Systems of Nonlinear Hyperbolic Equations

Professor Dr. E F Toro

Laboratory of Applied MathematicsUniversity of Trento, Italyeleuterio.toro@unitn.it

http://www.ing.unitn.it/toro

August 24, 2014

1 / 31

Overview

• We study some of the basic theory of non-linear hyperbolic equations

• We give three examples, the isothermal-gas equations, the non-linearshallow water equations and a non-linear model for blood flow incompliant vessels

2 / 31

First-order systems of equations

Here we study some very basic properties of nonlinear hyperbolic systems,starting from the general setting of m hyperbolic balance laws in threespace dimensions, written in differential conservative form

∂tQ + ∂xF(Q) + ∂yG(Q) + ∂zH(Q) = S(Q) , (1)

where

Q =

q1q2...qm

; F =

f1f2...fm

; G =

g1g2...gm

; H =

h1h2...hm

; S =

s1s2...sm

.

(2)Independent variables: x, y, z and t. Q(x, y, z, t): vector of dependent variables,called conserved variables; F(Q): flux vector in x-direction; G(Q): flux vector iny-direction; H(Q) flux vector in z-direction; S(Q): vector of source terms.

Fluxes and sources are prescribed functions of Q(x, y, z, t).

3 / 31

Note that each component k of the vectors in (2) is a function of thecomponents qk = qk(x, y, z, t) of the vector of unknowns Q(x, y, z, t)

qi = qi(x, y, z, t) ,fi = fi(q1(x, y, z, t), . . . , qm(x, y, z, t)) ,gi = gi(q1(x, y, z, t), . . . , qm(x, y, z, t)) ,hi = hi(q1(x, y, z, t), . . . , qm(x, y, z, t)) ,si = si(x, y, z, t, q1(x, y, z, t), . . . , qm(x, y, z, t)) .

(3)

The model linear advection-reaction equation is an example of (1)

∂tq(x, t) + λ∂xq(x, t) = βq(x, t) , λ : constant , β : constant , (4)

with f(q) = λq and s(q) = βq(x, t), both linear functions of q(x, t).

An equation of the form (1) is called a system of balance laws. WhenS(Q) = 0 we call (1) a system of conservation laws (homogeneousequations); otherwise we speak of inhomogeneous equations.

4 / 31

One-dimensional systems

A general m×m one-dimensional non-linear system with source terms,written in differential conservation-law form reads

∂tQ + ∂xF(Q) = S(Q) . (5)

Q: conserved variables, F(Q): fluxes and S(Q): sources. The integralform is:∫ xR

xL

Q(x, t2) dx =

∫ xR

xL

Q(x, t1) dx

+

∫ t2

t1

F(Q(xL, t)) dt−∫ t2

t1

F(Q(xR, t)) dt

+

∫ t2

t1

∫ xR

xL

S (Q(x, t)) dxdt .

(6)

5 / 31

We note that an alternative integral form is

d

dt

∫ xR

xL

Q(x, t) dx = F(Q(xL, t))− F(Q(xR, t)) +

∫ xR

xL

S (Q(x, t)) dx .

(7)Rankine-Hugoniot conditions. Defining the jumps

∆F = F(QR)− F(QL) ; ∆Q = QR −QL (8)

across a discontinuity of speed S, the Rankine-Hugoniot conditions for asystem read

∆F = S∆Q . (9)

Here QL and QR are the limiting states from the left and right of thediscontinuity. Note that unlike the scalar case, now it is not possible tosolve directly for the speed S in terms of the jumps ∆F and ∆Q.

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Quasi-linear form

It is convenient to express (5) in quasi-linear form

∂tQ + A(Q)∂xQ = S(Q) , (10)

where A(Q) is the Jacobian matrix of F(Q), yet to be defined.

To illustrate the procedure we consider the special case

∂tQ + ∂xF(Q) = 0 ,

Q =

[q1q2

], F(Q) =

[f1(q1, q2)f2(q1, q2)

].

(11)

Applying the chain rule to the flux term for each equation we have

∂tq1 +∂

∂q1f1(q1, q2)∂xq1 +

∂

∂q2f1(q1, q2)∂xq2 = 0 ,

∂tq2 +∂

∂q1f2(q1, q2)∂xq1 +

∂

∂q2f2(q1, q2)∂xq2 = 0 .

(12)

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In matrix form system (12) reads

∂t

q1

q2

+

∂f1∂q1

∂f1∂q2

∂f2∂q1

∂f2∂q2

∂x q1

q2

=

0

0

, (13)

or∂tQ + A(Q)∂xQ = 0 , (14)

where

A(Q) =∂F

∂Q=

∂f1∂q1

∂f1∂q2

∂f2∂q1

∂f2∂q2

(15)

is called the Jacobian matrix.

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For an m×m one-dimensional non-linear, homogeneous, system (5) hasquasi-linear form

∂tQ + A(Q)∂xQ = 0 , (16)

with Jacobian matrix

A(Q) =∂F

∂Q=

∂f1∂q1

∂f1∂q2

. . .∂f1∂qm

∂f2∂q1

∂f2∂q2

. . .∂f2∂qm

. . . . . . . . . . . .

∂fm∂q1

∂fm∂q2

. . .∂fm∂qm

. (17)

As done for linear systems we are now in a position to study the eigenstructure offirst-order systems.

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Eigenstructure: eigenvalues and eigenvectors

• The concepts of eigenstructure and hyperbolicity for a non-linearsystem like are the same as for linear systems.

• For non-linear systems the Jacobian, the eigenvalues and eigenvectorsare functions Q.

• The eigenvalues A(Q) are the roots of the characteristic polynomial

P (λ) = |A(Q)− λI| = 0 , (18)

where I is the identity matrix and λ is a parameter.

• We assume the m eigenvalues are written in increasing order

λ1(Q) ≤ λ2(Q) ≤ . . . ≤ λm(Q) . (19)

• We write eigenvectors in the order corresponding to their associatedeigenvalues

R1(Q) ;R2(Q) ; . . . ;Rm(Q) (20)

L1(Q) ;L2(Q) ; . . . ;Lm(Q) . (21)

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Hyperbolic system:

An m×m system (10) is hyperbolic if the Jacobian A(Q) has m realeigenvalues λi(Q) (i = 1, . . . ,m) and a corresponding set of m linearlyindependent eigenvectors Ri(Q)(i=1, . . . ,m).

The system is said to be strictly hyperbolic if it is hyperbolic and alleigenvalues are distinct.

A three-dimensional m×m system (1) is hyperbolic if the matrix

D(Q) = ω1A(Q) + ω2B(Q) + ω3C(Q) (22)

has m real eigenvalues λ1, λ2, . . . , λm and a corresponding set of m linearlyindependent eigenvectors R1,R2, . . . ,Rm, for all linear combinations(22), where the coefficients ω1, ω2, ω3 define a non-zero vector, that is√

ω21 + ω2

2 + ω23 > 0 . (23)

A, B, C are the Jacobian matrices associated to F, G and H in (1).

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Nature of characteristic fields

For a hyperbolic system the characteristic speed λi(Q) defines acharacteristic field, the λi-field; we also speak of the Ri-field or simplythe i-field.Recall that the gradient of an eigenvalue λi(Q) is given by

∇λi(Q) =

[∂

∂q1λi ,

∂

∂q2λi , . . . ,

∂

∂qmλi

]. (24)

A λi-characteristic field is said to be linearly degenerate if

∇λi(Q) ·Ri(Q) = 0 , ∀Q ∈ <m , (25)

where <m is the set of real-valued vectors of m components, called phasespace, state space. For a m×m system we speak of phase plane.A λi-characteristic field is said to be genuinely non-linear if

∇λi(Q) ·Ri(Q) 6= 0 , ∀Q ∈ <m . (26)

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Example 1: isentropic gas dynamics

∂tQ + ∂xF(Q) = 0 ,

Q =

[q1q2

]≡[ρρu

], F(Q) =

[f1f2

]≡[

ρuρu2 + p

],

(27)

where ρ is density, u is velocity and p is pressure. These quantities arecalled the physical, or primitive, variables, as distinct from the conservedvariables, which are the components of Q. As there are two equations andthree unknowns one requires a closure condition, called an equation ofstate, that expresses one variable in terms of the others.A simple closure is the isentropic equation of state that defines thepressure as

p = Cργ , C = constant , γ = constant . (28)

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To find the Jacobian we identify the conserved variables and express theflux components in terms of these.

q1 = ρ , q2 = ρu , f1 = ρu = q2 , f2 = ρu2 + p =q22q1

+ Cqγ1 . (29)

Now we calculate the components of the Jacobian matrix. We have

∂

∂q1f1(q1, q2) = 0 ,

∂

∂q2f1(q1, q2) = 1 ,

∂

∂q1f2(q1, q2) = −q

22

q21+ a2 ,

∂

∂q2f2(q1, q2) = 2

q2q1,

(30)

a2 =d

dρp(ρ) = p′(ρ) = γCqγ−11 =

γp

ρ. (31)

Now the Jacobian in terms of u and the sound speed a =√

γpρ becomes

A(Q) =

0 1

−u2 + a2 2u

. (32)

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The eigenvalues of the system are the roots of the characteristicpolynomial from (18)

P (λ) = λ2 − 2uλ+ u2 − a2 = 0 , (33)

from which the eigenvalues are found to be

λ1 = u− a , λ2 = u+ a . (34)

Exercise: Verify that the right and left eigenvectors are

R = α1

[1

u− a

], R2 = α2

[1

u+ a

]. (35)

and

L1 = β1[

1, −(u+ a)], L2 = β2

[−(u− a), 1

]. (36)

Here the coefficients α1, α2, β1, β2 are scaling factors.

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Example 2: the nonlinear shallow water equations

The shallow water equations arise in the modelling of a wide variety ofphysical phenomena, such as water flows, atmospheric flows, dense gasdispersion, avalanches and even astrophysical flows. Here we study theaugmented one-dimensional case with source terms

∂tQ + ∂xF(Q) = S(Q) , (37)

with

Q =

hhuhψ

, F(Q) =

huhu2 + 1

2gh2

huψ

, S(Q) =

s1s2s3

. (38)

u(x, t): velocity, h(x, t): depth, ψ(x, t): a passive scalar, g: accelerationdue to gravity. Fig. 15 depicts the geometric situation. The depth h(x, t)is related to the free surface elevation H(x, t) and the bed elevationb(x) ≥ 0 above a horizontal datum via

H(x, t) = b(x) + h(x, t) . (39)

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z

xz = 0

Solid

Water

Air

u(x, t)

b(x)

h(x, t)

H(x, t) = b(x) + h(x, t)

Fig. 15. Geometry of shallow water equations. The fixed bottom elevationis described by b(x). The water depth is defined by h(x, t), while the free

surface position is given by H(x, t) = b(x) + h(x, t).

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• The function b(x) defines the bed profile and for the problems ofinterest here is prescribed and does not depend on time t;

• g is the acceleration due to gravity taken as g = 9.81 m/s2, aconstant.

• There are two distinct situations of practical interest, namely the wetbed case in which the water depth h is greater than zero and the drybed case, in which portions of the bed are dry, that is h = 0.

• S(Q) is a source term vector that accounts for various physical andgeometric effects. For instance, when the bed elevation is variablethen the source term vector becomes

S(Q) =

0−ghb′(x)

0

. (40)

For many practical applications there will be additional terms in the vectorS(Q) to account for Coriolis forces, wind forces, bottom friction, etc.

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Eigenstructure and characteristic fields

For the Jacobian we identify the conserved variables and write the flux asfunction of these.

Q =

q1q2q3

=

hhuhψ

, F(Q) =

f1f2f3

=

q2q22/q1 + 1

2gq21

q2q3/q1

.

(41)Simple calculations give all the entries of A(Q). For example

∂

∂q1f1(q1, q2) = 0 ,

∂

∂q2f1(q1, q2) = 1 ,

∂

∂q3f1(q1, q2) = 0 ,

∂

∂q1f2(q1, q2) = −q22/q21 + gq1 = a2 − u2 ,

where a is the celerity (analogous to the sound speed in gases) defined as

a =√gh . (42)

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The Jacobian matrix is then given as

A(Q) =

0 1 0

a2 − u2 2u 0

−uψ ψ u

. (43)

Eigenvalues. The eigenvalues of A are

λ1 = u− a , λ2 = u , λ3 = u+ a , (44)

where a is the celerity, as defined in (42). This is verified by finding theroots of the characteristic polynomial

P (λ) = (u− λ)[−λ(2u− λ)− (a2 − u2)

]= 0 , (45)

The eigenvalues are all real; they are also distinct under all circumstances,except for the case of dry bed h = 0, in which case a = 0 andλ1 = λ2 = λ3 = u.

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The right eigenvectors are given by

R1 = α1

1u− aψ

, R2 = α2

001

, R3 = α3

1u+ aψ

(46)

and the left eigenvectors are given by

L1 = β1[u+ a, −1, 0

], L2 = β2

[−ψ, 0, 1

], L3 = β3

[u− a, −1, 0

].

(47)Here the coefficients α1, α2, α3, β1, β2 and β3 are scaling factors.

Bi-orthonormality. The reader can easily verify that the left and righteigenvectors (46), (47) are bi-orthonormal, that is they satisfy the relations

Li Rj =

1 if i = j ,

0 if i 6= j ,(48)

provided the scaling factors are chosen thus:

β1 =1

2aα1, β2 =

1

α2, β3 = − 1

2aα3. (49)

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Example 3: equations for blood flow

We consider here the case of constant material properties for bloodvessels. The governing equations are

∂tA+ ∂x(uA) = 0 ,

∂t(uA) + ∂x(αAu2) +A

ρ∂xp = −Ru .

(50)

• A(x, t) is cross-sectional area of vessel at position x and time t

• u(x, t) is averaged velocity of blood at a cross section

• p(x, t) is pressure

• Blood density ρ is assumed constant

• R > 0 is viscous resistance, a prescribed function

• α, coefficient that depends on assumed velocity profile

• There are 3 unknowns, 2 equations and thus a tube law is needed toclose the system

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Closure condition: the tube law

The tube law relates pressure p(x, t) to wall displacement via A(x, t).Here we adopt a very simple tube law of the form

p = pext(x, t) + ψ(A;β) (51)

Here we adopt the simple tube law

ψ(A;β) = β(x)(√

A−√A0

),

β(x) =

√π

(1− ν2)h0(x)E(x)

A0(x).

(52)

• A0(x) is the equilibrium cross-sectional area

• h0(x) is the vessel wall thickness

• E(x) is the Young’s modulus

• ν is the Poisson ratio, taken to be ν = 1/2

• ψ(A;β) = p− pext ≡ ptrans: transmural pressure

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Simplified model: assumptions

Assume constant material properties:

• h0 = constant• A0 = constant• E = constant• pext = constant• R = 0• α = 1

Therefore β(x) = β in (80) is constant and the term Aρ ∂xp in (78)

becomesA

ρ∂xp =

β

3ρ∂xA

3/2 . (53)

The equations now read

∂tA+ ∂x(uA) = 0 ,

∂t(uA) + ∂x(Au2) +β

3ρ∂xA

3/2 = 0 .

(54)

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Conservation-law form

Now the equations can be written in conservation-law form

∂tQ + ∂xF(Q) = S(Q) , (55)

where

Q =

q1

q2

≡ A

Au

, S(Q) =

s1

s2

≡ 0

−Ru

(56)

and the flux vector is

F(Q) =

f1

f2

≡

Au

Au2 +β

3ρA3/2

. (57)

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Jacobian matrix

The principal part of (55) written in quasi-linear form becomes

∂tQ +∂F

∂Q∂xQ = 0 , (58)

where

A(Q) =∂F

∂Q: Jacobian matrix of the system (59)

To find A(Q) one first expresses F(Q) in terms of Q, namely

F(Q) =

f1

f2

=

Au

Au2 +β

3ρA3/2

=

q2

q22q1

+β

3ρq3/21

(60)

A(Q) =

∂f1(q1, q2)

∂q1

∂f1(q1, q2)

∂q2∂f2(q1, q2)

∂q1

∂f2(q1, q2)

∂q2

=

[0 1

β2ρ

√A− u2 2u

].

(61)26 / 31

Eigenvalues:

Proposition: The eigenvalues of (61) are all real and given by

λ1 = u− c , λ2 = u+ c , (62)

where

c =

√β√A

2ρ(63)

is the wave speed, analogous to the sound speed in gas dynamics.

Proof. By definition, the eigenvalues of system are the eigenvalues of thematrix A, which in turn are the roots of the characteristic polynomial

P (λ) = Det(A− λI) = 0 . (64)

I is the identity matrix and λ is a parameter. Simple calculations give

P (λ) = λ2 − 2uλ+ u2 − β√A

2ρ= 0 ,

from which the result (62) follows.27 / 31

Right eigenvectors:

Proposition: The right eigenvectors of A corresponding to theeigenvalues (62) are

R1 = β1

[1

u− c

], R2 = β2

[1

u+ c

], (65)

where β1 and β2 are arbitrary scaling factors.

Proof. For an arbitrary right eigenvector R = [r1, r2]T we have

AR = λR , (66)

which gives the algebraic system

r2 = λr1 ,(c2 − u2)r1 + 2ur2 = λr2 .

}(67)

By substituting λ in (67) by the appropriate eigenvalues in (62) in turn, wearrive at the sought result.

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Formulation in terms of primitive variables

It can be easily shown that equations (82), in terms of the variables A andu, may be written in quasi-linear form as

∂tV + M∂xV = S(V) , (68)

V =

[Au

], M =

[u A

c2/A u

], S(V) =

[0

−Ru/A

], (69)

with

c =

√β√A

2ρ: wave speed . (70)

This is accomplished by

• Expanding derivatives in the continuity equation (assuming smoothsolutions)

• Expanding derivatives in the momentum equation and using thecontinuity equation

Exercise: Verify that (83)-(84) can be obtained from (82).29 / 31

Concluding Remarks

• We have studied some basic mathematical properties of systems ofnon-linear hyperbolic equations

• We have given examples examples of non-linear systems that have aphysical meaning

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Exercises for non-linear hyperbolic systems

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Problem 1: isentropic gas dynamics. Consider the isentropic equations ofgas dynamics

∂tQ + ∂xF(Q) = 0 ,

Q =

[q1q2

]≡[ρρu

], F(Q) =

[f1f2

]≡[

ρuρu2 + p

],

(71)

along with the isentropic equation of state

p = Cργ , C = constant , γ = constant . (72)

1 Find conditions on scaling parameters for the left and righteigenvectors to be orthonormal.

2 Determine the nature of the characteristic fields.

Problem 2: eigenstructure in terms of primitive variables. Consider theaugmented shallow water equations on a non-horizontal channel

∂tQ + ∂xF(Q) = S(Q) , (73)31 / 31

with

Q =

hhuhψ

, F(Q) =

huhu2 + 1

2gh2

huψ

, S =

0−ghb′(x)

0

.

(74)Here b(x) defines the channel bed above a horizontal datum and b′(x) isits gradient.

1 Assuming smooth solutions, by expanding the time and spatial partialderivatives show that the equations can be expressed in terms of thethree variables h(x, t), u(x, t), ψ(x, t), called primitive variables, andthat the governing equations are

∂th+ u∂xh+ h∂xu = 0 ,

∂tu+ u∂xu+ g∂xh = −gb′(x) ,

∂tψ + u∂xψ = 0 .

(75)

31 / 31

2 Write these equations in matrix form

∂tW + B∂xW = S(W) (76)

3 Find the eigenvalues and the left and right eigenvectors.

4 Is the system hyperbolic ?

5 Determine the nature of the characteristic fields.

Problem 3: eigenstructure in terms of conserved variables.

1 Find the Jacobian matrix

2 Carry out the same tasks as in problem 2 but now in terms of theconserved variables, departing directly from (73).

Problem 4: homogeneity property of a system. A system of the form (73)is said to be homogeneous if the flux satisfies

F(Q) = A(Q)Q , (77)

where A(Q) is the Jacobian matrix.

1 Verify that the shallow water equations do not satisfy thehomogeneity property.

31 / 31

2 Find an approximate Jacobian matrix A for the shallow waterequations such that the homogeneity property is satisfied.

3 Check if the equations of isentropic gas dynamics satisfy thehomogeneity property.

4 Verify that a linear hyperbolic system always satisfies the homogeneityproperty.

Blood flow equations. We consider the following one-dimensional,time-dependent non-linear mathematical model for blood flow

∂tA+ ∂x(uA) = 0 ,

∂t(uA) + ∂x(αAu2) +A

ρ∂xp = −Ru ,

(78)

where A(x, t) is cross-sectional area of vessel at position x and time t;u(x, t) is averaged velocity of blood at a cross section; p(x, t) is pressure;ρ is assumed constant; R > 0 is viscous resistance, a prescribed function;α, coefficient that depends on assumed velocity profile. Here we adopt avery simple tube law of the form

p = pext(x, t) + ψ(A;K) ,}

(79)31 / 31

with

ψ(A;K) = K(x)(√

A−√A0

), K(x) =

√π

(1− ν2)E(x)h0(x)√

A0(x). (80)

Here A0(x) is the equilibrium cross-sectional area; h0(x) is the vessel wallthickness; E(x) is the Young’s modulu; ν is the Poisson ratio, taken to beν = 1/2. Assume h0 = constant; A0 = constant; E = constant;pext = constant; R = 0; α = 1.Problem 5

1 Show that the term Aρ ∂xp in (78) becomes

A

ρ∂xp =

K

3ρ∂xA

3/2 . (81)

2 Show that the equations become

∂tA+ ∂x(uA) = 0 ,

∂t(uA) + ∂x(Au2) + K3ρ∂xA

3/2 = 0 .

(82)

31 / 31

3 Write equations (82) in conservation-law form

4 Find the Jacobian matrix

5 Find (i) the eigenvalues, (ii) left and right eigenvectors

6 Is the system hyperbolic ?

7 Determine the nature of the characteristic fields

8 Determine if the blood flow equations (78) are homogeneous.

Problem 6: equations in terms of primitive variables.

1 Show that equations (82), in terms of the variables A and u, may bewritten in quasi-linear form as

∂tQ + A∂xQ = 0 , (83)

Q =

[Au

], A =

u Ac2

Au

, (84)

with

c =

√K√A

2ρ: wave speed . (85)

31 / 31

2 Find eigenvalues and left and right eigenvectors

3 Is the system hyperbolic ?

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