Characteristic Lines. Hyperbolic Systems. Riemann Invariants

S.12.3. Characteristic Lines. Hyperbolic Systems. Riemann Invariants

Let us show that some systems of conservation laws can be represented as systems of ordinary differential equations along curves x = x(t) called characteristic curves.

Differentiating both sides of system (1) yields

where A = e G −1 (u)e F(u), e F(u) is the matrix with entries ∂F i

∂u j ,e G(u) is the matrix with entries ∂u j , and

∂G i

G e −1 is the inverse of the matrix e G. Let us multiply each scalar equation in (7) by b i = b i (u) and take the sum. On rearranging terms under the summation sign, we obtain

i,j=1

where the a ij = a ij (u) are the entries of the matrix A.

λ = λ(u), so that

b j a ji = λb i ,

j=1

then equation (8) can be rewritten in the form

Thus, system (7) is transformed to a linear combination of total derivatives of the unknowns u i with respect to t along the direction (λ, 1) on the plane (x, t), i.e., the total time derivatives are taken along the trajectories having the velocity λ:

b i = b i (u), λ = λ(u), x = x(t),

dt ∂x Equations (10) are called differential relations on characteristics. The second equation in (10) explains why an eigenvalue λ is called a characteristic velocity.

dt

∂t

The system of quasilinear equations (7) is called hyperbolic if the following two conditions are satisfied:

1 ◦ . All eigenvalues λ k = λ k (u) ( k = 1, . . . , n) of the matrix A(u) are real.

2 ◦ . There is a basis {b 1 , ...,b n }⊂E n formed by n left eigenvectors of A(u) and subjected to a normalization condition; the symbol E n stands for the n-dimensional Euclidean space.

Let us assume that the n × n hyperbolic system (7) has n distinct eigenvalues λ k (u), k = 1, . . . , n.

A trajectory x(t) with velocity λ k (u) that is a solution of system (10) is called the kth characteristic direction. The eigenvectors b k (u) that correspond to the eigenvalues λ k (u), respectively, are linearly independent.

⊂R n , they can be enumerated in order of increasing values, so that λ 1 (u) < ···<λ n (u), and system (7) is called strictly hyperbolic.

If all eigenvalues are distinct for any u = ( u 1 , ...,u n ) T

If all characteristic velocities λ=λ k of the hyperbolic system (7) are positive, the following initial-boundary value problem can be posed:

u=u L at t = 0,

u=u R at x = 0.

Remark 1. If the hyperbolic system (7) is linear and the coefficients of the matrix A are constant, then the eigenvalues λ k are constant and the characteristic lines in the ( x, t) plane become straight lines:

x=λ k t + const .

Since all eigenvalues λ k are different, the general solution of system (7) can be represented as the sum of particular solutions as follows:

(11) where the φ k ( ξ k ) are arbitrary functions, ξ k = x−λ k t, and r k is the right eigenvector of A corre-

u= 1 ( φ 1 x−λ

1 t)r

n ( x−λ n t)r +···+φ n ,

sponding to the eigenvalue λ k , k = 1, . . . , n. The particular solutions u k = φ k ( x−λ k t)r k are called traveling wave solutions. Each of these solutions represents a wave that travels in the r k -direction with velocity λ k .

Remark 2. The characteristic form (9) of the hyperbolic system (7) forms the basis for the numerical characteristics method which allows the solution of system (7) in its domain of continuity.

Fu ()

of increasing wave

direction

in rarefaction

arctan l

Figure 6. Characteristic velocity for a single quasilinear (hyperbolic) equation (2).

Suppose that we already have a solution u( x, t) for all values of x and a fixed time t. To construct a solution at a point ( x, t + ∆t), we find the points (x − λ k ∆ t, t) from which the characteristics arrive at the point ( x, t + ∆t). Since the u(x − λ k ∆ t, t) are known, relations (10) can be regarded as a system of n linear equations in the n unknowns u(x, t + ∆t). Thus, a solution for the time t + ∆t can be found.

Consider small perturbations of a solution to system (7). Substitute u = u 0 + δu into (7), where u 0 =u 0 ( x, t) is a solution of system (7) and δu = (δu 1 , . . . , δu n ) T is a small perturbation, |u 0 | ≫ |δu|. Neglecting the terms of higher order than the first term in | δu|, we obtain a system of linear equations in the form

i = 1, . . . , n, (12)

∂t

∂x

j,k=1 ∂u k ∂x

j=1

where the a ij = a ij (u 0 ) are the entries of the matrix A at the point u 0 . If u 0 is a constant vector, then the right-hand side of the linearized equation (12) is zero and its general solution can be represented as a superposition of n traveling waves; see formula (11).

Example 3. For the case of a single hyperbolic equation (2), relations (10) become

It has been taken into account here that λ=F ′ ( u); the prime denotes the derivative with respect to u. The second equation in (13) shows that the characteristic velocity equals the tangent to the flux function at the point u = u(x, t); see Fig. 6 . There is one characteristic velocity for one equation, and the unknown function is constant along the characteristic (first equation in (13)). Therefore, the characteristic velocity is also constant (second equation in (13)), and the characteristic is a straight line. This allows the construction of an exact solution to a Cauchy problem for (2) whenever the

characteristic velocity of the initial condition (5) increases monotonically in x, [F ′ ( ϕ(x))] ′ = F ′′ ( ϕ)ϕ ′ ( x) > 0. In this case, a unique characteristic straight line crosses an arbitrary point ( x, t), and the solution is constant along this line. As a result, the solution can be represented in the parametric form

x=ζ+F ′ ϕ(ζ) t,

u = ϕ(ζ).

The first equation in (14) is a transcendental equation in the unknown ζ = ζ(x, t), and the second one allows the calculation of the unknown u = u(x, t) from the initial condition (5).

Example 4. Adiabatic gas flow is governed by the system of equations (3)–(4). The vector u and the matrix A(u), which arise in the transformed system (7), become

where p ′ = p ′ ( ρ). The eigenvalues and the corresponding left eigenvectors are

b= p p ′ , ❴ ρ .

λ=v ❴

The linear combination of the equations (3)–(4) with coefficients b i is:

Z√

p ′ dρ ❴ ρ dv ≡ ❴ ρ d v ❴

p ′ dρ = 0.

dt

dt

dt

R 1 = const

R 2 = const

Figure 7. Loci of points where Riemann invariants are constant.

For an ideal polytropic gas, with p = Aρ γ , the eigenvalues and the corresponding left eigenvectors are:

In this case, the differential relations on the characteristics (10) become

The relations on the characteristics (10) can be simplified if system (7) admits Riemann invari- ants. Consider the differential b k i (u) du i , where b k (u) is a left eigenvector corresponding to the eigenvalue λ k (u). Assume that this differential admits an integrating multiplier µ k (u) or, in other words, the differential can be represented in the form

i (u) du i = µ (u) dR k (u).

i=1

The function R k (u) is called the kth Riemann invariant. The integrating multiplier µ k (u) can be found from Maxwell’s relations:

∂u j µ k

∂u i µ k

From (10) it follows that each Riemann invariant is constant along the corresponding characteristic curve.

Two Riemann invariants can always be constructed for a system of two equations, since the differential of two variables always admits an integrating multiplier. In this case, the change of variables

u i = u i ( R 1 , R 2 ),

i = 1, 2,

brings the hyperbolic system to

Example 5. As follows from (13), the Riemann invariant for the single equation (2) is the density u(x, t), which is constant along characteristics.

Example 6. Let us consider an adiabatic gas flow (see Example 4). From (15) it follows that the Riemann invariants are:

For an ideal polytropic gas, with p = Aρ γ , the Riemann invariants are constant along characteristics:

2 Aγρ γ−1

along dx

Figure 7 shows lines of R i = const on the phase plane ( v, ρ).

the equations of balance of mass and momentum:

∂u − ∂v = 0,

∂x ∂v − ∂σ(u) = 0.

Here, u is the deformation gradient (strain), v is the strain rate, and σ(u) is the stress. The eigenvalues and the corresponding left eigenvectors are given by

The Riemann invariants are constant along characteristics:

Zp ′ ( along dx = ❛ R=v p σ u) du = const σ ′ ( u).

dt

References for Subsection S.12.3: I. M. Gelfand (1959), P. Lax (1973), G. B. Whitham (1974), A. Jeffrey (1976), F. John (1982), B. L. Rozhdestvenskii and N. N. Yanenko (1983), J. Smoller (1983), R. Courant and D. Hilbert (1985), D. Serre (1996), C. M. Dafermos (2000), A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov (2001), R. J. LeVeque (2002).