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.
Consider the set of conservation equations:
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
To simplify these characteristic equations we can make the transformations such that
An integrating factor can be multiplied in to help integrate this. So the system now has the characteristic form
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
and the eigenvectors are found to be
where the Riemann invariants are
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.
- 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.