Shock and rarefaction curves

In the previous section we have seen waves named “contact discontinuities”, “rarefactions”, and

“shocks”. We would like to describe in somewhat more details what these waves are and in which problems they arise. In practice, we need to be able to construct the discontinuity curves. To this aim, we start from a point uo ∈ R^d and try to find the state vector u that satisfies the Rankine-Hugoniot equations (2.44):

λ(u− u⁰) = f (u)− f(u⁰) = A(u₀, u)(u− u⁰).

Obviously, the trivial identity is obtained for u = u₀. We want to find a different solution in a neighborhood of u₀. We note that this is a n-dimensional nonlinear system with n + 1 unknowns (u, λ) and cannot be solved as is.

The bi-orthogonality condition of the left and right eigenvalue systems of A(u₀, u) yields:

hlj(u₀, u), (u− u0)i = 0 for all j 6= i, i = 1, . . . , n (2.50) This is a n− 1-dimensional nonlinear system in the n unknowns given by the components of u. We can linearize this system by noting that by definition of average matrix (2.43) A(u0, u₀) = A(u₀), and hence we can write the following n− 1-dimensional linear system:

hlj(u₀), (u− u0)i = 0 for all j 6= i.

Because of the bi-orthogonality condition, the previous equation tells us that the vector u− u0 must be perpendicular to all the left eigenvectors lj j 6= i, and thus parallel to rⁱ:

u = u₀+ αr_i(u) for all i.

Thus, the state vector u along a shock curve must be tangent to the curve, or stated in a different way, the solution to (2.50) is a smooth curve tangent to i-th eigenvector r_i of the Jacobian matrix A(u₀) at u0, a condition that gives us a (inconvenient but) practical way to calculate the curve given u0. Thus, at least in principle, the equation of the characteristic curve can be written in parametric form with parameter σ as:

u = S_i(σ)(u₀). (2.51)

We note that the curve at each point σ must be orthogonal to li and tangent to ri(see Figure 2.24).

On the other hand, we can define the “i-th rarefaction curve” as the integral curve of the vector field r_i(u) passing through u₀. In other words, to determine the i-th rarefaction curve we can define a differential equation in parametric space imposing the tangent condition on the right eigenvalue:

du

dσ = r_i(u), u(0) = u₀.

The solution to this equation gives a curve (again in parametric space):

u = R_i(σ)(u₀), (2.52)

which results close to the shock curve u = Si(σ)(u0) up to second order around u0 (Figure 2.24).

Hence, at least locally around u0 we can confound Si(σ)(u₀) with R_i(σ)(u₀). In fact, the following result can be proved:

R_i(σ)(u₀) = u₀+ σr_i(u₀) +O σ²
S_i(σ)(u₀) = u₀+ σr_i(u₀) +O σ²
,

x u1

u2

x = 0 u^L

u^R u(t)

x t x = λi(u^L)t x = λi(u^R)t

u = u^L

u = u^R

FIGURE 2.25: Solution of the Riemann problem in the case of a rarefaction wave for a two-dimensional system. The left panel shows the situation in theu₁, u₂− x space. The states u^Landu^R are connected by a curve in the u₁− u², x space representing the rarefaction wave after each wave point has been advected with the corresponding speedλ_i (solid lines propagating from left to right).

The right panel shows the characteristics in thex− t space.

so that the error between Siand Riis given by:

| Rⁱ(σ)(u₀)− Sⁱ(σ)(u₀)|≤ Cσ³

while the speed of propagation λ_ialong the line can be expressed as:

λ_i(S_i(σ)(u₀), u₀) = λ_i(u₀) + σ

2h∇ λi(u₀), r_i(u₀)i + O σ²
.

Before proceeding with the description of different elementary waves, we report a definition due to Lax [8] which is useful to classify some simple situations that occur in practice. We call “genuinely nonlinear” an i-th wave for which the directional derivative of the i-th eigenvalue along the corre-sponding right eigenvector r_i is strictly greater than zero, i.e.,h∇ λi(u), r_i(u)i > 0 for all u. When h∇ λⁱ(u), r_i(u)i = 0 the i-th wave is called “linearly degenerate”. In the first case, the speed of prop-agation λ_i along the characteristic curve S_i(σ)(u₀), is always increasing with u, since its derivative along the curve is greater than zero. In the second case, we have that the speed of propagation is constant long the characteristics, and thus similar to the “linear” case. Obviously the two cases can co-exist and co-evolve depending on the nature of the Jacobian matrix A(u) of the system.

Centered Rarefaction Waves. We consider a genuinely nonlinear wave and assume that u^R = R_i(σ)(u^L), i.e., u^Rlies in the i-th rarefaction curve emanating from u^L. For some s∈ [0, σ], λⁱ(s) = λ_i(R_i(s)(u^L). If the wave is genuinely nonlinear, we can without loss of generality assume that λ_i is

increasing with s and is injective (for each value of λ ∈ [λⁱ(u^L), λ_i(u^R)] there is a unique value of s such that λ = λ_i(s)). The solution of the Riemann problem is then given by:

u(x, t) =







u^L if x < λi(u^L)t,

Ri(s)(u^L) if x = λi(s)t, λi(s)∈ [λⁱ(u^L), λi(u^R)], u^R if x > λi(u^R)t.

Figure 2.25 summarizes the behavior. The construction of the solution u(x, t) for a fixed time t proceeds as follows. The rarefaction wave at x = 0 connecting the left and right states u^Land u^Ris drawn on the spanning u and passing through x = 0 (shown in the figure with a dotted curve). Each point of this curve is moved horizontally by the amount λ_i(u)t.

Shock Waves. We look now at the case in which the u^Rstate is connected t the u^Lstate by a shock curve u^R§ⁱ(σ)(u^L). This occurs if the system is genuinely nonlinear. We can define the Rankine-Hugoniot speed λ = λ_i(u^R, u^L). The (piecewise constant) solution to the Riemann problem is then:

u(x, t) =

(u^L if x < λt u^R if x > λt.

Note that for σ > 0, we have that λ_i(u^R) > λ_i(u^L) and thus the Lax admissibility condition is violated. Obviously, the Lax condition is satisfied, i.e., u(x, t) is “entropy-stable” for σ < 0 since then λi(u^R) < λ_i(u^L, u^R) < λ_i(u^L).

Contact Discontinuity Waves. A contact discontinuity occurs for a linearly degenerate wave field.

Let u^R = Ri(σ)(u^L), i.e., the states u^L and u^R lie on a rarefaction curve, along which the speed is a constant (by assumption intrinsic in the Riemann problem formulation). The Rankine-Hugoniot condition writes:

f (u^R)− f(u^L) = Z σ

0 J^ff R_i(s)(u^L)

r_i R_i(s)(u^L)
ds

= Z σ

0

λ_i(u^L) r_i R_i(s)(u^L)
ds

= λ_i(u^L)R_i(σ)(u^L)− u^L .

The Lax entropy condition is satisfied as λ_i(u^L) = λ_i(u^L, u^R) = λ_i(u^R) and, moreover, the shock and rarefaction curves coincide S_i(σ)(u^L) = R_i(σ)(u^L) for all σ.

The three elementary wave types are exemplified by the graphs in Figure 2.26 drawn in the x− t plane. The structure of the different waves can be identified by the direction of the left and right characteristics (drawn as dashed arrows) that are converging towards the discontinuity line in the case of a shock, parallel to the discontinuity line in the case of a contact discontinuity, and parallel to the left and right characteristics lines external to the rarefaction fan.

x

t x = λit

u = u^L

u = u^R

x

t x = λit

u = u^L

u = u^R

x t x = λi(u^L)t x = λi(u^R)t

u = u^L

u = u^R

FIGURE2.26: Qualitative behavior of the characteristic fields for the three types of elementary waves used in the solution of the Riemann problem: shock (left), contact discontinuity (center), rarefaction (right). The dash arrows indicate the directions of the characteristic curves outside the discontinuity region.

General solution of the Riemann problem The general solution of the Riemann problem can now be obtained as a superposition of elementary waves. To this aim, we want to find intermediate states u^L = α₀, α₁, . . . , α_d = u^R such that each pair of adjacent states can be connected by one of the previously described elementary waves. In mathematical terms we can say that the intermediate state α_i is connected to the previous state αi−1via a curve αi = Ψ_i(σ_i)(α_i−1), so that the u^Land u^Rstates can be written as a composition of the d functions:

u^R= Ψ_d(σ_d)◦ . . . Ψ1(σ₁)(u^L).

If u^Rand u^Lare not too far apart, it can be shown that the above equation is well-posed and that there exist unique values of the wave strengths σ_i such that the above equation holds. Hence we proceed by starting from the u^L = α0state and solve a sequence of Riemann problems for i = 1, . . . , n:

u_t+ f (u)_x = 0 u(x, 0) =

(α_i−1 if x < 0, α_i if x > 0,

as seen for example in the case of a linear system in Figure 2.23. Each of the Riemann problems has an entropy solution in the i-th characteristic family.

3 Numerical Solution of Hyperbolic Conservation Laws

In this section we used the previous developments to work the details of numerical schemes for the solution of Hyperbolic Conservation Laws. We will address essentially the method of Finite Volumes and digress only shortly on the Discontinuous Galerkin method, without touching at all the important field of purely Lagrangian methods.

We will be using most of the theoretical developments described in the previous sections as funda-mental building blocks for our numerical approach. We will focus on standard finite volume schemes maintaining and discussing as much as possible multidimensional problems. However, as truly mul-tidimensional methods are still open questions, we will spend much time looking at one-dimensional approximations, but we will try to use multidimensional approximations. For the more recent devel-opments we refer the reader to more specialized books.

3.1 The Differential Equation

Problem 3.1. Given a domain Ω ∈ R^d, which we assume possesses sufficient regularity, we want to find a vector function u ∈ R^m, u = (u1, . . . , um), ui = uu(x, t) : Ω × R⁺ 7→ R, that satisfies the following equation:

ut+ div f (u) = 0 for all x∈ Ω and t ∈ [0, T ], (3.1)

u(x, 0) = u₀ for all x∈ Ω and t = 0, (3.2)

u(0, t) = u_b for all x∈ ∂Ωin and t∈ [0, T ], (3.3)

where f (u) = (f1(u), . . . , f_d(u)), f_i(u) : R^m 7→ R^m.

We notice here that the ”inflow” boundary where boundary conditions are imposed must satisfy the compatibility conditions discussed in section 2.3. We start by giving some basic definitions of the domain discretizations and its property and define the geometric quantities that will be needed in the development.