1. Trang chủ >
2. Công Nghệ Thông Tin >
3. Kỹ thuật lập trình >

# 5 Cauchy's Problem for a Quasi-Linear First-Order System

Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (6.6 MB, 489 trang )

208

8 Wave Propagation

It is necessary to clearly specify the meaning of indices in Aij h .x; u/:

i D 1; : : : ; n refers to the independent variable xi I

j D 1; : : : ; m refers to the equation;

h D 1; : : : ; m refers to the function uh :

If the matrices

0

Ai D @

Ai11

Ai1m

Aim1

Aimm

1

0

A;

cD@

c1

1

A;

(8.18)

cm

are introduced, system (8.16) can be written in the following concise way:

A1

@u

C

@x1

C An

@u

D c;

@xn

or in the still more compact form

Ai

@u

D c:

@xi

(8.19)

The Cauchy problem for the system (8.19) consists of finding a solution of (8.19)

which satisfies Cauchy’s data

u.x/ D u0 .x/

8x 2 †n 1 ;

(8.20)

where †n 1 is a regular surface of
By proceeding as in the previous section, we introduce the coordinates

. 1 ; : : : ; n /, so that †n 1 is represented by the equation n D f .x/ D 0. When

expressed in these coordinates, system (8.19) becomes

Ai

@u

@u @ h

D Ai

Dc

@xi

@ h @xi

or

Ã

Â

@ h @u

D c;

Ai

@xi @ h

(8.21)

whereas Cauchy’s data assume the form

u. 1 ; : : : ;

n 1 ; 0/

D uQ 0 . 1 ; : : : ;

n 1 /:

(8.22)

8.6 Classification of First-Order Systems

209

According to the assumption that Aij h .x; u/ and ci .x; u/ are analytic functions of

their arguments, the solution can be expanded in a power series of n ; i.e.,

Â

u.x/ D u0 . 1 ; : : : ;

n 1/

C

@u

@ n

Ã

x0

1

nC

2

Â

@2 u

@ n2

Ã

2

n

C

;

(8.23)

x0

where x0 2 †n 1 . The system (8.21) can be written in the form

Â

Ã

@u

i @f

A

D F;

@xi x0 @ n

where F is expressed from Cauchy’s data. This form highlights the fact that

.@u=@ n /x0 is determined if and only if

i @f

det A

Ô 0:

@xi x0

(8.24)

It is easy to verify that this condition allows us to determine all the derivatives

@r u=@ nr at any x0 2 †n 1 . Moreover, it can be proved that the series (8.23) is

uniformly convergent toward a solution of the Cauchy problem (8.21), (8.22).

8.6 Classification of First-Order Systems

An .n 1/-dimensional hypersurface †n 1 of equation f .x/ D 0 is called a

characteristic surface for Cauchy’s problem (8.21), (8.22), if it is a solution of the

equation

Ã

Â

i @f

D 0:

det A

@xi

(8.25)

As we saw for a second-order PDE, if the system (8.19) is quasi-linear, then the

matrices Ai depend on the solution u of the system as well as on the point x.

Therefore (8.25) allows us, at least in principle, to find the function f .x/ if a solution

u0 is assigned. As a consequence, the classification of a quasi-linear system depends

on the solution u0 , i.e., on Cauchy’s data.

Once again, the determination of characteristic surfaces requires that we solve

a nonlinear first-order PDE. We remark that if a solution f of (8.25) exists, then

the vector N normal to the characteristic surface f D const, has components Ni

proportional to @f =@xi , and the following condition holds:

det.Ai Ni / D 0:

(8.26)

210

8 Wave Propagation

In addition, the vectors N lean against a cone, since if N satisfies (8.26), then so

does N for any real .

If

det A1 Ô 0I

(8.27)

then from (8.26) it follows that

det.A1 /

1

det.Ai Ni / D det..A1 / 1 Ai Ni / D 0

i.e.,

det IN1 C

n

X

!

D 0; B˛ D .A1 / 1 A˛ :

B˛ N˛

(8.28)

˛D2

Given an arbitrary vector .N˛ / of
in x exist if the algebraic equation of order m (8.28) admits a real solution N1 .

Moreover, (8.28) represents the characteristic equation of the following eigenvalue

problem:

n

X

!

B N˛ v D

˛

N1 v;

(8.29)

˛D2

so that the roots of (8.28) are the opposite eigenvalues of the matrix

n

X

B˛ N˛ ;

(8.30)

˛D2

which, in general, is not symmetric. According to all the previous considerations,

system (8.19) can be classified as follows:

1. The system (8.19) is called elliptic at x (or for the solution u0 when it is quasilinear), if, for any N, (8.28) does not admit any real solution N1 , i.e., there are no

vectors normal to characteristic surfaces passing through an arbitrary point x 2

. As a consequence, elliptic systems do not have real characteristic surfaces.

2. The system (8.19) is hyperbolic at x (or for the solution u0 when it is quasilinear), if, for any N, (8.28) has only real roots, some of them eventually

coincident, and the eigenvectors span
3. The system (8.19) is totally hyperbolic at x if, for any N, all m roots of (8.28) are

real, distinct, and the corresponding eigenvectors form a basis of
4. The system (8.19) is parabolic at x if, for at least one root of (8.28), the

dimension of the corresponding subset of eigenvectors is less than the algebraic

multiplicity of the root itself.

8.7 Examples

211

N1

N

Fig. 8.2 Cone with several nappes

Q D .N2 ; : : : ; Nn / 2
We remark that in the last three cases, for any vector N

there are several values of N1 (with a maximum of m) which satisfy (8.28). This

means that the cone Ax has several nappes (see Fig. 8.2).

8.7 Examples

1. A second-order PDE is equivalent to a system of first-order differential equations.

In order to prove that the transformation preserves the character (elliptic,

hyperbolic, or parabolic), we observe that Laplace’s equation

@2 u

@2 u

C

D 0;

@x12

@x22

with the substitution

@u

D v;

@x1

@u

D w;

@x2

is equivalent to the system

@v

@w

C

D 0;

@x1

@x2

@v

@w

D 0;

@x2 @x1

where the second equation expresses that the second mixed derivatives of the

function u are equal.

212

8 Wave Propagation

By comparing to (8.19), the previous system can be written as

Â

0 @v 1

1

0

Â

Ã @v

Â Ã

1 0 B @x1 C

0 1 B @x2 C

0

;

@ @w A C

@ @w A D

0 1

10

0

@x1

@x2

Ã

and the characteristic equation (8.25) is given by

ÂÂ

det

1 0

0 1

Ã

@f

C

@x1

Â

01

10

Ã

@f

@x2

Ã

D0

or

Â

@f

@x1

Ã2

Â

C

@f

@x2

Ã2

D 0:

Then (8.28) becomes

ÂÂ

det

Ã

Â

Ã Ã

10

0 1

N1 C

N2 D 0I

01

10

i.e.,

N12 C N22 D 0

and this equation admits complex roots for any real value of N2 .

By using this procedure, we ask the reader to verify that D’Alembert’s

equation is equivalent to a hyperbolic system.

2. The diffusion equation

@u

@x1

@2 u

D0

@x22

is equivalent to the system

@w

D v;

@x2

@v

@x2

@w

D 0;

@x1

8.7 Examples

213

where @u=@x1 D v and @u=@x2 D w. In matrix form, this is

Â

0 @v 1

1

0

Â

Ã @v

Â Ã

0 0 B @x1 C

0 1 B @x2 C

v

;

@ @w A C

@ @w A D

0 1

10

0

@x1

@x2

Ã

and multiplying on the left by the inverse of

Â

01

10

Ã

gives

Â

0

1

0 @v 1

Â

Ã @v

Â Ã

1 0 B @x2 C

0

0 1 B @x1 C

:

@ @w A D

@ @w A C

01

v

0 0

@x1

@x2

Ã

As a consequence, (8.29) becomes

Â

0 N1

0 0

ÃÂ

v1

v2

Ã

Â

C N2

10

01

ÃÂ

v1

v2

Ã

D 0:

It is now rather easy to verify that N2 D 0 is an eigenvalue of multiplicity 2

and the corresponding eigenvector is .v1 ; 0/; so that it spans a one-dimensional

subspace. Therefore, the system is parabolic.

3. Finally, we introduce an example related to fluid mechanics (a topic addressed

in Chap. 9). Let S be a perfect compressible fluid of density .t; x/ and velocity

v.t; x/ (oriented parallel to the axis Ox). The balance equations of mass (5.22)

and momentum (5.30) of S are

@

@

Cv

C

@t

@x

@v

@v

C v

C p0.

@t

@x

@v

D 0;

@x

@

D 0;

/

@x

where p D p. / is the constitutive equation.

The above system can be reduced to the form (8.19) as follows:

Â

10

0

Ã

0

0

1

1

@

Ã @

Â

v

B @t C

B @x C

@ @v A C

@ @v A D 0;

p0 v

@t

@x

214

8 Wave Propagation

so that the characteristic equation (8.25) is given by

ÂÂ

det

10

0

Ã

@f

C

@t

Â

v

p0 v

Ã

@f

@x

Ã

D 0:

It can also be written in the form

1

@f

@x

Â

ÃC

@f

@f A D 0

Cv

@t

@x

0

@f

@f

Cv

B @t

@x

det @

@f

p0

@x

and the expansion of the determinant gives

Â

Ã2

@f

@f

Cv

@t

@x

p

0

Â

@f

@x

Ã2

D 0:

Furthermore, (8.28) can be written as

ÂÂ

det

Ã

Ã Ã

Â

10

v

N1 C

N2 D 0;

01

p0= v

i.e.,

N12 C 2N2 vN1 C N22 .v2

p 0 / D 0:

This equation has two distinct roots

N1 D

p Á

p 0 N2 ;

which are real if p 0 . / > 0. The corresponding eigenvectors

Â

1;

p

p0

Ã

;

Â p 0Ã

p

1;

;

are independent, so that the system is totally hyperbolic.

8.8 Second-Order Systems

In many circumstances, the mathematical model of a physical problem reduces to a

second-order quasi-linear system

8.8 Second-Order Systems

ij

AHK

215

@2 uK

C fH .x; u; ru/ D 0;

@xi @xj

H; K D 1; : : : ; m;

(8.31)

of m equations in the unknown functions u1 ; : : : ; um depending on the variables

x1 ; : : : ; x n .

In this case, the classification of the system can be pursued by two procedures.

The procedure discussed in the previous section can be still applied by transforming (8.31) into a first-order system of (m C mn) equations by adding the mn

auxiliary equations

@uH

D vHj ;

@xj

H D 1; : : : ; m; j D 1; : : : ; n;

with mn auxiliary unknowns and by rewriting (8.31) in the form

ij

AHK

@vKj

C fH .x; u; v/ D 0;

@xi

H; K D 1; : : : ; m:

Such a procedure has the disadvantage that there is a huge increase in the number

of equations: as an example, if m D 3 and n D 4, the transformed system gives rise

to 15 equations.

This shortcoming calls for a different approach, so that the system (8.31) is

written in the matrix form

Aij

@2 u

C f.x; u; ru/ D 0;

@xi @xj

(8.32)

where

ij

ij 1

A11

A1m

Aij D @ : : : : : : : : : : : : A ;

ij

ij

Am1

Amm

0

0

fD@

f1

1

A:

fm

We remark that the matrices Aij are generally not symmetric. By proceeding as in

the previous two sections, instead of (8.14), (8.24), we find that the characteristic

surfaces are now given by the equation

Â

@f @f

det A

@xi @xj

ij

Ã

D 0:

(8.33)

As a consequence, the vector N normal to the characteristic surface f D const,

satisfies the condition

det.Aij Ni Nj / D 0;

(8.34)

216

8 Wave Propagation

which defines, at any x, a cone with multiple nappes. This result allows us to extend

the classification rule of first-order systems to second-order ones.

8.9 Ordinary Waves

Let C be a region of
into two parts C and C C , where C C is that part which contains the unit vector N

normal to †n 1 . If u.x/ is a C 1 function in C †n 1 as well as a solution of the

system (8.21) in each one of the regions C and C C , then the following theorem

can be proved:

Theorem 8.2. The surface †n 1 is a first-order singular surface with respect to

the function u.x/ if and only if it is a characteristic surface for the Cauchy problem

(8.21)–(8.22).

Proof. Let †n 1 be a first-order singular surface with respect to u.x/. Then the

following jump condition holds across †n 1 (see (2.49)):

ÄÄ

@u

@xi

D aNi

8x0 2 †n 1 ;

(8.35)

x0

where a is a vector field with m components. In addition, since u.x/ satisfies the

system (8.19) in both C and C C regions, for x ! x0 2 †n 1 , we find that

Â

i

A

@u

@xi

Ã˙

D c;

x0

so that

ÄÄ

Ai

@u

@xi

D 0:

(8.36)

x0

Owing to (8.35), when we note that Ni D @f =@xi , where f .x/ D 0 is the equation

of the surface †n 1 , we conclude that

Ã

Â

@f

Ai

a D 0:

(8.37)

@xi x0

If the surface †n 1 is singular, then the field a is nonvanishing at some point x0 2

†n 1 . Equivalently, the coefficient determinant of the system (8.37) in the unknown

vector a has to be equal to zero. Therefore, (8.25) holds and †n 1 is a characteristic

surface.

Conversely, if (8.25) holds, then (8.35) is satisfied for a nonvanishing vector a

and †n 1 is a first-order singular surface.

t

u

8.9 Ordinary Waves

217

In its essence, the previous theorem states that the characteristic surfaces of the

system (8.19) are coincident with the singular first-order surfaces of the solution

u.x/ of (8.19). System (8.37) is called the jump system associated with (8.19).

In order to highlight the role of the previous theorem in wave propagation,

a physical phenomenon is supposed to be represented by the first-order quasilinear system (8.21), whose unknowns are functions of the independent variables

.x1 ; x2 ; x3 ; x4 / D .t; x/ 2 <4 , where t is the time and x is a spatial point. A moving

surface S.t /, of equation f .t; x/ D 0, is supposed to subdivide a region V

<3

C

into two parts V .t / and V .t /, with the normal unit vector to S.t / pointing toward

V C .t /.

If the solution u.t; x/ of (8.21) exhibits a discontinuity in some of its first

derivatives across S.t /, then u.t; x/ is said to represent an ordinary wave. If the

function itself exhibits a discontinuity on S.t /, then u.t; x/ represents a shock

wave. In both cases, S.t / is the wavefront and V C .t / is the region toward which

the surface S.t / is moving with normal speed cn . Accordingly, regions V .t / and

V C .t / are called the perturbed and undisturbed region, respectively.

It follows that a wavefront S.t / of an ordinary wave is a first-order singular

surface with respect to the solution u.t; x/ of the system (8.19), or, equivalently, a

characteristic surface.

With these concepts in mind, we can determine the wavefront f .t; x/ D 0 by

means of the theory of singular surfaces as well as by referring to characteristic

surfaces. The relevant aspect relies on the fact that the system (8.19) predicts the

propagation of ordinary waves if and only if its characteristics are real, i.e., the

system is hyperbolic.

It could be argued that the definition of a wave introduced here does not

correspond to the intuitive idea of this phenomenon. As an example, the solution

of D’Alembert’s equation can be expressed as a Fourier series of elementary waves.

In any case, when dealing with ordinary waves of discontinuity, it is relatively easy

to determine the propagation speed of the wavefront S.t / and its evolution as well

as the evolution of the discontinuity. Furthermore, in all those cases in which we

are able to construct the solution, it can be verified a posteriori that the propagation

characteristics are coincident with those derived from the theory of ordinary waves.

If the evolution is represented by a first-order system of PDEs

@u X i

@u

C

B .x; t; u/

D b;

@t

@x

i

iD1

3

(8.38)

then the associated jump system is

cn a C

3

X

Bi .x; t; u/r ni a D 0;

(8.39)

iD1

where cn is the propagation speed of the wavefront f .t; x/ D 0, n is the unit vector

normal to the surface, and a is the vector of discontinuities of first derivatives. If we

introduce the m m matrix

218

8 Wave Propagation

Q.x; t; u; n/ D

3

X

Bi .x; t; u/ni ;

(8.40)

iD1

where m is the number of unknowns, i.e., the number of components of u.t; x/, then

system (8.39) becomes

Q.x; t; u; n/a Dcn a:

(8.41)

state uC .x; t / toward which the ordinary wave propagates, the matrix Q is a known

function of r and t , due to the continuity of u.x; t / on S.t /. Furthermore, for a given

direction of propagation n, speeds of propagation correspond to the eigenvalues of

the matrix Q, and discontinuities of first derivatives are the eigenvectors of Q.

The existence of waves requires that eigenvalues and eigenvectors be real.

Once the propagation speed cn .x; t; uC ; n/ has been determined, the evolution of

the wavefront can be derived by referring to the speed of propagation of S.t / (see

Chap. 4):

@f

D

@t

cn .x; t; uC ; n/ jrf j :

(8.42)

Since S.t / is a moving surface, @f =@t ¤ 0; furthermore, n D rf = jrf j and cn

is a homogeneous function of zero order with respect to @f =@xi . Therefore, (8.42)

reduces to the eikonal equation:

Â

Ã

@f

@f

C cn x; t; uC ;

D 0:

@t

@xi

(8.43)

Finding the solution of this equation is far from being a simple task.

Here again it may happen that the physical problem can be represented by m

second-order PDEs with m unknowns depending on x1 ; : : : ; xn . This system could

be transformed into a new first-order system of mn equations with mn unknowns,

to which the previous results apply, by introducing the new unknowns

@ui

D vij ;

@xj

@vij

@vih

D

;

@xh

@xj

expressing the invertibility of second derivatives of functions ui . Because such a

procedure greatly increases the number of equations, the extension of the previous

Xem Thêm
Tải bản đầy đủ (.pdf) (489 trang)

### Tài liệu bạn tìm kiếm đã sẵn sàng tải về

Tải bản đầy đủ ngay
×