Ebook quasilinear hyperbolic systems, compressible flows, and waves part 2

Chapter 5 Asymptotic Waves for Quasilinear Systems In this chapter, we shall assume the existence of appropriate asymptotic ex pansions, and derive asymptotic equations for hyperbolic PDEs These equa[.]

Chapter 5

Asymptotic Waves for Quasilinear

Systems

In this chapter, we shall assume the existence of appropriate asymptotic ex-pansions, and derive asymptotic equations for hyperbolic PDEs These equa-tions display the qualitative effects of dissipation or dispersion balancing non-linearity, and are easier to study analytically In this way, the study of com-plicated models reduces to that of models of asymptotic approximations ex-pressed by a hierarchy of equations, which facilitate numerical calculations; this is often the only way that progress can be made to analyze complicated systems Essential ideas underlying these methods may be found in earlier publications; see for example Boillat [19], Taniuti, Asano and their coworkers ([7], [194]), Seymour and Varley [166], Germain [61], Roseau [155], Jeffrey and Kawahara [82], Fusco, Engelbrecht and their coworkers ([58], [59]), Cramer and Sen [43], Kluwick and Cox [94], and Cox and Kluwick [42] An account of some of the rigorous results which deal with convergence of such expansions may be found in [47]

5.1 Weakly Nonlinear Geometrical Optics

Here we shall discuss the behavior of certain oscillatory solutions of quasi-linear hyperbolic systems based on the theory of weakly nonquasi-linear geometri-cal optics (WNGO), which is an asymptotic method and whose objective is

to understand the laws governing the propagation and interaction of small amplitude high frequency waves in hyperbolic PDEs; this method describes asymptotic expansions for solutions satisfying initial data oscillating with high frequency and small amplitude The propagation of the small amplitude waves, considered over long time-intervals, is referred to as weakly nonlinear Most waves are of rather small amplitude; indeed, small amplitude high frequency short waves are frequently encountered Linearized theory satisfactorily de-scribes such waves only for a finite time; after a sufficiently long time-interval, the cumulative nonlinear effects lead to a significant change in the wave field the basic principle involved in the methodology used for treating weakly non-linear waves lies in a systematic use of the method of multiple scales Each

133

dissipative mechanism present in the flow, such as rate dependence of the medium, or geometrical dissipation due to nonplanar wave fronts, or inhomo-geneity of the medium, defines a local characteristic length (or time) scale, that arises in a natural way Indeed, WNGO is based on the assumption that the wave length of the wave is much smaller than any other characteristic length scale in the problem When this assumption is satisfied, i.e., the time (or length) scale defined by the dissipative mechanism is large compared with the time (or length) scale associated with the boundary data, the wave is referred to as a short wave, or a high frequency wave For instance, consider the motion excited by the boundary data u(0, t) = af (t/τ ) on x = 0, where a represents the size of the boundary data, while τ is the applied period or pulse length When we normalize the boundary data, by a suitable nondimensional quantity and t by the time scale τr, which is defined by the dissipative mech-anism present in the flow, the geometrical acoustics limit then corresponds

to the high frequency condition  = τ /τr  1; in fact, there is a region in the neighborhood of the front where the nonlinear convection associated with the high frequency characteristics is important For both significant nonlin-ear distortion and dissipation, in this high frequency limit, we must have the nondimensional amplitude |a| = O() In terms of these normalized variables, the boundary condition becomes u(0, t) = g(t/), which describes oscillations

of small amplitude  and of frequency −1 For data of size , the method of WNGO yields approximations, which are valid on time intervals typically of size −1 Following the pioneering work of Landau [99], Lighthill [108], Keller [91] and Whitham [210], a vast amount of literature has emerged on the de-velopment, both formal and rigorous, in the theory of WNGO However, the development of these methods through systematic self-consistent perturba-tion schemes in one and several space dimensions is due to Chouqet-Bruhat [34], Varley and Cumberbatch ([204], [206]), Parker [140], Mortell and Varley [128], Seymour and Mortell [165], Hunter and Keller [75], Fusco [60], Mazda and Rosales [115], and Joly, Metivier and Rauch [88] For results based on numerical computations of model equations of weakly nonlinear ray theory, the reader is referred to Prasad [145] Using the systematic procedure, alluded

to and applied by the above mentioned authors, we study here certain aspects

of WNGO, which are largely based on our papers ([146], [171], [173], [190], and [192])

5.1.1 High frequency processes

Consider a quasilinear system of hyperbolic PDEs in a single space variable

u,t+ A(u)u,x+ b(u) = 0, −∞ < x < ∞, t > 0 (5.1.1) where u and b are n-component vectors and A is a n × n matrix Let uo be

a known constant solution of (5.1.1), such that b(uo) = 0 We consider small amplitude variations in u from the equilibrium state u = uo, which are of the size , described earlier; and look for a small amplitude high frequency wave

5.1 Weakly Nonlinear Geometrical Optics 135 solution of (5.1.1), representing a single wave front, and valid for times of the order of O(1/) Then the WNGO ansatz for the situation described above is the formal expansion for the solution of (5.1.1)

u = uo+ u1(x, t, θ) + 2u2(x, t, θ) + (5.1.2) where θ is a fast variable defined as θ = φ(x, t)/ with φ as the phase function

to be determined The wave number k and wave frequency ω are defined by

k = φ,xand ω = −φ,t By considering the Taylor expansion of A and b in the neighborhood of uo, and taking into account (5.1.2), equation (5.1.1) implies that

O(o) : (Ao− λI)u1,θ= 0

O(1) : (Ao− λI)u2,θ= −(∇A)ou1u1,θ− {u1,t+ Aou1,x+ (∇b)ou1}φ−1

,x , (5.1.3) where the subscript o refers to the evaluation at u = uo, I is the n × n unit matrix, λ = −φ,t/φ,x, and ∇ is the gradient operator with respect to the components of u The equation (5.1.3)1implies that for a particular choice of the eigenvalue λo(assuming that it is simple),

u1= π(x, t, θ)Ro, (5.1.4) where π is a scalar oscillatory function to be determined, and Ro is the right eigenvector of Ao corresponding to the eigenvalue λo The phase φ(x, t) is determined by

φ,t+ λoφ,x= 0 (5.1.5)

It may be noticed that if φ(x, 0) = x, then (5.1.5) implies that φ(x, t) = x−λot Let Lo be the left eigenvector of Ao corresponding to its eigenvalue λo, satisfying the normalization condition Lo·Ro= 1 Then if (5.1.3)2is multiplied

by Lo on the left, we obtain along the characteristic curves associated with (5.1.5) the following transport equation for π ;

π,τ + ν(π2/2),θ= −µπ, (5.1.6) where ∂/∂τ = ∂/∂t + λo∂/∂x, and

µ = Lo· ∇b|o· Ro, ν = Lo· ∇A|o· RoRo (5.1.7)

If the initial condition for π is specified, i.e., π|τ =0 = πo(xo, θo) with xo = x|τ =o and θo= −1φ|τ =0, then the minimum time taken for the smooth solu-tion to breakdown can be computed explicitly In fact, the initial condisolu-tions lead to a shock only when ν∂πo/∂θo < 0 and −ν∂πo/∂θo > µ; in passing,

we remark that the presence of source term b in (5.1.1) makes the solution exist for a longer time relative to what it would have been in the absence of

a source term Further, if (5.1.1) has an associated conservative form, then (5.1.6), which is the proper conservation form, can be used to study the prop-agation of weak shocks (see [75])

5.1.2 Nonlinear geometrical acoustics solution in a relaxing

gas

The foregoing asymptotic analysis can be used to study the small ampli-tude high frequency wave solution to the one dimensional unsteady flow of a relaxing gas described by the system (4.6.1), i.e., ui

,t+ Aijuj

,x+ Bi= 0, i, j =

1, 2, 3, 4, where the symbols have the same meaning, defined earlier Here we are concerned with the motion consisting of only one component wave associ-ated with the eigenvalue λ = u + a The medium ahead of the wave is taken

to be uniform and at rest The left and right eigenvectors of A corresponding

to this eigenvalue are

L = (0, 1/(2a), 0, 1/(2ρa2)), R = (ρ, a, 0, ρa2), (5.1.8) and the phase function φ(x, t) is given by φ(x, t) = x−aot, where the subscript

o refers to the uniform state uo= (ρo, 0, σo, po) ahead of the wave The trans-port equation for the wave amplitude π is given by (5.1.6) with coefficients µ and ν determined as

µ = (α + (aoΩ)/2) and ν = (γ + 1)ao/2, (5.1.9) where α is the same as in (4.6.14)1 The characteristic field associated with π

in (5.1.6) is defined by the equations

dx/dt = ao, dθ/dt = ((γ + 1)aoπ)/2 (5.1.10)

In view of (5.1.10), equation (5.1.6) can be written as dπ/dt = −ao(α+(Ω/2))π along any characteristic curve belonging to this field, and yields on integration

π = πo(xo, θo)(A/Ao)−1/2exp{−αaot}, (5.1.11) along the rays x − aot = xo(constant) We look for an asymptotic solution of the hyperbolic system (4.6.1)

u = uo+ u1(x, t, θ) + O(2), (5.1.12) satisfying the small amplitude oscillatory initial condition

u(x, 0) = g(x, x/) + O(2), (5.1.13) where g is smooth with a compact support; indeed, expansion (5.1.12) with u1

given by (5.1.4), where the wave amplitude π is given by (5.1.11), is uniformly valid to the leading order until shock waves have formed in the solution (see Majda [116]) Using (5.1.11) in (5.1.10)2we obtain the family of characteristic curves parametrized by the fast variable θoas

θ −(γ + 1)2 aoπo(xo, θo)

Z t A(xo+ aoˆt) A(xo)

−1/2

exp (−αaoˆt)dˆt = θo (5.1.14)

5.2 Far Field Behavior 137

It may be noticed that equations (5.1.11) and (5.1.14) are similar to the equa-tions (4.6.16) and (4.6.17), and therefore the discussion with regard to the shock formation and its subsequent propagation follows on parallel lines It may be recalled that for plane (m = 0) and radially symmetric (m = 1, 2) flow configurations, (A/Ao) = (x/xo)m, and therefore the integral in (5.1.14) converges to a finite limit as t → ∞ Thus, an approximate solution (5.1.12), satisfying (5.1.13), is given by

u = u0+ Roπo(xo, θo)(A(xo+ aot)/A(xo))−1/2exp (−aoαt),

where R is given by (5.1.8)2; the fast variable θo, given by (5.1.14), is chosen such that θo = x/ at t = 0, and the initial value of π is determined from (5.1.13) as πo(x, ξ) = g(x, x/)/ao This completes the solution of (4.6.1) and (5.1.13); any multivalued overlap in this solution has to be resolved by introducing shocks into the solution

5.2 Far Field Behavior

When the characteristic time τ associated with the boundary data is large compared with the time scale τrdefined by the dissipative mechanism present

in the medium (i.e., δ = τr/τ << 1), the situation corresponds to the low frequency propagation condition This means that the time and distances con-sidered are large in comparison to the relaxation time or relaxation length Since at large distances, away from the source, any nonlinear convection is associated with the low frequency characteristics, and as the principal signal

in this region is centered on the equilibrium or low frequency characteristics,

it is possible to introduce a reduced system of field equations which provides

an approximate description of the wave process Based on Whitham’s ideas,

it was shown by Fusco [60] that for a quasilinear first order system of PDEs with several space variables involving a source term, it is possible to introduce

a reduced system which, in an asymptotic way, brings out the dissipative ef-fects produced by the source term against the typical nonlinear steepening of the waves In fact, the wave motion is asymptotically described by a trans-port equation of Burger’s type that holds along the characteristic rays of the reduced system We illustrate this procedure for the one dimensional nonequi-librium gas flow described by the system (4.6.1)

As the principal signal in the far field region is centered on the equilibrium characteristic, the system (4.6.1) is approximated by the following reduced system

ρ,t+ uρ,x+ ρu,x+ Ωρu = 0, u,t+ uu,x+ ρ−1p,x= 0,

p,t+ up,x+ ρa2

∗(u,x+ Ωu) = 0, Q(p, p, σ) = 0, (5.2.1)

where a∗= (∂p/∂ρ)|1/2S,σ=σ∗is the equilibrium speed of sound with S as specific entropy of the gas

In order to study the influence of nonequilibrium relaxation in (4.6.1) on the wave motion, associated with (5.2.1), we consider the following stretching

of the independent variables ˜x = δ2x, ˜t = δ2t When expressed in terms

of ˜x and ˜t, and then suppressing the tilde sign, the system (4.6.1) yields the following set of equations

ρ,t+ uρ,x+ ρu,x+ Ωρu = 0, u,t+ uu,x+ ρ−1p,x= 0,

p,t+ up,x+ γp(u,x+ Ωu) = −(γ − 1)ρQδ−2, (σ,t+ uσ,x) = Qδ−2

(5.2.2)

In the limit δ →0, the above system yields the reduced system (5.2.1), and, therefore, the transformed system (5.2.2) may be regarded as a perturbed problem of an equilibrium state characterized by (5.2.1); see Fusco [60]

We now look for an asymptotic solution of (5.2.2) exhibiting the character

of a progressive wave, i.e.,

f (x, t) = f0+ δf1(x, t, θ) + δ2f2(x, t, θ) + · · · , (5.2.3) where f may denote any of the dependent variables ρ, u, σ and p; f0 refers to the known constant state, θ = φ(x, t)/δ is a fast variable, and φ(x, t) is the phase function to be determined

By introducing (5.2.3) and the Taylor’s expansion of Q about the uniform state (ρ0, 0, σ0, p0) into the transformed equations (5.2.2)1 and (5.2.2)2 and canceling the coefficients of δ0, δ1, we obtain the following system of first order partial differential equations for the first and second order variables

O(δ0) : ρ1 ,θφ,t+ ρou1 ,θφ,x= 0, ρou1 ,θφ,t+ p1 ,θφ,x= 0,

O(δ1) : ρ2,θφ,t+ ρou2,θφ,x= −ρ1,t− ρou1,x−mρo u 1

x + (u1ρ1,θ+ ρ1u1,θ)φ,x, O(δ1) : ρou2 ,θφ,t+ p2 ,θφ,x= −ρ1u1 ,θφ,t− ρou1u1 ,θφ,x− p1,x− ρou1 ,t

(5.2.4) Similarly, equations (5.2.2)3 and (5.2.2)4, on equating the coefficients of δ0

and δ−1, yield the following equations

O(δ0) : p1,θφ,t+ ρoa2u1,θφ,x= −(γ − 1)ρoQ2, σ1,θφ,t= Q2,

O(δ−1) : σ1= c(p1− (a2/γ)ρ1)/ρo, (5.2.5) where Q2=

 c

τ ρ2



poρ2

ρo − p1ρ1

 + cp2

τ ρo−cpoρ2

τ ρ2 −σ2

τ

 Eliminating Q2 between (5.2.5)1 and (5.2.5)2, and using (5.2.5)3 in the resulting equation, we get

γ{1 + (γ − 1)c}p1 ,θφ,t+ ρoa2γu1 ,θφ,x− (γ − 1)ca2ρ1 ,θφ,t= 0 (5.2.6) Equation (5.2.6), together with (5.2.4)1 and (5.2.4)2, constitutes a system

of three equations for the unknowns ρ1 ,θ, u1 ,θ and p1 ,θ The necessary and sufficient condition for this system to have a nontrivial solution is that the

5.2 Far Field Behavior 139 determinant of the coefficient matrix must vanish, i.e., −φt/φx = 0, ±Γ , where Γ = a0

 (cγ + γ − c)/(cγ2− cγ + γ) 1/2 For −φt/φx= +Γ, the phase function φ is determined as φ(x, t) = t − ((x − xo)/Γ) when φ(xo, t) = t; as the vector of unknowns is collinear to the right null vector of the coefficient matrix, we get

ρ(1)= Γ−2p(1), u(1)= (ρ0Γ)−1p(1), σ(1) = (γ−1)c {ρ0(γ + (γ − 1)c)}−1p(1)

(5.2.7)

It may be noticed that Γ is a characteristic speed related to the reduced system (5.2.1) The system of equations for the second order variables, on multiplying by the left null vector of the coefficient matrix, corresponding to the value −φ,t/φ,x = Γ, and taking into account the relations (5.2.7), yields the following transport equation for p(1) in the moving set of coordinates X and ξ,

∂Xp(1)− λ p(1)∂ξp(1)+ mp(1)/2X = ω∂ξξ2 p(1), (5.2.8) where ∂X = ∂x+ Γ−1∂t and the nonlinear and dissipation coefficients λ and

ω are given by

λ = (γ/ρ0ΓΛ1) {γ + 1 + 2c(γ − 1)} , ω = 2cτγΓa20{(γ − 1)/Λ1}2, with Λ1= 2a2(cγ + γ − c)

Equation (5.2.8) is known as the generalized Burgers equation which allows

us to study in detail various effects that appear in the propagation of plane (m = 0), cylindrical (m = 1) and spherical (m = 2) waves in a dissipative medium with a quadratic nonlinearity

In contrast to the high frequency nonlinear solution, discussed earlier, a significant feature of the low frequency solution in the far field is its continuous structure, i.e., any convective steepening is always diffused by the dissipative nature of the relaxation The reader is referred to Crighton and Scott [44] and Manickam et al [117] for a detailed discussion of the analytical and numerical solutions of such an equation

Remarks 5.2.1: In a special case where the coefficient ω vanishes in (5.2.8), the source term B involved in (4.7.1) does not cause dissipation; in this sit-uation, the asymptotic development (5.2.3) needs to be modified in order to see how the higher order terms in (4.7.1) may influence the wave motion as-sociated with the characteristic speed of the reduced system The transport equation for the wave amplitude in this case is the generalized Korteveg-de Vries (K dV) equations While considering the diffraction of a weakly nonlin-ear high frequency wave in a direction transverse to its rays for a first order quasilinear system involving a source term, it can be shown that the wave am-plitude is governed by the Zabolotskaya-Khokhlov equation or by a modified Kadomtsev-Petviashvili (KP) equation (see [59] and [60])

5.3 Energy Dissipated across Shocks

We have noticed that the classical solutions of nonlinear hyperbolic con-servation laws, in one or several unknowns, break down in a finite time be-yond which the solutions have to be interpreted in the sense of distributions, known as weak solutions These weak solutions often contain discontinuities supported along lower dimensional manifolds The strength of the discontinu-ity is constrained through the Rankine-Hugoniot formula, which relates the jump in the solution to the unit normal to the discontinuity locus Recently, motivated by their study on gravity currents, Montgomery and Moodie [125] have analyzed the effect of singular forcing terms on the Rankine-Hugoniot conditions in systems involving one space dimension The singular forcing function considered, is a sum of a surface density and a volume density and the latter does not contribute to the jump relation Their results to several space dimensions were generalized in [191] using pull back of distributions; in-deed, the analysis in [191] sheds some light on the mechanism through which the terms involving the volume density cancel out, thereby not contributing to the jump conditions The analysis in this section is related to another problem

in the system of conservation laws namely, the manipulation of conservation laws through multiplication by polynomial nonlinearities The resulting sys-tems contain distributional terms supported along singularity loci; these may

be regarded as singular forcing terms considered in [125] Here we derive an expression for these singular terms which have been interpreted as the energy dissipated at the shocks As an application, the expression for these singular terms is obtained from the transport equation governing the propagation of small amplitude high frequency waves in hyperbolic conservation laws

5.3.1 Formula for energy dissipated at shocks

We consider an autonomous scalar conservation law without source term for a scalar valued function u on a domain Ω in IRn+1 namely,

n

X

j=0

∂jFj(u) = 0 (5.3.1)

Let the system (5.3.1), when multiplied by P (u), a polynomial or a smooth function of one variable, transform into

n

X

∂j( ˜Fj(u)) = 0, (5.3.2)

5.3 Energy Dissipated across Shocks 141 for certain densities ˜Fj(u) defined on Ω The systems (5.3.1) and (5.3.2) are equivalent for smooth solutions u of (5.3.1), namely the identity

P (u)

n

X

j=0

∂jFj(u) =

n

X

j=0

∂jF˜j(u),

holds for all smooth functions u However, the class of weak solutions of this systems involving discontinuities, are different since the corresponding Rankine-Hugoniot conditions differ To restore the equivalence of the two sys-tems for a class of distributional solutions which are piecewise smooth with a jump discontinuity along a locus Φ(x) = 0, we modify the equation (5.3.2) by adding a distributional term to (5.3.2) namely,

P (u)

n

X

j=0

∂jFj(u) =

n

X

j=0

∂j( ˜Fj(u)) + E, (5.3.3)

where the distribution E is supported on Φ(x) = 0 In the context of nonlinear geometrical optics, this singular term is interpreted as the energy dissipated across the shock locus Φ(x) = 0 We proceed to determine explicitly the distribution E Let φ be any arbitrary test function, then (5.3.3) gives

n

X

j=0

Z

Ω

P (u)∂jFj(u)φ =

n

X

j=0

hP (u)∂jFj(u), φi

= hE, φi +

n

X

j=0

h∂jF˜j(u), φi, (5.3.4)

in the sense of distributions We proceed to simplify the term Σn

j=0h∂jF˜j(u), φi

on the right-hand side of (5.3.4):

n

X

j=0

h∂jF˜j(u), φi = −

Z

Ω

n

X

j=0

˜

Fj(u)∂jφ

= − Z

Φ=0

n

X

j=0

[ ˜Fj]njφdS +

Z

Ω

n

X

j=0

∂j ˜Fj(u)φ, (5.3.5)

where [f (u)] denotes the jump of f (u) across the locus of discontinuity Φ = 0, ˆ

n = (n1, n2, , nn) is the unit normal vector to Φ = 0 and dS is the area measure on the surface Since the function u is of class C1up to the boundary

on either side of the discontinuity locus, the left-hand side of (5.3.4) transforms as

n

XZ

Ω

P (u)∂jFj(u)φ =

n

XZ

Ω

∂jF˜j(u)φ, (5.3.6)

and so (5.3.4) and (5.3.5) prove that the distribution E is a smooth density supported on the discontinuity locus given by

E : φ 7→

Z

φ=0

φ

n

X

j=0

[ ˜Fj(u)]njdS (5.3.7)

We now prove a general result concerning the distribution E arising out of multiplying (5.3.1) by P (u)

Theorem 5.3.1 Assume that the conservation law

∂tF0(u) + ∂xF1(u) + G(x)u = 0,

on multiplication by P (u) becomes

∂tF˜0(u) + ∂xF˜1(u) + G(x)uP (u) + E = 0.

The distribution E, supported on the discontinuous locus t = Ψ(x), is given

by the density

−

 [ ˜F0][F1] [F0]− [ ˜F1]



Φ∗(δ0), (5.3.8) where Φ∗(δ0) denoted the pull back of δ0 by Φ with Φ(x, t) = t − Ψ(x) Proof: In the special case of one space variable, the discontinuity locus is taken to be the graph of the function t = Ψ(x) So, ˆndS = (−Ψ0, 1)dx which on substitution in (5.3.7) and using the Rankine-Hugoniot condition (see (5.3.12)

Remark 5.3.1: The distribution E is of order zero and so involves only the pull back of δ0 Presumably for higher order equations, the distribution E may be of higher order involving the pull back of derivatives of δ0

5.3.2 Effect of distributional source terms

Recall the formula for pull back of a distribution by a submersion [74]

∂jΦ∗H = (∂Φj)Φ∗δ0= (∂jΦ)dS

|∇xΦ| , (5.3.9) where ∇xis the spatial gradient operator, and H denotes the Haviside function

so that Φ∗(H) is the characteristic function of the set {x : Φ(x) > 0} = Ω+, implying thereby

∂jχΩ += (∂jΦ)dS

Similarly, we define Ω−{x : Φ(x) < 0} Let us now consider a weak solution u

of a scalar conservation law (5.3.1) in n independent variables in a region Ω