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    41

    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

    >= =