**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.^{[1]}

## Mathematical theory

Consider the set of conservation equations:

where and are the elements of the matrices and where and 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 plane defined by the vector field . The term in the brackets will be rewritten in terms of a total derivative where are parametrized as

comparing the last two equations we find

which can be now written in characteristic form

where we must have the conditions

where can be eliminated to give the necessary condition

so for a nontrivial solution is the determinant

For Riemann invariants we are concerned with the case when the matrix is an identity matrix to form

notice this is homogeneous due to the vector being zero. In characteristic form the system is

- with

Where is the left eigenvector of the matrix and is the characteristic speeds of the eigenvalues of the matrix which satisfy

To simplify these characteristic equations we can make the transformations such that

which form

An integrating factor can be multiplied in to help integrate this. So the system now has the characteristic form

- on

which is equivalent to the diagonal system^{[2]}

The solution of this system can be given by the generalized hodograph method.^{[3]}^{[4]}

## Example

Consider the one-dimensional Euler equations written in terms of density and velocity are

with being the speed of sound is introduced on account of isentropic assumption. Write this system in matrix form

where the matrix 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

( and are the widely used notations in gas dynamics). For perfect gas with constant specific heats, there is the relation , where is the specific heat ratio, to give the Riemann invariants^{[5]}^{[6]}

to give the equations

In other words,

where and 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 so long as the matrix determinant of is not zero.

## See also

## References

**^**Riemann, Bernhard (1860). "Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite" (PDF).*Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen*.**8**. Retrieved 2012-08-08.**^**Whitham, G. B. (1974).*Linear and Nonlinear Waves*. Wiley. ISBN 978-0-471-94090-6.**^**Kamchatnov, A. M. (2000).*Nonlinear Periodic Waves and their Modulations*. World Scientific. ISBN 978-981-02-4407-1.**^**Tsarev, S. P. (1985). "On Poisson brackets and one-dimensional hamiltonian systems of hydrodynamic type" (PDF).*Soviet Mathematics - Doklady*.**31**(3): 488–491. MR 2379468. Zbl 0605.35075.**^**Zelʹdovich, I. B., & Raĭzer, I. P. (1966). Physics of shock waves and high-temperature hydrodynamic phenomena (Vol. 1). Academic Press.**^**Courant, R., & Friedrichs, K. O. 1948 Supersonic flow and shock waves. New York: Interscience.