# HYPERBOLIC EQUATIONS

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 ∂_{t}q + u∂_{x}q = 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 > t_{0} 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 = t_{0} is fully given. The characteristics of this problem are straight lines: x_{char}(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}).

## 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:∂_{t}Q + A∂_{x}Q = 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 e_{i}, 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^{−1}Q 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 = (q_{1}, ··· , q_{m}) and F = (f_{1}, ··· , f_{m}). 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.