1
Chien-Hong Cho ( 卓建宏 )(National Chung Cheng University)( 中正大學 )
2010/11/11 in NCKU
On the finite difference approximation for hyperbolic blow-up problems
2
Contents A brief introduction for blow-up problems. Review on the parabolic blow-up problem. CLM equation
Spectral method Finite difference method
Numerical results of the semi-linear wave equation. Sufficient condition for blow-up. Numerical scheme for semi-linear wave
equation and blow-up result Future work.
3
Blow-up problem for Broadly speaking, singularity occurs in the
solution, its derivative, or its high-order derivatives in finite time.
Here, we use blow-up in a narrow sense. We say that the solution of an initial-value problem blows up in finite time T if the solution becomes infinity as where T is called the blow-up time.
),,( xxxt uuuFu
,Tt
4
Example
The solution of the ODE
is .
The solution tends to infinity as t tends to 1.
1)0(,)()(' 2 ututu
ttu
1
1)(
5
More examples
)1( uuu xxt
Semi-linear heat equation (Fujita, Weissler, Friedman, Mcleod, …)
The porous medium equation with a nonlinear source (Samarskii, Galaktionov, …)
)1()( uuu xxt
Semi-linear wave equation (John, Kato, Glassey, Levine, …)
)1( uuu xxtt
CLM equation (Constantin, Lax, Majda, …)
Huuut Hyperbolic type(H denotes the Hilbert transform)
Problem Occurred in Numerical Calculation for a Blow-up Problem
Ex: Consider the ODE 1)0(,2 uuu
21
nnn
ut
uu
and the difference scheme
The numerical solution exists for all That is, the numerical solution exists globally, or equivalently, the numerical solution does not blow up in finite time T.
.0n
.21 nnn utuu or equivalently,
7
Main purposeWe want to construct an appropriate finite difference scheme for blow-up problems. By the word ‘appropriate’ we mean a scheme that satisfies the following:
• convergence.
• blow-up in finite time.
• convergence for the numerical blow-up time.
8
1-dim semi-linear heat equation
)0,10(,1 txuuu xxt
)10(0)(,0)(),0(:.. 00 xxuxuxuCI
0)1,()0,(:.. tutuCB
where .1
Then the solution blows up in finite time T :
(Fujita(1966), Weissler(1984), Friedman and Mcleod(1985), etc..)
.),( Ttastu
9
Finite difference scheme
.,1,0,0
.1,...,2,1),(
.,1,0;1,,1,)(2
0
00
12
111
nuu
Njxuu
nNjuh
uuu
t
uu
nN
n
jj
nj
nj
nj
nj
n
nj
nj
where ,1Nh nju ),( jn xtu: the approximation of at ),( jn xt
;)(
1,1min
p
nnuH
t p
nu
pu
puh
njNj
pN
j
pnj
|,|max
1,||
11
11
1
jhx j : the spatial mesh point.
nt
1
0
0,0n
knt
n
: the time grid point.2
12
h
: fixed.
10
0t1t
1 nt
0
lim),(n
nnn
tthT
1
01
n
knn tt
0t1t
nt1nt
(numerical blow-up time)
)(
1,1min
p
nnuH
t
))((1
,1min: ssHu
texp
nn
Blow-up cannot be reproduced if we use uniform nt )( nt
11
The semi-linear case 0
.)(lim
sHs
)(
1
sH
sss
ds
ssHH )1(1 )(
1
(a) H is monotone increasing and
(b) is monotone increasing.
(c)
Then
Namely, the finite difference solution blows up in finite
time & the numerical blow-up time converges to T.
.),(lim&),(0
0
ThTthTh
nn
Let T be the blow-up time. Assume that H satisfies
12
Remark This was a toy problem. Although the
convergence of the numerical blow-up time is proved, the convergence rate still remains open.
More fluid related problem? The finite difference approximation becomes much more difficult for blow-up problems of hyperbolic type than the parabolic ones. We consider two simple equations -- CLM equation and the semi-linear wave equation, which show the difficulties. Our recent results will also be reported.
13
CLM equation
The Constantin-Lax-Majda equation
)0,(
txHuut
u
Here H denotes the Hilbert transform:
)()(
2cot
2
1)( xdyyu
yxxHu
where the integral denotes Cauchy’s principal part.
)()(),0( 0 xxuxu
u is periodic in x and
0),( dxxtu
14
Remark A 1-dim model for the vorticity equation.
(see Constantin, Lax, Majda (1985))
The solution to CLM equation is given explicitly by
Thus, the solution blow-up in finite time if and only if
is nonempty, where
20
220
0
))(())(2(
)(4),(
xutxtHu
xutxu
0)(&0)(| 00 xHuxux
)/2( MT
.|)(sup 0 xxHuM
15
Let
Then
Thus, the solution is given by
.cos)(0 xxu
.sin)(0 xxHu
.cos)sin2(
cos4),(
222 xtxt
xtxu
0t
8.1t
16
Spectral method for CLM equationLet N be a positive integer.
Consider the following spectral approximation :
)( NNNN HuuPt
u
where, for is given by
N
nnnN nxtbnxtatuP
1
)cos)(sin)(()(
Nn
nn Pnxtbnxtatu ,)cos)(sin)(()(1
Nu denotes the approximation of the solution u.
17
For simplicity, we consider the case of odd function:
1
0 sin),0(n
n nxaxu
1
sin)(),(n
n nxtaxtuwhence
Then we have
,)(,2
1)(,0)( 213
2121 aata
dt
data
dt
dta
dt
d
which tells that is a polynomial in t of order (n-1).
,
2
)()(,)(
2010
22011 t
aataata
)(tan
Theorem does not blow up for all N.),( xtuN
18
Finite difference method
Let .),(),(),( xtiHuxtuxt
Then one has .2
1 2idt
d (Constantin, Lax, Majda, 1985)
We define a finite difference approximation in such a way that approximates for . Here, is by definition the imaginary part of . Then we have
)(tn )/2,( NntNn 1
)(tnnHu
.)0(2
)0(2)(
ti
it
n
nn
The solution does not blow up.
19
Other difference scheme Discretize the Hilbert transform in a different way. Difficulties: (a) The equalities derived from the properties of
the Hilbert transform are not true in the discrete version.
(b) It is difficult to show that the maximum values of the finite difference solution propagate to the zero of I.D. due to the complexity of the discretized Hilbert transform.
20
1-dim semi-linear wave equation
The solution blows up in finite time T. That is,
Many sufficient conditions for blow-up were given. For example, Glassey, John, Kato, Levine, etc..
)0,10(0,1 txuuu xxtt
10),(),0(;)(),0(:.. xxgxuxfxuCI t
0,0)1,()0,(:.. ttutuCB
2,1 u
.),( Ttastu
21
Levine’s Result (1974)
Let the nonlinear term be and the initial data satisfy2u
Then the solution blows up in finite time T.
.,|),(|max),(..:..10
TtasxtututsTeix
1
0
21
0
21
0
3 ))(())('(2
1))((
3
1dxxgdxxfdxxf
22
Another sufficient condition for blow-up (Cho)
1
0)sin(),(
2)( dxxxtut Let
Assume that
.0)0('
)0(0)sin()(,2)sin()(
21
0
1
0
dxxxgdxxxf
Then blows up in finite time T.
)(lim..:.. ttsTeiTt
)(t
Independent of the Levine’s condition.
More convenient for numerical analysis.
23
By Jensen’s inequality,
22 ))(()()('' ttt
This implies the blow-up of .
2
1
3))((3
2)('
Ktat
Sketch of the proof
Multiply to both sides, one has)(' t
where K is a constant decided by the initial data.
)(t
24
Numerical Scheme
)1,,2,1()()(,)(
)2,1,0(0
)2,1,0,1,2,1(
21
010
0
2
2
11
1
11
Njxgtxfuxfu
nuu
nNj
uh
uuu
t
uu
t
uu
jjjjj
nN
n
nj
nj
nj
nj
n
nj
nj
n
nj
nj
n
nju
1nju
1nju
nt
1 nt
nn
nj
nj
t
uu
1
1
1
n
nj
nj
t
uu
;2
1 nnn
tt
where ,1Nh nju ),( jn xtu: the approximation of at ),( jn xt
;)(
1,1min
p
nnuH
t p
nu
pu
puh
njNj
pN
j
pnj
|,|max
1,||
11
11
1
jhx j : the spatial mesh point.
nt
1
0
0,0n
knt
n
: the time grid point.
1/ h : fixed.
25
Remark
If we put , then the scheme
22
11
1
11 21 nj
nj
nj
nj
n
nj
nj
n
nj
nj
n
uh
uuu
t
uu
t
uu
nt
is in fact
22
11
2
11 22 nj
nj
nj
nj
nj
nj
nj u
h
uuuuuu
26
Moreover, assume that converges to u (t, x) while u is smooth. Then the numerical blow-up time also converges. Namely,
Theorem (Cho)
Define
Assume that H satisfies
nku
.),(lim0
ThTh
c
dssHsG
sGtsc .
)()(
)('..0
(H1) is monotone increasing.(H2) (H3) .)(lim
sH
s
)(
)(
sH
sGss
)2/30(,)(: ssHEX
.),(0
nnthT
Then the solution blows up in finite time. That is,
002
23 ,
46
1)( KKzzzG
: constant decided by I.D.
27
Convergence while u is smooth Suppose the solution u blows up at t = T.
u is smooth in
ujn converges to u in That is,
as long as
.)(0 1 TTt
.0 1Tt
.1Ttn
0|),(|maxlim1,,10
jn
nj
Njhxtuu
28
Numerical examples
0)(;)sin(300)(,0016.0,1
,1min,002.0 2/1
xgxxfu
thn
n
29
Numerical examples
)2sin(200)(
;)sin(10)(
)(
,8.0/,5002/1
xxg
xxf
ssH
hN
30
Difficulty in proving convergence
Two level time meshes appear in the scheme and thus their relation plays an important role in the stability.
To show the convergence (while u is smooth), we need some a priori estimates or stability in some norms, which can be derived from the well-known “energy conservation property” of the wave equations for the uniform time mesh ( ), while the energy need not be conserved for non-uniform time mesh.
nt
nn tt ,1
31
Remark
In fact, to prove the convergence (while u is smooth) , we only need the stability for the finite difference solution of the linear wave equation. But for the non-uniform time mesh, only a little is known.
Samarskii & Matus’s scheme(2001); Matsuo’s ( 松尾宇泰 ) scheme(2007): strong restriction on the spatial part of their scheme.
32
Linear wave equation xxtt uu
10,)(),0(,)(),0(
0,0)1,()0,(
0,10,),(),(
xxgxuxfxu
ttutu
txxtuxtu
t
xxtt
We consider the initial-boundary-value Problem
Then we have
0),(2
1),(
2
11
0
22
dxxtuxtu
dt
dxt
That is,
1
0
221
0
22 ),0(2
1),0(
2
1),(
2
1),(
2
1dxxuxudxxtuxtu xtxt
,0 t
33
Finite difference scheme
We consider
2
11
1
11 21
h
uuu
t
uu
t
uu nj
nj
nj
n
nj
nj
n
nj
nj
n
and the well-known discrete energy
,2
1 11
1
11
11
1
21
h
uu
h
uuh
t
uuhE
nj
nj
N
j
nj
nj
N
j n
nj
njn
h
which corresponds to the energy .2
1 1
0
21
0
2
dxudxu xt
Then we have .
,
,
,
11
11
11
nnnh
nh
nnnh
nh
nnnh
nh
ttifEE
ttifEE
ttifEE
34
Samarskii’s and Matsuo’s scheme Samarskii et. al considered a difference equation
in a finite dimensional space and then applied to the linear wave equation.
Matsuo used the so-called discrete variational method. Namely, he defined the discrete energy first, and then derived the finite difference scheme whose solution conserves the given discrete energy.
Neither of which can be applied to our scheme.
0)()()( ˆ nnt
ntt ytAytBytD
Semi-discrete scheme
0)()(
))(()()(2)(
)(
0
22
11
2
2
tutu
tuh
tutututu
dt
d
N
jjjj
j
Moreover, we have that, for any
hT
h
N
jjj Ttasxtu
1
1
sin)(
There exist such that
hTCxtutu jjNj
)(),()(max 01,,1
,0 TT
and that .lim0
TThh
Theorem (Cho)
Remark It should be noted that we can prove the
convergence of the numerical solution for the semi-discrete scheme by using the energy conservation property of the linear wave equation,
which does not hold in the full-discrete case.
0)()()(
1
2
1
21
1
N
n
jjN
n
j
h
tutuh
dt
tduh
dt
d
2-nd order ODE
1001
00
2
1
11
,)(1
atau
auu
t
uu
t
uu n
n
nn
n
nn
n
We consider the 2-nd ODE blow-up problem
.0)0(',0)0(),()('' 102 auaututu
where
and the finite difference analogue
.2
1,
)(
1,1min 1
nnnnn ttuH
t
2)('' utu
Theorem (Cho)Under certain assumption on H, we have
,)( 21 CTTCT
where
T denotes the blow-up time of
1
)(n
ntT denotes the numerical blow-up time.
).()('' 2 tutu
0, 21 CC are constants independent of .
C. H. Cho, On the convergence of numerical blow-up time for a 2nd order nonlinear ordinary differential equation, Appl. Math. Lett., 24, 2011, 49-54.
39
Future work Stability for finite difference schemes of the linear
wave equation with non-uniform time meshes.
A rigorous proof for the convergence of the finite difference solution to the semi-linear wave equation.
Convergence order for the numerical blow-up time.
40
0)(),sin(300)( xgxxf )2sin(20)(),sin(500)( xxgxxf
41
Thank you for your attention.
References C.-H. Cho, S. Hamada, and H. Okamoto, On the finite
difference approximation for a parabolic blow-up problem, Japan J. Indust. Appl. Math., 24 (2007), pp 131-160.
C.-H. Cho, A finite difference scheme for blow-up problems of nonlinear wave equations, Numerical Mathematics:TMA, 3 (2010), pp. 475-498.
C.-H. Cho, On the convergence of numerical blow-up time for a second order nonlinear ordinary differential equation, Appl. Math. Lett., vol.24 no.1 (2011), pp 49-54.
42