Riemann invariant

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Riemann invariants are mathematical transformations made on a system of quasi-linear first order partial differential 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 hyperbolic partial differential equations of the form

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 nontrival 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


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


which is equivalent to the diagonal system[2]

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


Consider the shallow water equations

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

In shallow water equations there is the relation to give the riemann invariants

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 so long as the matrix determinant of is not zero.


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