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3. Some Notions on Hyperbolic Partial Differential Equations

All of the mathematical models for fluid flows in which we are interested in

this introductory book are systems of Partial Differential Equations (PDEs).

Furthermore, almost all of the governing equations are of hyperbolic type or

include a hyperbolic part. Numerically, it is generally accepted that these

hyperbolic terms of the PDEs of fluid flow are the terms that pose the most

stringent requirements on the numerical algorithms. Due to these arguments,

in this Chapter we deal almost exclusively with hyperbolic PDEs and

hyperbolic conservation laws.

3.1 Definitions and Examples

In this section we study systems of first-order partial differential equations

of the form

1 1

1

( , , ,..., ) ( , , ,..., ) 0m

jiij m i m

j

uu a x t u u b x t u ut x=

+ + =

(3.1)

for mi ,...,1= . This is a system of m equations in m unknowns iu that

depend on space x and a time-like variable t. Here iu are the dependent

variables and x, t are the independent variables; this is expressed via the

notationt

utxuu iii

= );,( denotes the partial derivative of ),( txui with

respect to t; similarlyx

ui

denotes the partial derivative of ),( txui with

respect to x. We also make use of subscripts to denote partial derivatives.System (3.1) can also be written in matrix form as:

0t x+ + =U AU B (3.2)

with


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1 1 11 1

2 2 21 2

1

.. ..

.. .., ,

: : : :

.. ..

m

m

m m m mm

u b a a

u b a a

u b a a

= = =

U B A

(3.3)

If the entries ija of the matrix A are all constant and the components jb of

the vector B are also constant the system is linear with constant

coefficients. If ),(and),( txbbtxaa jjijij == the system is linear with

variables coefficients. The system is still linear ifB depends linearly on Uand is called quasi-linear if the coefficient matrix A is a function of the

vector U, that is A = A(U) Note that quasi-linear systems are in general

systems of non-linear equations. System (3.2) is called homogeneous if

0=B .

For a set of PDEs one needs to specify the range of variation of the

independent variablesx and t. Usuallyx lies in a subinterval of the real line,

namely ;1 rxxx


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Definition. (Conservation Laws).

Conservation laws are systems of partial differential equations that can bewritten in the form:

0t x+ =U F(U) (3.6)

where

1

2 2

1

,

m m

u f

u f

u f

= =

U F(U)

(3.7)

U is called the vector of conserved variables, F = F(U) is the vector of

fluxes and each of its componentsfi is a function of the component uj ofU.

Definition. (Jacobian Matrix).

TheJacobian of theflux functionF(U) is the matrix:

1 1 1

2 1 2

1

m

m

m m m

f u f u

f u f u

f u f u

= =

FA(U)U

(3.8)

Note that conservation laws can also be written in quasi-linear form, by

applying the chain rule to the flux term in (3.6), namely:

( )x x

=

F U F U

U(3.9)

and thus (3.6) becomes:

0t x+ =U A(U)U (3.10)

Definition. (Eigenvalues).

The eigenvalues i of a matrix A are the solutions of the characteristic

polynomial:

( )det 0 =A I (3.11)


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whereIis the identity matrix. The eigenvalues of the coefficient matrixA of

a system are called the eigenvalues of the system.

Physically, eigenvalues represent speeds of propagation of information.

Speeds will be measured positive in the direction of increasing x and

negative otherwise.

Definition. (Eigenvectors).

A righteigenvectorof the matrixA corresponding to an eigenvalue i ofA

is a column vectorr(i)

satisfying( ) ( )i i

i=Ar r .Similarly, a left eigenvectorof a matrix A corresponding to an eigenvalue

i ofA is a row vectorl(i)

r such that( ) ( )i i

i=l A l .

Next, as an example, we find eigenvalues and eigenvectors for a system of

PDEs.

The linearised equations of Gas Dynamics (see Chapter 2) are the 2 x 2

linear system:

0

2

0

0

0

u

t xu a

t x

+ =

+ =

(3.12)

where the unknowns are the density and the speed, ( ),u ; 0 is a constantreference density and a is the sound speed (a positive constant). When

written in the matrix form this systems becomes:

0t x+ =U AU (3.13)

with:

0

2

0

0,0au

= =

U A (3.14)

The eigenvalues of the system are the zeros of the characteristics

polynomial:

( ) 0 2 220

0det 0

0a

a

= = =

A I (3.15)


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The two real and distinct solutions of the algebraic equation are the two

eigenvalues of the system, namely:

1 2,a a = = + (3.16)

We are now able to find the right eigenvectors corresponding to the

eigenvalues 1 and 2 . The eigenvector(1)r for the first eigenvalue is a

found as follows: we look for a vector [ ](1) 1 2,T

r r=r such that (1)r is a right

eigenvector ofA, that is (1) (1)1

=Ar r :

0 1 1

2

0 2 2

0

0r ar

a r ar

=

,

which produces two linear algebraic equations for the unknowns r1 and r22

0 2 1 1 2

0

, .a

r ar r ar

= ==

It is obvious that these two algebraic equations are equivalent and thus we

have a single linear algebraic equation in two unknowns. This gives a one-

parameter family of solutions. Thus we select an arbitrary non-zero

parameter 1 , a scaling factor , and set 1 1r = in any of the equations to

obtain 2 1 0r a = for the second component and hence the first righteigenvector becomes:

(1)

1

0

1

a

=

r .

The eigenvectorr(2) for the eigenvalue2

a = + is found in a similar manner

and we obtain (using a second scaling factor denoted 2 )>

(2 )

2

0

1

a

=

r .

The two scaling parameters can be taken arbitrarily. Taking the scaling

factors to be 01 = and 02 = gives the right eigenvectors>

0 0(1) (2),a a

= =

r r (3.17)


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Definition. (Hyperbolic, Elliptic or Parabolic System).

A system of PDEs is said to be hyperbolic at a point (x,t) ifA has m real

eigenvalues m ,,1 and a corresponding set of m linear independent right

eigenvectors (1) ( ), , mr r . The system is said to be strictly hyperbolic if the

eigenvalues i are all distinct.The system is said to be elliptic at a point

(x,t) if none of the eigenvalues i of A are real. A system of PDEs is said to

be parabolic at a point (x,t) ifA has m real eigenvalues m ,,1 and but

the corresponding set of m right eigenvectors (1) ( ), , mr r are not linear

independent.

Note that strict hyperbolicity implies hyperbolicity, because real and distinct

eigenvalues ensure the existence of a set of linearly independent

eigenvectors. Both scalar examples given before are trivially hyperbolic.

The linearised gas dynamic equations are also hyperbolic, since 1 and 2

are both real at any point (x, t). Moreover, as the eigenvalues are also

distinct this system is strictly hyperbolic.

An interesting example of a first-order system, with t replaced by x and x

replaced byy is the Cauchy-Riemann equations:0

0

u v

x y

v u

x y

=

+ =

(3.18)

where ),(1 yxuu = and ),(2 yxvu = . When written in matrix notation these

equations become:

0x y+ =U AU

with

0 1, .

1 0

u

v

= =

U A

the characteristic polynomial 0= IA gives 012 =+ , which has no real

solutions for and thus the system is elliptic.

A second useful example is represented by the small perturbation steady

equations of Aerodynamics:


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2 0

0

x y

x y

u a v

v u

+ =

+ =(3.19)

where:

1

12

2

=

Ma .

Here denotes the constant free-stream Mach number and

),(),,( yxvyxu are small perturbations of the x and y velocity components

respectively. In matrix notation these equations read

0x y+ =U AU

with:20

,1 0

u a

v

= =

U A

The character of these equations depends entirely on the value of the Mach

number M . For subsonic flow 1M and the system is strictly hyperbolic, with

eigenvalues:

aa +== 21 , .and the corresponding right eigenvectors:

(1) (2)

1 2

1 1,

1a a

= =

r r ,

where 1 and 2 are two non-zero scaling factors.

3.2 The Initial Value Problem for the Linear Advection Equation

In this section we study in detail the initial-value problem (IVP) for the

special case of the linear advection equation:

0

: 0

: , 0,

: ( ,0) ( )

t xPDE u au

with x t

IC u x u x

+ =

< < > =

(3.20)

where a is a constant wave propagation speed. The initial data at time 0=t is a function ofx alone and is denoted by )(0 xu . Generally, we shall not be


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explicit about the conditions 0; >


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Therefore the rate of change of u along the characteristic curve )(txx =

satisfying (3.22) is zero, that is, u is constant along the characteristic

curve )(txx = . The speed a is called the characteristics speed and it is the

slope of the curve )(txx = in the t plane. We notice here that in practice

it is more common to use the t plane to sketch the characteristics, inwhich case the slope of the curves in question is a1 .

The family of characteristic curves )(txx = given by the ODE (3.22) are a

one-parameter family of curves. A particular member of this family is

determined when an initial condition (IC) at time 0=t for the ODE (3.22)is added. Suppose we set

0(0) x= (3.24)

then the single characteristic curve that passes through the point )0,( 0x is:

0x x at = + (3.25)

Now we may regard the initial positionx0 as a parameter and in this way we

reproduce the full one-parameter family of characteristics. The fact that the

curves are parallel is typical of linear hyperbolic PDEs with constant

coefficients.

Recall the conclusion from (3.23) that u remains constant along

characteristics. Thus, if u is given the initial value )()0,( 0 xuxu = at time

0=t , then along the whole characteristic curve atxtx += 0)( that passes

through the initial point 0x on the x-axis, the solution is:

0 0 0( , ) ( ) ( )u x t u x u x at = = (3.26)

The second equality follows from (3.25). The interpretation of the solution

of the IVP for PDE is this: given an initial profile )(xuo , the PDE will

simply translate this profile with velocity a to the right if a>0 and to the leftif a


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3.2.2 The Riemann Problem

By using only geometric arguments we have constructed the analytical

solution of the general IVP (3.20) for the linear advection equation. This is

given by (3.26) in terms of the initial data )(0 xu . Now we study a special

IVP called the Riemann problem:

0

: 0.

0,: ( ,0) ( )

0,

t x

L

R

PDE u au

u if xIC u x u x

u if x

+ =

(3.27)

where Lu (left) and Ru (right) are two constant values. Note that the initial

data has a discontinuity at 0=x . The IVP (3.27) is the simplest initial-valueproblem one can pose. The trivial case would result when RL uu = .

From the previous discussion on the solution of the general IVP we expect

any point on the initial profile to propagate a distance atd= in time t. Inparticular, we expect the initial discontinuity at 0= x to propagate adistance atd= in time t. This particular characteristic curve at= will

then separate those characteristic curves to the left, on which the solutiontakes the value Lu , from those curves to the right, on which the solution

takes on the value Ru . So the solution of the Riemann problem (3.40) is

simply:

0

if 0,( , ) ( )

if 0.

L

R

u x at u x t u x at

u x at

(3.28)

Solution (3.28) also follows directly from the general solution (3.26),

namely )(),( 0 atxutxu = . From the Riemann problem (3.27)

L

uatxu = )(0

if 0 atx . The solution

of the Riemann problem can be represented in the t plane, as shown inFig 3.1. Through any point 0x on the x-axis one can draw a characteristic.

As a is constant these are all parallel to each other. For the solution of the

Riemann problem the characteristic that passes through 0=x is significant.This is the only one across which the solution changes.


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Fig. 3.1. The solution of the Riemann problem (a>0).

3.3 Linear Hyperbolic Systems

In the previous section we studied in detail the behaviour and the general

solution of the simplest PDE of hyperbolic type, namely the linear advection

equation with constant wave propagation speed. Here we extend the analysis

to sets ofm hyperbolic PDEs of the form:

0t x+ =U AU (3.29)

where the coefficient matrix A is constant. From the assumption of

hyperbolicity A has m real eigenvalues i and m linearly independent

eigenvectors ( ) , 1, ,i i m=r .

3.3.1 Diagonalisation and Characteristic Variables

In order to analyse and solve the general IVP for the system (3.29) it is

found useful to transform the dependent variables U(x,t) to a new set of

dependent variables W(x,t). To this end we recall the following definition.


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Definition. (Diagonasible System).

A matrixA is said to be diagonisable ifA can be expressed as:1 1or = =A KK K AK (3.30)

in terms of a diagonal matrix and a matrixR. The diagonal elements of

are the eigenvalues i ofA and the columns ofR are the right

eigenvectors ofA corresponding to the eigenvalues i, that is

1

(1) ( ) ( )

0

0 0, , , , ( )

0

m ii

m

i

= = =

R r r Ar r

(3.31)

A system (3.29) is said to be diagonalisable if the coefficient matrix A is

diagonalisable. We notice that (3.30) implicitly states the existence of 1R .

Based on the concept of diagonalisation one often defines a hyperbolic

system as a system with real eigenvalues and diagonalisable coefficient

matrix. The existence of the inverse matrix 1R makes it possible to define

a new set of dependent variables 1 2( , , . )Tmw w w=W

Definition. (Characteristic variables).

The new variablesT

mwwwW ).,,( 21 = obtained via the transformation

= -1W R U (3.32)are calledcharacteristic variables.

We also have the inverse transform:

=U RW (3.33)

The introduction of the characteristic variable is a useful tool to be used forthe solution of the linear system (3.30). The basic idea is to use the

diagonalisable property of the Jacobian matrix A. This can be done by

changing the variables from U to W so that the new system to be easier to

solve. When expressed in terms of the characteristic variables W we need

the partial derivatives tU and xU . Since A is constant, R is also constant

and therefore these derivatives are


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t t

x x

=

=

U RW

U RW(3.34)

Direct substitution of these expressions into equation (3.29) gives

0t x+ =RW ARW (3.35)

Multiplication of this equation from the left by 1R and use of (3.30) gives

0t x+ =W W (3.36)

This is called the canonical form orcharacteristic form of system (3.42).When written in full this system becomes:

1 1 1

2 2 2

0 .. 0

0 .. 00

.. .. .. ..

0 .. ..m m mt x

w w

w w

w w

+ =

(3.37)

Clearly the i-th PDE of this system is

0, 1, ,i iw w

i i mt x

+ = =

(3.38)

and involves the single unknown ),( txwi . The system is therefore

decoupled and each equation is identical to the linear advection equation in

(3.27); now the characteristic speed is i and there are m characteristic

curves satisfying m ODEs:

, for 1, ,idx

i mdt

= = (3.39)

3.3.2 The General Initial-Value Problem

We now study the IVP for the PDEs (3.29). The initial condition is nowdenoted by superscript (0), namely(0) ( ) (0)

1( , , )o Tmu u=U (3.40)

We find the general solution of the IVP by first solving the corresponding

IVP for the canonical system (3.36) in terms of the characteristic variables

W and initial condition (0) ( ) (0)1

( , , )o Tmw w=W such that:


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(0) 1 (0)=W R U (3.41)

The solution of the IVP for (3.36) is direct. By considering each unknown

),( txwi satisfying (3.38) and its corresponding initial data)0(

iw we write its

solution immediately as(0)( , ) ( ), for 1, ,i i iw x t w x t i m= = (3.42)

The solution of the general IVP in terms of the original variables U is now

obtain by transforming back according to =U KW . When written in fullthis expression becomes

(1) (2) ( )

1 1 1 2 1 1

(1) (2) ( )

1 2

(1) (2) ( )

1 2

,

....

,

....

.

m

m

m

i i i m i

m

m m m m m

u w r w r w r

u w r w r w r

u w r w r w r

= + + +

= + + +

= + + +

(3.43)

or(1) (2) ( )

1 1 1 1

(1) (2) ( )2 2 2 2

1 2

(1) (2) ( )

m

m

m

mm m m m

u r r r

u r r rw w w

u r r r

= + + +

(3.44)

or, more concisely

( )

1

( , ) ( , ) .m

i

i

i

x t w x t =

= U r (3.45)

This means that the function ),( txwi is the coefficient of( )ir in an

eigenvector expansion of the vector U. But according to (3.42),

)(),( )( txwtxw ioii = and hence the solution to the IVP is:

( )0 ( )

1

( , ) ( )m

i

i i

i

x t w x t =

= U r (3.46)

Thus, given a point (x, t) in the tx plane, the solution ( , )tU at this point

depends only on the initial data at the m point txx ii =)(0 . These are the


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intersections of the characteristics of speed i with the x-axis. Further, the

solution forU can be seen as the superposition ofmwaves, each of which

is advected (or convected) independently without change in shape. The i-th

wave has shape (0) ( )( ) iiw x r and propagates with speed i .

3.3.3. A Representative Example: The IVP for the Linearised Equations

of Gas Dynamics

As a simple example lets now study the general IVP for the linearised

equations of Gas Dynamics (3.12), namely

01 1

1 22

02 2

00, ,

0t x

u uu u u

au u

+ =

,

with initial condition

=

)(

)(

)0,(

)0,()0(

2

)0(

1

2

1

xu

xu

xu

xu.

We define characteristic variables1

1 2( , )Tw w = =W R U ,

where R is the matrix of right eigenvectors and 1K is its inverse, both

given by

00 0 1

00

1,

2

a

aa a a

= = K K .

In terms of the characteristic variables we have

00

0

2

1

2

1 =

+

xtw

w

a

a

w

w

or in full

[ ][ ])()(

2

1),(

)()(2

1

),(

)0(

20

)0(

1

0

2

)0(

20

)0(

10

1

atxuatxaua

txw

atxuatxauatxw

+=

++=


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This is the solution in terms of the characteristic variables. In order to obtain

the solution to the original problem we transform back using U = K W . This

gives the final solution as

[ ] [ ]

[ ] [ ].)()(2

1)()(

2

1),(

,)()(2

1)()(

2

1),(

)(

20

)0(

1

)0(

20

)0(

12

)0(

20

)0(

1

)0(

20

)0(

11

atxuatxaua

atxuatxaua

txu

atxuatxaua

atxuatxaua

txu

o++++=

++++=

3.4 The Riemann Problem for Linear Hyperbolic Systems

In this section we study the Riemann problem for the hyperbolic, constant

coefficient system (3.29). This is the special IVP:

(0)

: 0,

, 0,

, 0,: ( ,0) ( )

, 0

t x

L

R

PDEs

x t

xIC x x

x

+ =

< < >

U AU

UU U

U

(3.47)

Clearly, this is a generalisation of the IVP (3.20) for the scalar hyperbolic

equation.

3.4.1 The General Solution

We assume that the system is strictly hyperbolic and we order the real and

distinct eigenvalues as

m


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As the eigenvectors (1) ( ), , mr r are linearly independent, we can expand the

data LU , constant left state, and RU , constant right state, as linear

combinations of the set (1) ( ), , mr r , that is:

( )

1

( )

1

,

.

mi

L i

i

mi

R i

i

=

=

=

=

U r

U r

(3.48)

with constant coefficients ii , , for mi ,,1 = . Formally, the solution of

the IVP (3.47) is given by (3.46) in terms of the initial data )()0(xwi for the

characteristic variables and the right eigenvector ( )ir . In terms of the

characteristic variables we have m scalar Riemann problems for the PDEs

0i iiw w

t x

+ =

(3.49)

with the initial data formally obtained from

( ) ( )

( )( ) ( )( )( ) ( )

1 1

1 2

0 01

1 1

1 2

.. 0

.. 0

Ti

L i m

i

Ti

R i m

i

if x

if x

= =

= + U r r (3.53)

The first term in the right hand side of (3.53) represents the contribution at

the point (x, t) of the information carried along the characteristics for which

0ix t < . The second term has a similar meaning.

A particularly important case for numerical applications is that in which the

solution is determined on the t-axis, i.e. at a point (x=0,t). We have,

successively:

( ) ( ) ( )00

0,0

i i

i i i

i i

ifw t w t

if

>= =

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