 The hydrodynamics equations are nothing more than signal-propagation equations, and equations of this kind are called hyperbolic equations. The equations of hydrodynamics are only a member of the more general class of hyperbolic equations, and there are many more examples of hyperbolic equations than just the equations of hydrodynamics.

## The simplest form of a hyperbolic equation

Consider the equation tq + u∂xq = 0.

where q = q(x, t) is a function of one spatial dimension and time, and u is a velocity that is constant in space and time. This is called an advection equation, as it describes the time dependent shifting of the function q(x) along x with a velocity u. The solution at any time t > t0 can be described as a function of the state at time t0: q(x, t) = q(x − ut, 0).

This is a so-called initial value problem in which the state at any time t > t0 can be uniquely found when the state at time t = t0 is fully given. The characteristics of this problem are straight lines: xchar(t) = x(0)char + ut. This is a family of lines in the (x, t) plane, each of which is labeled by its own unique value of (x(0)char). Advection of a function q(x, t) with constant velocity u (left) and space-varying velocity u(x) (right). The space-varying velocity problem comes in two versions: the conserved form (dashed) and the non-conserved simple advection form (dotted).

## Hyperbolic sets of equations: the linear case with constant Jacobian

Let us consider a set of linear equations that can be written in the form:∂tQ + A∂xQ = 0, where Q is a vector of m components and A is an m × m matrix. This system is called hyperbolic if the matrix A is diagonalizable with real eigenvalues. The matrix is diagonalizable if there exists a complete set of eigenvectors ei, i.e. if any vector can be written as:

In this case one can write

We can define a matrix in which each column is one of the eigenvectors:

Then we can transform the first equation into:

which with Q˜ = R−1Q then becomes:

where A˜ = diag (λ1, ··· , λm). Not all λi must be different from each other. This system of equations has in principle m sets of characteristics. But any set of characteristics that has the same characteristic velocity as another set is usually called the same set of characteristics. So in the case of 5 eigenvalues, of which three are identical, one typically says that there are three sets of characteristics.

## Hyperbolic equations: the non-linear case

Let us focus on the general conservation equation:

where, as ever, Q = (q1, ··· , qm) and F = (f1, ··· , fm). In general, F is not always a linear function of Q, i.e. it cannot always be formulated as a matrix A times the vector Q (except if A is allowed to also depend on Q, but then the usefulness of writing F = AQ is a bit gone). So let us assume that F is some non-linear function of Q. Let us, for the moment, assume that F = F(Q, x) = F(Q), i.e. we assume that there is no explicit dependence of F on x, except through Q. Then we get

where ∂F/∂Q is the Jacobian matrix, which depends, in the non-linear case, on Q itself. We can nevertheless decompose this matrix in eigenvectors (which depend on Q) and we obtain

Here the eigenvalues λ1, ··· , λm and eigenvectors (and hence the meaning of Q˜) depends on Q. In principle this is not a problem. The characteristics are now simply given by the state vector Q itself. The state is, so to speak, self-propagating. We are now getting into the kind of hyperbolic equations like the hydrodynamics equations, which are also non-linear self-propagating.

Summary Article Name
HYPERBOLIC EQUATIONS
Description
The hydrodynamics equations are nothing more than signal-propagation equations, and equations of this kind are called hyperbolic equations. The equations of hydrodynamics are only a member of the more general class of hyperbolic equations, and there are many more examples of hyperbolic equations than just the equations of hydrodynamics.
Author
Publisher Name
Soul Of Mathematics
Publisher Logo