Systems of first-order equations

2.6.1 The linear case

We have seen several times now that the wave equation is in general re-written as a system of two first-order equations. For this reason it is fundamental to study systems of linear first-order ODEs. To this aim, we let u = (u1, u2, . . . , ud)^T = u(x, t)∈ R^dbe a column vector satisfying:

A(x, t)∂u

∂t + B(x, t)∂u

∂x = C(x, t)u + d(x, t), (2.32)

where A, B, and C, are real n× n matrices and d ∈ R^d. We prescribe Cauchy data along the curve t = φ(x). In this first order case we prescribe values of u:

u(x, φ(x)) = f (x).

Differentiating the previous equation, we have:

u_x+ φ⁰u_t= f⁰, and substituting into (2.32):

(A− φ⁰B)u_t = Cf + d− Bf⁰.

The curve φ is a characteristic curve if the derivatives of u cannot be found from the Cauchy data alone. In this case this amounts to saying that the matrix multiplying utis singular:

det (A− φ⁰B) = 0.

Noting that φ⁰ = dt/dx, we can write the following differential equation of order n:

det (Adx− Bdt) = 0,

called the characteristic differential equation.

(X, T ) t

x P_n . . .

P₃ P₂

P₁

FIGURE 2.12: Characteristic curves obtained by solving (backwards) the differential equation (2.34) for each eigenvaluei starting from the point (X, T ).

For example, assuming matrix A invertible at t = 0, we can prescribe initial data on the x-axis:

u(x, 0) = f (x).

For small t, we can multiply equation (2.32) by A⁻¹ to obtain:

u_t+ ˜Bu_x = ˜Cf + ˜d,

where ˜B = A⁻¹B, ˜C = A⁻¹C, and ˜e = A⁻¹d. The characteristic differential equation becomes then:

det



B˜−dx dtI



= 0, i = 1, . . . , n,

which states that the speed of propagation of the i-th wave is the i-th eigenvalue of matrix ˜B:

dx_i/dt = λ_i(x, t), i = 1, . . . , n, (2.33)

which is a set of n differential equations. Note that, dx/dt = λi, i = 1, . . . , n are the eigenvalues of matrix ˜B(x, t). The system (2.32) is said to be hyperbolic if ˜B has a set of linearly independent (right) eigenvectors and corresponding real eigenvalues. In other words, ˜B is such that

Br˜ _i = λ_ir_i, i = 1, . . . , n,

where the eigenvectors ri are all linearly independent (form a basis for R^dand depend continuously on x, t). In the hyperbolic case, matrix ˜B can be diagonalized as:

B = U ΛU˜ ⁻¹ = U ΛV^T,

where matrices U and V contain the ordered (with respect to the eigenvalues λ_i) right and left eigen-vectors r_i and l_i as columns and the diagonal matrix Λ contains the corresponding eigenvalues λ_i (see Appendix D.3). Since U⁻¹ = V^T we can transform system (2.32) in canonical form by setting u = U v, and thus v = V^Tu, called the “characteristic variables”:

vt+ Λvx = ˜Cv + ˜d,

with ˜C = V^TCU − V^TU_t− V^TBU_x and ˜d = V^Td. A few basic facts about linear algebra and left and right eigenvectors are reviewed in Appendix D. Also the initial conditions are transformed:

V^Tf (x) = g(x) for t = 0. This way we have achieved a complete decomposition of the system obtaining a set of first order differential equations for each component vi of the vector function v that are coupled only on the right hand side. The i-th equation takes the form:

dv_i

Thus, we can use the method of characteristic to solve each equation obtaining a family of charac-teristics for each i = 1, . . . , n. The specific characteristic curve of the i-th family is obtained by

“back-tracking”, i.e., solving the differential equation starting from a generic point (X, T ) to obtain points P_i on the x-axis (Figure 2.12). We would like to remark that the (qualitative) curves on the left of point (X, T ) are relative to positive eigenvalues λ_i > 0, while the ones on the right correspond to negative λi. Indicating with x(t) = γi(t, X, T ) the backtracked characteristic curve of the i-th family, the general solution of (2.34) is obtained by integration along the curve γ_i:

v_i(X, T ) = g_i(γ_i(0, X, T )) +

We remark here that the main difference that we note with respect to the scalar case is that now the i-th wave (in the new variables v_i) is obtained as a superposition of all other waves v_j, j = 1, . . . , n traveling along different curves and with different characteristic speeds. In the integral above the domain of integration changes with the wave number i, and for this reason in the coefficients ˜c_ij and ˜d_i the substitution x(τ ) = γ_i(τ, X, T ) has been made where the curve γ_i is indexed with the wave number. We remark that, as seen in Section 2.2 for Burgers equation, the time interval where this solution is actually well defined cannot be too large, and we say that this solution is valid for sufficiently small T , usually without being more specific as done with Burgers equation.

In the previous paragraphs we have looked at a pure initial value problem, where initial data were prescribe at t = 0 on the entire x-axis. Now we want to consider the presence of a boundary. For this we assume that the equation is defined for x > 0, so that we need to impose a boundary condition at x = 0 for all times t > 0. However, we have to specify these boundary conditions only for those wave families corresponding to positive eigenvalues. For the other characteristic curves the domain of dependence does not include the t-axis (Figure 2.12). If we rank the eigenvalues from the

(X, T ) t

Pn x . . .

Pr

Pj

P1

FIGURE 2.13: Characteristic curves obtained by solving (backwards) the differential equation (2.34) for each eigenvaluei starting from the points (X, T ) and from the boundary point P_j.

smallest (λ_d) to the largest (λ₁), i.e., λ_d < λ_d−1 < . . . < 0 < λ_r < λ_r−1 < . . . < λ₁, we have the situation shown in Figure 2.13, where from point Pj a new set of characteristics has been identified by backtracking. We recall again that all these qualitative considerations work only in the case of smooth solutions, when u is continuous together with its first derivative. In more complex situations, as we have seen for the Burgers equation where shocks develop, we can still proceed with a similar analysis, but to do so we need to properly define how discontinuous solutions have to be interpreted.

Example 2.13. To exemplify the above development we address here the case of a system of first order PDEs with constant coefficients. Let u(x, t), f (x) ∈ R^d. Then our constant coefficient system takes the form:

u_t+ Au_x = 0, u(x, 0) = f (x), (2.35)

where matrix A is a hyperbolic matrix, i.e., it has real eigenvalues and eigenvectors and is diagonal-izable. We denote by r_i, l_i, λ_i the (constant) right and left eigenvectors and the eigenvalues of A, respectively, and as before we collect the right eigenvectors as columns of the matrix U and the left eigenvectors as columns of the matrix V . Recalling (D.11), we can express the component of u using the basis U as u_i =hlⁱ, ui. Then scalar multiplication of (2.35) by l¹, . . . , l_dyields equations for the i-th wave:

hlⁱ, u_ti + hlⁱ, Au_xi = u^i,t+ λ_iu_i,x = 0 u_i(x, 0) =hlⁱ, f (x)i = fⁱ(x).

We obtain a decoupled system of Cauchy problems that can be solved independently using the for-mula (2.3):

u_i(x, t) = f_i(x− λⁱt).

t

x t

x FIGURE 2.14: Superposition of two linear waves (left) and complex interactions of two nonlinear waves (right).

Then the solution to our problem is given by:

u(x, t) =

n

X

i=1

f_i(x− λit)r_i. (2.36)

This can be seen easily by taking time and space derivatives to obtain:

u_t(x, t) =

n

X

i=1

−λihli, f_x(x− λit)i ri =−Aux(x, t).

Again, we note that the solution is the superposition of n waves each traveling with characteristic speed λi.

2.6.2 The quasi-linear case

We have seen that in a linear system of conservation laws the different waves do not interact and the solution u is a linear superposition of the different waves. On the other hand, if the system is nonlinear the different waves may interact in a complex manner. The situation is qualitatively shown in Figure 2.14, where the linear waves move undeformed along the characteristic lines, while nonlinear waves move along curves and their interaction may generate new waves.

That this is true can be seen from the following discussion [2]. Consider the quasi-linear system of conservation laws:

u_t+ A(u)u_x = 0.

The matrix A(u) is a function of u and so are its eigenvalues λ_i(u) and its left and right eigenvectors l_i(u) and r_i(u). As usual we denote with U (u) = {r1(u), . . . , r_d(u)} the matrix of the basis vectors formed with the right eigenvectors, with V (u) = {l¹(u), . . . , l_d(u)} the matrix of the dual basis, i.e.

the left eigenvectors, and with Λ(u) the diagonal matrix of the eigenvalues of A. Using (D.11), we can express the spatial gradient∇ u = uxas a linear combination of the right eigenvectors r_iof A as:

u_t= (V^Tu_t)^TU =

n

X

i=1

hlⁱ(u), u_ti rⁱ =−Au^x = A(V^Tu_x)^TU = −

n

X

i=1

λ_i(u)hlⁱ(u), u_ti rⁱ(u).

Differentiating ux with respect to t and ut with respect to x, equating the mixed derivatives, after some calculations (for details see [2]), we obtain:

(u_x,i)_t+ (λ_iu_x,i)_x =

n

X

j=1,k=1 j>k

G_ijk, (2.37)

with G_ijk given by:

G_ijk = (λ_j − λ^k)hlⁱ, [r_j, r_k]i u^x,ju_x,k,

where [rj, rk] = J^rkrj− J^rjrk = [directional derivative of rkalong rj]- [directional derivative or rj

along rk] is called the Lie bracket. The quantityJ^rk denotes the Jacobian matrix of vector rj. There-fore,J^rkr_jindicates the directional derivative of r_kin the direction of r_j. We note that equation (2.37) is in conservation form and decoupled on the left-hand-side. The source term, however, couples all the waves. In fact, Gijk can be interpreted as the amount of i-waves originating from the interaction of j-waves with k-waves [2]. The quantity λ_j − λ^k describes the wave speed difference, the term u_x,ju_x,k the product of the densities of the j-waves and the k-waves, and hli, [r_j, r_k]i gives the i-th component of the Lie bracket in the U basis .