finit volum methods
TRANSCRIPT

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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 firstorder 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 timelike 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 quasilinear if the coefficient matrix A is a function of the
vector U, that is A = A(U) Note that quasilinear systems are in general
systems of nonlinear 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 quasilinear 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 nonzero
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 firstorder system, with t replaced by x and x
replaced byy is the CauchyRiemann 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 freestream 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 nonzero scaling factors.
3.2 The Initial Value Problem for the Linear Advection Equation
In this section we study in detail the initialvalue 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
oneparameter 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 oneparameter 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 xaxis, 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 initialvalueproblem 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 xaxis 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 ith 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 InitialValue 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 xaxis. Further, the
solution forU can be seen as the superposition ofmwaves, each of which
is advected (or convected) independently without change in shape. The ith
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 taxis, 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
>= =