Riemann invariant

Riemann invariants are mathematical transformations made on a system of conservation equations to make them more easily solvable. Riemann invariants are constant along the characteristic curves of the partial differential equations where they obtain the name invariant. They were first obtained by Bernhard Riemann in his work on plane waves in gas dynamics.
Mathematical theory.
Consider the set of conservation equations:
where formula_2 and formula_3 are the elements of the matrices formula_4 and formula_5 where formula_6 and formula_7 are elements of vectors. It will be asked if it is possible to rewrite this equation to
To do this curves will be introduced in the formula_9 plane defined by the vector field formula_10. The term in the brackets will be rewritten in terms of a total derivative where formula_11 are parametrized as formula_12
comparing the last two equations we find
which can be now written in characteristic form
where we must have the conditions
where formula_18 can be eliminated to give the necessary condition
so for a nontrival solution is the determinant
For Riemann invariants we are concerned with the case when the matrix formula_4 is an identity matrix to form
notice this is homogeneous due to the vector formula_23 being zero. In characteristic form the system is
Where formula_26 is the left eigenvector of the matrix formula_4 and formula_28 is the characteristic speeds of the eigenvalues of the matrix formula_4 which satisfy
To simplify these characteristic equations we can make the transformations such that formula_31
which form
An integrating factor formula_33 can be multiplied in to help integrate this. So the system now has the characteristic form
which is equivalent to the diagonal system
The solution of this system can be given by the generalized hodograph method.
Example.
Consider the one-dimensional Euler equations written in terms of density formula_38 and velocity formula_39 are
with formula_42 being the speed of sound is introduced on account of isentropic assumption. Write this system in matrix form
where the matrix formula_5 from the analysis above the eigenvalues and eigenvectors need to be found. The eigenvalues are found to satisfy
to give
and the eigenvectors are found to be
where the Riemann invariants are
(formula_50 and formula_51 are the widely used notations in gas dynamics). For perfect gas with constant specific heats, there is the relation formula_52, where formula_53 is the specific heat ratio, to give the Riemann invariants
to give the equations
In other words, 
where formula_59 and formula_60 are the characteristic curves. This can be solved by the hodograph transformation. In the hodographic plane, if all the characteristics collapses into a single curve, then we obtain simple waves. If the matrix form of the system of pde's is in the form
Then it may be possible to multiply across by the inverse matrix formula_62 so long as the matrix determinant of formula_4 is not zero.