Given a system of *n* conservation equations for the *n* conserved
variables in *m*-dimensional space:

Note: the choice of conserved variables is unique.
The conservation equations can be transformed into ``primitive'',
wave-like equations for a corresponding set of *n* non-unique primitive
variables ,

where the are matrices. The choice of primitive variables is not unique, and could be taken to be the conserved variables. Assume the two systems are related by

Then the conservation equations can be transformed as follows:

where

The eigenvalues of are the solution of

The system is hyperbolic if all the eigenvalues of all the are real. The left (row), , and right (column), , eigenvectors of , satisfy

The left and right eigenvectors are orthogonal,

The matrices can be individually, but not simultaneously, diagonalized by a similarity transformation

where the columns of are the right eigenvectors and rows of its inverse are the left eigenvectors , and is the diagonal matrix, .

The **Characteristic Equations** for a single direction can be found
as follows: Combine all the divergence terms except that for the desired
direction, say 1, with the source term, so the conservation equations become

and the primitive equations become

The 1D **Characteristic Equations** are obtained by
multiplying this last equation by

or in component form

which gives the variation of along the characteristic curve in the -plane defined by

tangent to the eigenvector with eigenvalue .

Sometimes it is possible to define a CHARACTERISTIC VARIABLE or
**Riemann Invariant** by

if the expression is a perfect differential (the coefficients satisfy Pfaff's condition for the integrability of differential forms or no more than two differentials appear on the right hand side). Then the characteristic equation (13) becomes

These are wave equations with characteristic speeds and
with **Riemann Invariant** constant along the characteristic
.

Wed Sep 19 18:16:24 MET DST 2001