can obtain
(1)
v̂(k1 ,k2 )
= (2k1 + 1)
∞
X
v̂(p,k2 ) + (2k2 + 1)
p=k1 +1
k1 +p odd
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∞
X
v̂(k1 ,q) .
q=k2 +1
k2 +q odd
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Specially, for any v ∈ PN , we have
(3.1)
(1)
v̂(k1 ,k2 )
= (2k1 + 1)
N
X
p=k1 +1
k1 +p odd
v̂(p,k2 ) + (2k2 + 1)
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N
X
q=k2 +1
k2 +q odd
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v̂(k1 ,q) .
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Let Jk,N = {j k + 1 ≤ j ≤ N, k + j odd}, then
(1)
v̂(k1 ,k2 ) = (2k1 + 1)
X
v̂(p,k2 ) + (2k2 + 1)
p∈Jk1 ,N
X
v̂(k1 ,q)
∀ v ∈ PN
q∈Jk2 ,N
Theorem 3.2. Consider the SV operator Q = Qm , (2.1) with the parametrization and operator R, (2.2). For any φ ∈ PN , the following inequalities hold:
k∂x (Rφ)k2 ≤ cm4 ln N kφk2
k∂x φk2 ≤ 2k∂x (Qφ)k2 + cm4 ln N kφk2
k∂x (Qφ)k2 ≤ 2k∂x φk2 + cm4 ln N kφk2
Normic Inequality of
Two-Dimensional Viscosity
Operator
Yuehui Chen
Proof. We decompose ∂x (Rφ(x)) = A1 (x) + A2 (x), where




m
N
X
X
A1 (x) := ∂x 
r̂k φ̂k Lk (x) , A2 (x) := ∂x 
r̂k φ̂k Lk (x) .
|k|=0
|k|=m+1
By Lemma , k∂x φk ≤ cN 2 kφk, ∀ φ(x) ∈ PN , and hence kA1 (x)k2 ≤ cm4 kφk2 .
(0)
(d)
Further let Jk,N = {j j ∈ Jk,N , j > m}, Jk,N = {j j ∈ Jk,N , max{j, kd } >
m}, here, m ∈ N , d = 1, 2. Then

N
X
X
kA2 (x)k2 ≤
(2k
+
1)
r̂(p,k2 ) φ̂(p,k2 )
 1
|k|=0
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(2)
p∈Jk ,N
1
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2
+(2k2 + 1)
X

r̂(k1 ,q) φ̂(k1 ,q)  kLk k2
(1)
2 ,N
q∈Jk
≤ 8 (A2,1 + A2,2 ) ,
in which
A2,1
=
N X
N
X
 X

(2k1 + 1)2 
r̂(p,k2 ) φ̂(p,k2 ) 
k1 =0 k2 =0
=
N
X
N
X
k1 =0 k2
(2)
1 ,N
Yuehui Chen
2

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2k1 + 1  X

r̂(p,k2 ) φ̂(p,k2 ) 

2k2 + 1
(2)
=0
Contents
p∈Jk
1 ,N


N X
N
X
X
2k1 + 1  X

2
−2  
|r̂
|
kL
k
|φ̂(p,k2 ) |2 kL(p,k2 ) k2 


(p,k2 )
(p,k2 )
2k
+
1
2
(2)
(2)
k =0 k =0
1
≤
1
(2k1 + 1)(2k2 + 1)
p∈Jk

≤
Normic Inequality of
Two-Dimensional Viscosity
Operator
2

N
X
2
p∈Jk
1 ,N
N
X
k1 =0 k2
p∈Jk

2k1 + 1  X m4 (2p + 1)(2k2 + 1) X

|φ̂(p,k2 ) |2 kL(p,k2 ) k2 

2 2
2
2k2 + 1
4(p + k2 )
(2)
(2)
=0
1 ,N
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1 ,N

p∈Jk
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≤ c1 m4
≤ c2 m
4

N
X


N
X
k2 =0
k1 =0
N
X
N
X
k2 =0
p=0
≤ 2c2 m4
N X
N
X


 X
(2k1 + 1) 
(2)
1 ,N
p∈Jk
2p + 1 

(p2 + k22 )2
!
2
2
|φ̂(p,k2 ) | kL(p,k2 ) k
N
X
m−2
k2 =0 p=0
≤ c3 m4 ln N
p=0
p−3
(0)
1 ,N
p∈Jk
m
X
N
X
(2k1 + 1) +
k1 =0
N X
N
X

!

|φ̂(p,k2 ) |2 kL(p,k2 ) k2 
X
(2k1 + 1)
k1 =0
|φ̂(p,k2 ) |2 kL(p,k2 ) k2
N
X
!
(2k1 + 1)k1−2
k1 =m+1
|φ̂(p,k2 ) |2 kL(p,k2 ) k2
Yuehui Chen
k2 =0 p=0
2
4
Normic Inequality of
Two-Dimensional Viscosity
Operator
≤ c3 m ln N kφk ;
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Similarly,
A2,2 =
N
X
N
X
k1 =0 k2
2

2k2 + 1  X

r̂(k1 ,q) φ̂(k1 ,q)  ≤ c4 m4 ln N kφk2 .

2k1 + 1
(1)
=0
q∈Jk
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Thus
kA2 (x)k2 ≤ c5 m4 ln N kφk2 .
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k∂x (Rφ)k2 ≤ 2kA1 (x)k2 + 2kA2 (x)k2 ≤ cm4 ln N kφk2 .
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Since ∂x φ ≡ ∂x (Qφ) + ∂x (Rφ), the desired estimates follow.
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Remark 1. The theorem shows the equivalence of the H 1 norm before and after
application of the SV operator , Q = Qm for moderate size of mN N 1/4 .
This holds despite the fact that for m = mN ∼ cN β −→ ∞, 0 < 4β < 1, the
corresponding SV operator Qm is spectrally small.
Normic Inequality of
Two-Dimensional Viscosity
Operator
Yuehui Chen
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References
[1] D. GOTTLIEB AND S.A. ORSZAG, Numerical analysis of spectral methods:Theory and applications, SIAM, Philadelphia, 1977.
[2] B.-Y. GUO, Spectral Methods and their Applications, Singapore: World
Scientific, 1998.
[3] E. TADMOR. Convergence of spectral methods for nonlinear conservation
laws, SIAM J. Numer. Anal., 26 (1989), 30–44.
[4] Y. MADAY, S.M. OULD KABER, AND E. TADMOR, Legendre pseudospectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., 30 (1993), 321–342.
[5] H.-P. MA, Chebychev-Legendre pseudospectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., 35 (1998), 869–892.
[6] B.-Y. GUO, H.-P. MA AND E. TADMOR, Spectral vanishing viscosity
method for nonlinear conservation laws, SIAM J. Numer. Anal., 39 (2001),
1254–1268.
Normic Inequality of
Two-Dimensional Viscosity
Operator
Yuehui Chen
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