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] Consider the set of conservation equations: where
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
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
In characteristic form the system is Where
is the left eigenvector of the matrix
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
[3][4] Consider the one-dimensional Euler equations written in terms of density
being the speed of sound is introduced on account of isentropic assumption.
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 (
are the widely used notations in gas dynamics).
For perfect gas with constant specific heats, there is the relation
is the specific heat ratio, to give the Riemann invariants[5][6] to give the equations In other words, where
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