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.
Contents
Mathematical theory
Consider the set of conservation equations:
where
To do this curves will be introduced in the
comparing the last two equations we find
which can be now written in characteristic form
where we must have the conditions
where
so for a nontrival solution is the determinant
For Riemann invariants we are concerned with the case when the matrix
notice this is homogeneous due to the vector
Where
To simplify these characteristic equations we can make the transformations such that
which form
An integrating factor
which is equivalent to the diagonal system
The solution of this system can be given by the generalized hodograph method.
Example
Consider the shallow water equations
write this system in matrix form
where the matrix
to give
and the eigenvectors are found to be
where the Riemann invariants are
In shallow water equations there is the relation
to give the equations
Which can be solved by the hodograph transformation. 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