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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

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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.

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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

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Example

The solution of the ODE

is .

The solution tends to infinity as t tends to 1.

1)0(,)()(' 2 ututu

ttu

1

1)(

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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,

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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.

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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

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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

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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

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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.

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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

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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

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220

0

))(())(2(

)(4),(

xutxtHu

xutxu

0)(&0)(| 00 xHuxux

)/2( MT

.|)(sup 0 xxHuM

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Let

Then

Thus, the solution is given by

.cos)(0 xxu

.sin)(0 xxHu

.cos)sin2(

cos4),(

222 xtxt

xtxu

0t

8.1t

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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.

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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

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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.

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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.

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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

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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

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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.

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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

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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.

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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

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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 ,

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1)( KKzzzG

: constant decided by I.D.

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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

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Numerical examples

0)(;)sin(300)(,0016.0,1

,1min,002.0 2/1

xgxxfu

thn

n

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Numerical examples

)2sin(200)(

;)sin(10)(

)(

,8.0/,5002/1

xxg

xxf

ssH

hN

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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

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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.

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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

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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

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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.

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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.

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0)(),sin(300)( xgxxf )2sin(20)(),sin(500)( xxgxxf

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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.

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